Objective Mathematics - User contributions [en]

Context theory

2026-02-22T20:06:25Z

Lfox: Created page with "What is set theory ''about''? What are its objects of study?"

What is set theory ''about''?

What are its objects of study?

Main Page

2026-02-22T20:03:19Z

Lfox:

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Category theory]]
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Russell's paradox]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Logic]]
* [[Against models (essay)]]
* [[Category theory: abstracting mathematical construction (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
* [[Perspective Theory]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]
* [[Context theory]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].

Russell's paradox

2025-08-25T01:34:17Z

Lfox: /* Implications */

'''Russell's paradox''' is a paradox that arises in a certain "naive" approach to set theory. The reasoning behind it is as follows.

We begin with the plausible-sounding premise that to any logical predicate <math>P(x)</math> there exists a set <math>\{x : P(x) \}</math>. Indeed, there is some way in which predicates and sets seem like they are the same thing. One way I can think about this object <math>x</math> on my table is to say that it is a banana, or <math>B(x)</math>; another way I can think about the same fact is to consider bananas as a set <math>\{ y : B(y) \} </math>, and say that <math>x</math> belongs to this set, or <math>x \in \{ y : B(y) \} </math>. If it makes sense to say that <math>x</math> is a banana, then it makes sense to consider the set of all bananas.

Now, let's consider the predicate <math>E(x) := (x \in x )</math>, which makes sense to state if <math>x</math> is a set. Although it is difficult to think of examples of sets that contain themselves, this predicate is not as ridiculous as it might seem. For example, the set of triangles (if appropriately defined) is itself a triangle. Or, for example, "<math>x</math> is a set" is a well-defined predicate, so therefore there exists a set of all sets -- and since it is a set, it must necessarily contain itself.

The paradox comes in when we consider the set <math>R = \{ x : x \not\in x \}</math>, and ask: does <math>R</math> contain itself, i.e. is <math>R \in R</math>? If the answer is yes, then <math>R\not\in R</math>, which would be a contradiction. If no, i.e. if <math>R \not\in R</math>, then <math>R \in R</math>, which is again a contradiction.

== The solution ==
The essence of the following solution was explained to me by Harry Binswanger in his ''Philosophy of Mathematics'' course (2024), and in his book ''How We Know''. I have added a few embellishments of my own.

The heart of the matter is that we have to be more careful about what we mean by a "set." Sets don't just exist "out there," but rather a set is some things out there, ''which someone is'' ''considering together as a single unit''. Sets are concepts of consciousness, and so they are relational. They are things out there, as viewed in a particular way by a particular man. A set is an action of awareness, or of referring.

To refer is to refer ''to something''. A reference to an act of referring can be fine in certain circumstances, e.g. I can unproblematically refer to the fact that I referred to Harry Binswanger's book ''How We Know''. But an act of referring that refers ''to nothing but itself'' is a contradiction in terms. It would be referring to its referring to its referring to its referring to ... etc. We would get an infinite regress, and so this supposed act of reference would in fact be referring to nothing at all, i.e. it would not be an act of reference. Harry Binswanger calls this "the fallacy of pure self-reference."

Now, consider the implications of this logical fact for statements in set theory. Self-referential sets, like "the set of all sets" cannot exist. Why? Because to consider ''all'' sets together as a set is to refer to this particular act of reference. And what is this particular act of reference referring to? Well in part, it is referring to its own referring (and its own referring is referring in part to its own referring, to its own referring, to its own referring, to ...). Again, we get an infinite regress.

== Implications ==
We have arrived at the surprising fact that the statement that "<math>x </math> is a set" is perfectly unproblematic, but that "the set of all sets" is not something which can be rationally considered. Usually, to form a set, it is sufficient just to know what you are referring to.

An implication of this is that the concept "set" is unlike most other concepts, in that its referents cannot be considered together as a single unity, for one of its referents would be ''that act of consideration itself''. This must have implications for generalizations about sets, but it's hard to say what.

Russell's paradox does have some implications for "set formation rules." Traditionally, the way it is dealt with is by only allowing sets to be formed that are subsets of some other larger set, and then postulating the existence of a universe (a very large set, that everything else can be a subset of). Naively, one would like this "universal" set to be existence, or the set of all things that exist, but that is [TODO].

Traditional math takes an intrinsic approach to constructing the universal set, but I take an objective approach to it. The "universal" set is: ''existence, as it is known by a particular consciousness''. It is the set of all the things that are known to exist by a particular consciousness. That is, in order for you to validly form a set, it must consist exclusively of things that ''exist'', and which you are ''aware of''. [TODO this needs work. I have a concept of chair, I can validly consider the set of all chairs, but I am not aware of---nor could I be aware of---every single individual chair.] Each individual has his own "universe," from which he is allowed to form sets; the universe is different for different individuals, and the scope of an individual's universe changes with his knowledge.

If, in forming sets, you only ever include things in your own "universal" set, then you cannot possibly commit the fallacy of pure self-reference. Part of the reason is that man's future is open-ended, and it cannot be said which acts of awareness will exist. Another part of the reason is that your present acts of awareness exist, but there are limits to how aware of them you can be: There must always exist some acts of awareness that you are not aware of, because to be aware of an act of awareness ''is a new act of awareness'', which itself is a new thing you'd have to be aware of, and yet you can only be aware of finitely many things.

Russell's paradox

2025-08-19T06:28:09Z

Lfox:

'''Russell's paradox''' is a paradox that arises in a certain "naive" approach to set theory. The reasoning behind it is as follows.

We begin with the plausible-sounding premise that to any logical predicate <math>P(x)</math> there exists a set <math>\{x : P(x) \}</math>. Indeed, there is some way in which predicates and sets seem like they are the same thing. One way I can think about this object <math>x</math> on my table is to say that it is a banana, or <math>B(x)</math>; another way I can think about the same fact is to consider bananas as a set <math>\{ y : B(y) \} </math>, and say that <math>x</math> belongs to this set, or <math>x \in \{ y : B(y) \} </math>. If it makes sense to say that <math>x</math> is a banana, then it makes sense to consider the set of all bananas.

Now, let's consider the predicate <math>E(x) := (x \in x )</math>, which makes sense to state if <math>x</math> is a set. Although it is difficult to think of examples of sets that contain themselves, this predicate is not as ridiculous as it might seem. For example, the set of triangles (if appropriately defined) is itself a triangle. Or, for example, "<math>x</math> is a set" is a well-defined predicate, so therefore there exists a set of all sets -- and since it is a set, it must necessarily contain itself.

The paradox comes in when we consider the set <math>R = \{ x : x \not\in x \}</math>, and ask: does <math>R</math> contain itself, i.e. is <math>R \in R</math>? If the answer is yes, then <math>R\not\in R</math>, which would be a contradiction. If no, i.e. if <math>R \not\in R</math>, then <math>R \in R</math>, which is again a contradiction.

== The solution ==
The essence of the following solution was explained to me by Harry Binswanger in his ''Philosophy of Mathematics'' course (2024), and in his book ''How We Know''. I have added a few embellishments of my own.

The heart of the matter is that we have to be more careful about what we mean by a "set." Sets don't just exist "out there," but rather a set is some things out there, ''which someone is'' ''considering together as a single unit''. Sets are concepts of consciousness, and so they are relational. They are things out there, as viewed in a particular way by a particular man. A set is an action of awareness, or of referring.

To refer is to refer ''to something''. A reference to an act of referring can be fine in certain circumstances, e.g. I can unproblematically refer to the fact that I referred to Harry Binswanger's book ''How We Know''. But an act of referring that refers ''to nothing but itself'' is a contradiction in terms. It would be referring to its referring to its referring to its referring to ... etc. We would get an infinite regress, and so this supposed act of reference would in fact be referring to nothing at all, i.e. it would not be an act of reference. Harry Binswanger calls this "the fallacy of pure self-reference."

Now, consider the implications of this logical fact for statements in set theory. Self-referential sets, like "the set of all sets" cannot exist. Why? Because to consider ''all'' sets together as a set is to refer to this particular act of reference. And what is this particular act of reference referring to? Well in part, it is referring to its own referring (and its own referring is referring in part to its own referring, to its own referring, to its own referring, to ...). Again, we get an infinite regress.

== Implications ==
We have arrived at the surprising fact that the statement that "<math>x </math> is a set" is perfectly unproblematic, but that "the set of all sets" is not something which can be rationally considered. Usually, to form a set, it is sufficient just to know what you are referring to.

An implication of this is that the concept "set" is unlike most other concepts, in that its referents cannot be considered together as a single unity, for one of its referents would be ''that act of consideration itself''. This must have implications for generalizations about sets, but it's hard to say what.

Russell's paradox does have some implications for "set formation rules." Traditionally, the way it is dealt with is by only allowing sets to be formed that are subsets of some other larger set, and then postulating the existence of a universe (a very large set, that everything else can be a subset of). Naively, one would like this "universal" set to be existence, or the set of all things that exist, but that is

Traditional math takes an intrinsic approach to constructing the universal set, but I take an objective approach to it. The "universal" set is: ''existence, as it is known by a particular consciousness''. It is the set of all the things that are known to exist by a particular consciousness. That is, in order for you to validly form a set, it must consist exclusively of things that ''exist'', and which you are ''aware of''. Each individual has his own "universe," from which he is allowed to form sets; the universe is different for different individuals, and the scope of an individual's universe changes with his knowledge.

If, in forming sets, you only ever include things in your own "universal" set, then you cannot possibly commit the fallacy of pure self-reference. Part of the reason is that man's future is open-ended, and it cannot be said which acts of awareness will exist. Another part of the reason is that your present acts of awareness exist, but there are limits to how aware of them you can be: There must always exist some acts of awareness that you are not aware of, because to be aware of an act of awareness ''is a new act of awareness'', which itself is a new thing you'd have to be aware of, and yet you can only be aware of finitely many things.

Russell's paradox

2025-08-19T06:21:01Z

Lfox: /* Implications for the relation between sets and predicates */

'''Russell's paradox''' is a paradox that arises in a certain "naive" approach to set theory. The reasoning behind it is as follows.

We begin with the plausible-sounding premise that to any logical predicate <math>P(x)</math> there exists a set <math>\{x : P(x) \}</math>. Indeed, there is some way in which predicates and sets seem like they are the same thing. One way I can think about this object <math>x</math> on my table is to say that it is a banana, or <math>B(x)</math>; another way I can think about the same fact is to consider bananas as a set <math>\{ y : B(y) \} </math>, and say that <math>x</math> belongs to this set, or <math>x \in \{ y : B(y) \} </math>. If it makes sense to say that <math>x</math> is a banana, then it makes sense to consider the set of all bananas.

Now, let's consider the predicate <math>E(x) := (x \in x )</math>, which makes sense to state if <math>x</math> is a set. Although it is difficult to think of examples of sets that contain themselves, this predicate is not as ridiculous as it might seem. For example, the set of triangles (if appropriately defined) is itself a triangle. Or, for example, "<math>x</math> is a set" is a well-defined predicate, so therefore there exists a set of all sets -- and since it is a set, it must necessarily contain itself.

The paradox comes in when we consider the set <math>R = \{ x : x \not\in x \}</math>, and ask: does <math>R</math> contain itself, i.e. is <math>R \in R</math>? If the answer is yes, then <math>R\not\in R</math>, which would be a contradiction. If no, i.e. if <math>R \not\in R</math>, then <math>R \in R</math>, which is again a contradiction.

== The solution ==
The essence of the following solution was explained to me by Harry Binswanger in his ''Philosophy of Mathematics'' course (2024), and in his book ''How We Know''. I have added a few embellishments of my own.

The heart of the matter is that we have to be more careful about what we mean by a "set." Sets don't just exist "out there," but rather a set is some things out there, ''which someone is'' ''considering together as a single unit''. Sets are concepts of consciousness, and so they are relational. They are things out there, as viewed in a particular way by a particular man. A set is an action of awareness, or of referring.

To refer is to refer ''to something''. A reference to an act of referring can be fine in certain circumstances, e.g. I can unproblematically refer to the fact that I referred to Harry Binswanger's book ''How We Know''. But an act of referring that refers ''to nothing but itself'' is a contradiction in terms. It would be referring to its referring to its referring to its referring to ... etc. We would get an infinite regress, and so this supposed act of reference would in fact be referring to nothing at all, i.e. it would not be an act of reference. Harry Binswanger calls this "the fallacy of pure self-reference."

Now, consider the implications of this logical fact for statements in set theory. Self-referential sets, like "the set of all sets" cannot exist. Why? Because to consider ''all'' sets together as a set is to refer to this particular act of reference. And what is this particular act of reference referring to? Well in part, it is referring to its own referring (and its own referring is referring in part to its own referring, to its own referring, to its own referring, to ...). Again, we get an infinite regress.

== Implications ==
We have arrived at the surprising fact that the statement that "<math>x </math> is a set" is perfectly unproblematic, but that "the set of all sets" is not something which can be rationally considered. Usually, to form a set, it is sufficient just to know what you are referring to.

An implication of this is that the concept "set" is unlike most other concepts, in that its referents cannot be considered together as a single unity, for one of its referents would be ''that act of consideration itself''. This must have implications for generalizations about sets, but it's hard to say what.

Russell's paradox does have some implications for "set formation rules." Traditionally, the way it is dealt with is by only allowing sets to be formed that are subsets of some other larger set, and then postulating the existence of a universe (a very large set, that everything else can be a subset of).

Traditional math takes an intrinsic approach to constructing the universal set, but I take an objective approach to it. The "universal" set is: ''existence, as it is known by a particular consciousness''. It is the set of all the things that are known to exist by a particular consciousness. That is, in order for you to validly form a set, it must consist exclusively of things that ''exist'', and which you are ''aware of''. Each individual has his own "universe," from which he is allowed to form sets; the universe is different for different individuals, and the scope of an individual's universe changes with his knowledge.

If, in forming sets, you only ever include things in your own "universal" set, then you cannot possibly commit the fallacy of pure self-reference. Your acts of awareness exist, but there are limits to how aware of them you can be.

Russell's paradox

2025-08-19T05:36:27Z

Lfox: Created page with "'''Russell's paradox''' is a paradox that arises in a certain "naive" approach to set theory. The reasoning behind it is as follows. We begin with the plausible-sounding premise that to any logical predicate <math>P(x)</math> there exists a set <math>\{x : P(x) \}</math>. Indeed, there is some way in which predicates and sets seem like they are the same thing. One way I can think about this object <math>x</math> on my table is to say that it is a banana, or <math>B(x)<..."

Main Page

2025-08-19T05:36:18Z

Lfox:

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Category theory]]
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Russell's paradox]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Logic]]
* [[Against models (essay)]]
* [[Category theory: abstracting mathematical construction (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
* [[Perspective Theory]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].

Set

2025-08-18T23:23:12Z

Lfox: /* My email to Ray 03/27/25 */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
This essay was prompted by Ray's suggestion that I put my problem with the diagonal argument into writing. My problem is not really with the diagonal argument per se, it is with the more basic concept of sets. Rather than saying something negative about set theory as it stands today, I will offer something positive. Starting from scratch, I will sketch what I think a rational theory of sets would look like.

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed in a particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups of the units of concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument to the uninitiated. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We may now see the subtlety with the idea of "the set of all points on O." Those points on O which are ''physical'' must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So regardless of which of those we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>. A function <math>f : X \rightarrow Y</math> is ''something you could do''.

A "problem" with this way of thinking about functions is that sometimes indeed you are doing something with <math>x</math>, and producing a new thing <math>f(x)</math>, but sometimes you haven't actually produced a new thing, you're just thinking about <math>x</math> in a different way. For example, all men are mortal, so there should be a function from the group of all men to the group of all mortal beings. It is arguable, however, that these functions represent something you could do, but it's something you do in consciousness---they represent a change of perspective. You aren't doing anything to <math>x</math> itself, you're just looking at <math>x</math> differently.

For an uncontroversial example of a function, it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Another type of function, one without a clear mathematization, is a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life, e.g. given a line segment on a piece of paper, pull out an amount of string equal to that length, then use the string to extend that line segment by drawing a new line segment of equal length. Mainstream math models the above process as the function <math>\mathbb{R} \rightarrow \mathbb{R}</math> which sends <math>x \mapsto 2x</math>, but that's not quite right, because <math>\mathbb{R}</math> is a number (or at least, something akin to a number), and not the length itself. It is very common that mainstream math thinks of <math>\mathbb{R}</math> as "lengths as they actually are in reality." It conflates length with measurement of length.

Is a function <math>\mathbb{R} \rightarrow \mathbb{R}</math> something you could do? It's not clear. What even are the elements of the group <math>\mathbb{R}</math>? The elements of <math>\mathbb{R}</math> are equivalence classes of Cauchy sequences, so those are two things we need to understand: equivalence classes, and Cauchy sequences.

Cauchy sequences are functions <math>\mathbb{N} \rightarrow \mathbb{Q}</math>. Is a Cauchy sequence something you could do? Some of them are, like <math>\{ 1/n \}_{n=1}^\infty </math>, but some of them aren't, like <math>\{ 1/h_n \}_{n=1}^\infty</math> where <math>h_n = </math> "the number of steps the <math>n</math>th Turing machine takes to halt." And some of them are very bizarre, like <math>\{ a_n \}_{n=1}^\infty </math> where <math>a_n = </math> "the number of stars in the Milky Way of mass <math>\leq n</math> kilograms."

Is it legitimate to talk about Cauchy sequences that are ''not'' something you could do? The answer of mainstream mathematics is a resounding "yes." My answer is a tentative "no": if it's not something which exists in reality, and it's not something which potentially could exist in reality, then your sequence doesn't refer to anything, and so it's meaningless. It's like talking about flying purple cats. The reason why my answer is ''tentative'' "no" is that sometimes we can learn things from impossible hypotheticals; sometimes they reveal things about the nature of our legitimate concepts. I don't want to police what sort of Cauchy sequences people can or cannot talk about, I just want to reorient math in the direction of reality.

What is an equivalence class? Well that's actually quite simple. First let's consider fractions: 1/3 and 2/6 are the same ''fraction'', but they are different ''expressions''. A fraction is an equivalence class of expressions <math>p/q</math> involving two integers <math>p,q\in \mathbb{Z}</math> (where <math>q \neq 0</math>), and where the equivalence relation is that <math>p/q = r/s</math> if <math>ps = qr</math>. The concept of equivalence classes extends far beyond mathematics, to any concept which is a special case of another concept. The two copies of ''Conformal Field Theory'' (some physics textbook) in my office are different ''copies'', but they are the same ''book''. Now, given two Cauchy sequences <math>\{ 1/n \}_{n=1}^\infty </math> and <math>\{ 1/n^2 \}_{n=1}^\infty </math>, which mind you are two things---two processes---actually out there in reality, we say they are different qua ''sequence'', but they are the same qua ''real number''. Generally, given two Cauchy sequences <math>\{ a_n \}_{n=1}^\infty </math>and <math>\{ b_n \}_{n=1}^\infty </math>, we say regard them as equivalent if for any <math>\epsilon \in Q, </math> such that <math>\epsilon > 0</math>, there exists an <math>N_\epsilon \in \mathbb{N}</math> such that whenever <math>n \geq N</math>, <math>| a_n - b_n| < \epsilon </math>.

So what real thing does <math>\mathbb{R}</math> refer to? It refers to a particular kind of potential action <math>\mathbb{N} \rightarrow \mathbb{Q}</math> (namely one that is "Cauchy"), from a particular perspective (namely the perspective from which two Cauchy sequences are the same if they are going to the same place).

Now, how about functions <math>\mathbb{R} \rightarrow \mathbb{R}</math>? Well it's just something that takes a real number and sends it to a real number. If you think of the real number as a sequence <math>\{ a_n \}_{n=1}^\infty </math>, that's fine, but the function needs to be defined for <math>\{ a_n \}_{n=1}^\infty </math> qua real number rather than <math>\{ a_n \}_{n=1}^\infty </math> qua sequence. Again, this point is logical and not mathematical. We could consider a function from the group of all books to the group of all strings, e.g. which sends a book to its author; such a function can be defined on ''copies'' of books (like maybe your way of implementing it is: purchase a copy of the book and look at who the author of the copy is), but it must necessarily depend on the properties of the copy ''qua book'' rather than its properties ''qua copy''. For example, it can't depend on the physical location of the book.

So, in conclusion, the answer to the question "Is a function <math>\mathbb{R} \rightarrow \mathbb{R}</math> something you could do?" is yes. But what sort of processes does it refer to in real life? Is it really practical?

== Group formation rules [analogue of ZF] ==
A group doesn't have to be specified in a completely unambiguous way. In real life, there is no such thing.

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

I actually don't like the term "form a set." It's vague. Really what we're doing is we are ''identifying'' sets; we are pointing to some things in reality, and saying "that's a set."

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there need not be a practical way to check that all the members of a set are the same.

Quotients / identity are actually what I have to think the most about.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Fraction

2025-08-15T21:54:23Z

Lfox:

A '''fraction''' is a pair <math>p/q</math> of integers <math>p,q : \mathbb{Z} </math>, <math>q \neq 0</math>, regarded from the perspective that <math>p / q =_{\mathbb{Q}} r / s </math> if <math>p s =_{\mathbb{Z}} q r</math>.

== Ratio ==
I want to say something like: A ratio is an actual relationship between quantities.

Suppose that <math>p/q > 0</math>, and that it is describing some quantity <math>Q</math>.

specifically, there's something about the ''symbols'' that is the same as <math>Q</math>.

is <math>Q</math> a quantity? or is it a relationship between quantities? Well it could be either: we could have 3/5ths of a pizza, OR: we could have the ratio of the circumference of a pizza to its diameter. The former has units, the latter doesn't. The former is a quantity, and as for the latter... I guess it is not a quantity? Yeah. It's a quantitative relationship, but not a quantity.

Let's stick with the former case, where <math>Q</math> is an actual quantity. What then does it mean to identify <math>Q</math> with <math>p/q</math>? Well first of all, there has to be some other quantity <math>\overline{Q}</math> that is being used as a foil. And then the idea is that <math>q</math> <math>Q</math>s would be the same quantity as <math>p</math> <math>\overline{Q}</math>s.

Now, for the latter case, suppose we have two quantities <math>Q</math> and <math>\overline{Q}</math>. To understand the relationship between these quantities, we understand it in terms of sameness. If <math>q</math> <math>Q</math>s is the same quantity as <math>p</math> <math>\overline{Q}</math>s, then we say that the ratio of <math>Q</math> to <math>\overline{Q}</math> is <math>p/q</math>.

So really these two cases are the same. We can say either that some quantity is <math>p/q</math>ths of another quantity, or else we can say that the ratio of the first to the second is <math>p/q</math>.

Now, what about negative fractions?

[OLD stuff below as of 08/15/25]

Ratios are very fundamental. Any measurement at all involves ratios.

A natural number is a set of several things, thought of in relation to a single thing.

A fraction is a set of one or more things, thought of in relation to a set of one or more things.

Just like how I defined an integer as an ordered pair of natural numbers, I could define a fraction as an ordered pair of natural numbers. The difference is that these pairs have a different equivalence relation, and different addition / multiplication operations.

== Rational numbers ==
Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios.

== n-ratios, projective space ==
[TODO]

Integer

2025-08-15T21:15:05Z

Lfox:

An '''integer''' is a difference between two multitudes, considered together as an [[Order|ordered]] [[Set|pair]]. The concept of integer is sometimes denoted by a symbol, <math>\mathbb{Z}</math>. [TODO old as of 08/15/25]

As a ''noun'', an integer is a symbol like -2, -1, 0, 1, etc.

As an ''adjective'', integers can be used to describe different things. For example, a signed difference -- see the rest of this article. Or, for example, a location on an oriented pointed line. Or, for example, a winding number.

== Examples of integers ==
In this section, I will give examples of concretes subsumed under specific integers, like -2, 0, 3, etc.

Consider two piles of apples. Suppose that the two piles are ordered, so that we may refer to one as "pile #1" and the other as "pile #2."

* If pile #1 contains 3 apples, and pile #2 contains 5 apples, then together they are a unit of the concept "-2." One may say that the signed difference between them is -2 apples.
* If pile #1 contains 71 apples, and pile #2 contains 71 apples, then together they are a unit of the concept "0." One may say that the signed difference between them is 0 apples.
* If pile #1 contains 3 apples, and pile #2 is empty (contains no apples), then together they are a unit of the concept "3." One may say that the signed difference between them is 3 apples.
Consider two sets of gold coins. "Set #2" is the set of gold coins which Bob has withdrawn from his bank account. "Set #1" is the set of gold coins which Bob has deposited in his bank account.

* If set #1 contains 6 gold coins, and set #2 contains 5 gold coins, then together they are a unit of the concept "1." His balance is said to be 1 gold coin.
* If set #1 contains 100,000 gold coins, and set #2 contains 200,000 gold coins, then together they are a unit of the concept "-100,000." His balance is said to be -100,000 gold coins.
Consider two [[Point|points]] A and B on a [[Line|ray]]. Let <math>L_A</math> denote the distance from A to the origin, and let <math>L_B</math> denote the distance from B to the origin. Choose the ordering of the set <math>\{ L_A, L_B \}</math> according to which <math>L_A</math> is first, and <math>L_B</math> is second. Then

* If <math>L_A</math> is found to be 17 inches, and <math>L_B</math> is found to be 31 inches, then together they are a unit of the concept "14." One may say that the signed distance between A and B is 14 inches.
* If B is the origin of the ray, and <math>L_A</math> is found to be 10 inches, then together they are a unit of the concept "-10." One may say that the signed distance between A and B is -10 inches.
* If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a [[rational number]].
Consider a point P on a line with an origin. Let <math>L</math> denote the number of centimeters between P and the origin which are to the ''left'' of the origin, and let <math>R</math> denote the number of centimeters between P and the origin which are to the ''right'' of the origin. Necessarily, either <math>L = 0</math> or <math>R = 0</math>. The tuple <math>(R,L)</math> is an integer (if L and R are integral), and it is the linear coordinate of P.

== Difference ==
The '''difference''' between two quantities A and B, is the quantity which would have to be added to the lesser quantity in order to make it equal to the greater quantity. In the case where A and B are the same quantity, we say that the difference between them is zero, or 0.

=== Examples ===
The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

The difference between a pile consisting of 5 apples, and a pile consisting of 3 apples, is 2 apples.

The difference between 3 and 5 is 2.

=== Non-examples ===
"A difference between me and my friend is that I like chocolate ice cream, but he doesn't." This is a perfectly valid use of the concept of difference, but it's not a difference of quantities.

== Signed differences ==
[TODO this section needs a lot of work]

A difference is something that is identified with respect to two quantities. For a difference, the ''order'' of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.

A '''signed difference''', which Objective Mathematics sometimes calls a '''sifference''', is a concept much like a difference, except that it keeps track of the ''order'' the two quantities under consideration. Let A and B denote two quantities, where A is greater than or equal to B. The sifference between A and B is the difference between A and B; the sifference between B and A is the difference between A and B, but with a slight asterisk to remind us about the order.

Describing things like I have, in the English language, may give the reader a slightly incorrect idea, because "and" is often considered to be symmetrical. Indeed, "Bob and Jane" usually means the same thing as "Jane and Bob." In our context, however, it is very important to distinguish between the two noun phrases. This emphasis on the order of "and" is not completely foreign to English, however: authors know that at times, the subtle change in emphasis between "Bob and Jane" and "Jane and Bob" matters.

Symbolically, we write the sifference between <math>X</math> and <math>Y</math> as <math>X-Y</math>, and we write the sifference between <math>Y</math> and <math>X</math> as <math>Y - X</math>.

=== Examples ===
The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

in fact there is a [[symmetry]] between them.

== Addition of integers ==
'''To add''' two integers <math>n</math> and <math>m</math> is to regard them as a single ordered pair <math>p</math>, such that the first element of <math>p</math> is the [[Natural number#Addition|sum]] of the multitudes in the first element of <math>n</math> and the first element of <math>m</math>, and such that the second element of <math>p</math> is the sum of the multitudes of the second element of <math>n</math> and the second element of <math>m</math>. The resulting integer is denoted by <math>n + m</math>.

In symbolic notation, <math>(n,m) + (a,b) := (n + a, m + b)</math>.

=== Examples ===
[TODO put my drawing]

== Negation of integers ==
'''To negate''' an integer <math>n</math> is to reverse the ordering of the pair to which it refers. The negation of <math>n</math> is denoted by <math>-n</math>.

The ordering is something that your consciousness added to the pair. So in some sense every integer (meaning every object referred to by an integer) carries a <math>\mathbb{Z}_2</math> [[symmetry]]. It's like a gauge symmetry (change in description) rather than a true symmetry.

=== Examples ===
[TODO put my drawing]

== Multiplication of integers ==
'''To multiply''' an integer <math>n</math> by a natural number <math>k</math>, is to add <math>n</math> to itself <math>k</math> times. The number measuring the resulting difference may be denoted as <math>kn</math> or <math>k \times n</math>.

We can extend this to multiplication of integers by integers, as follows. To multiply <math>n</math> by 0 is 0. To multiply <math>n</math> by a negative integer, <math>-k</math>, is to add the negation of <math>n</math> to itself <math>k</math> times.

In symbolic notation, we have described an operation<math>(n,m)\times (p,q) := (np + mq, nq + mp) </math>.

=== Examples ===
[TODO put tons and tons of examples here]

== The traditional concept ==
[TODO needs a lot of work]

The integers are sometimes taken as an irreducible primary in mathematics. The mathematician Leopold Kronecker said<ref>Bell, E. T. ''Men of Mathematics''. Seventh printing, Simon & Schuster, 1937, p. 477.</ref><blockquote>God made the integers, all the rest is the work of man. </blockquote>and I think this quote is famous because some mathematicians sympathize with it.

Integers are literally taken as an irreducible primary by ZFC set theory, which has the axiom of infinity (TODO write something else. Calling this the integers, and not---say, the natural numbers---is an oversimplification.)

=== The Grothendieck construction ===
There's a construction I have seen called the Grothendieck construction, which is in some ways similar to Objective Mathematics' way of viewing the integers. The idea is to view the integers as the set <math>\mathbb{N} \times \mathbb{N} / \sim</math>, where <math>(n,m) \sim (p,q) </math> if there exists some <math>k \in \mathbb{N}</math> such that <math>(p,q) = (n + k, m + k)</math>.

I am sure that versions of this construction long predate Grothendieck, but it was made famous by Grothendieck because it's the starting point of K theory.

== References ==

The block universe (temporary name)

2025-06-04T18:22:44Z

Lfox:

The aim of this essay is to argue against the metaphysical picture of the block universe. I'm writing this for people like Andrei, who recognize that there is some sort of contradiction with determinism, but have no clue how to reconcile it with what we know about physics.

Maybe I should title it "the arrogance of modern physics." It's so ridiculous for them to say we know everything except for like dark matter and confinement and quantum gravity and times earlier than the first 10^-47 seconds of the Big Bang. Everything else is jUsT aN eNgInEeRiNg pRoBlEm.

I'm only going to cover classical physics in this essay. Quantum physics makes everything way more confusing, and I don't think it ultimately would be relevant for the key points of my argument.

== The first initial condition ==
Physics doesn't explain the Big Bang

== The nature of time ==
Time is motion. A measurement of how much time has passed is a quantitative identification of how much motion has happened. To say "it has been 30 seconds" is to say that the second hand has moved halfway across the clock.

In reality, there is no such thing as an instant in time. No one has ever observed one, and no one ever could even in principle. An instant is an ''idealization'', corresponding to an arbitrarily small amount of motion. This idealization is clearly useful for many things, but no one knows precisely what its domain of validity is.

A corollary of the above is that in reality, there is no such thing as ''the state of a physical system'' at some instant in time. Again, the concept of an instantaneous state is a useful idealization, but its domain of validity remains unclear.

Obviously, I can't paint a precise picture of what it looks like when this picture breaks down. If I could, I would be the next Isaac Newton. But to jog the reader's imagination, I will concoct a fictional story:

Suppose we have a system consisting of many parts that are moving independently. Each part alternates between going through some fixed motion which takes time <math>\tau</math> to complete, and doing nothing at all. The parts act asynchronously, by which I mean they aren't all moving at the same time. Oops. Next, I want to say what the state of the system is at time <math>t= 0</math>, but I can't do that, because I'm keeping firmly in mind the fact that such a thing is an idealization.

There's no such thing as the state

aosdfjoasd

ok here's my idea:

what if it were the case that to predict the future of something, you need to know not just where it is now, but what its trajectory has been for the past hour?

The block universe (temporary name)

2025-05-29T17:35:20Z

Lfox: /* The nature of time */

The block universe (temporary name)

2025-05-29T17:28:22Z

Lfox: Created page with "The aim of this essay is to argue against the metaphysical picture of the block universe. I'm writing this for people like Andrei, who recognize that there is some sort of contradiction with determinism, but have no clue how to reconcile it with what we know about physics. Maybe I should title it "the arrogance of modern physics." It's so ridiculous for them to say we know everything except for like dark matter and confinement and quantum gravity and times earlier than..."

The aim of this essay is to argue against the metaphysical picture of the block universe. I'm writing this for people like Andrei, who recognize that there is some sort of contradiction with determinism, but have no clue how to reconcile it with what we know about physics.

Maybe I should title it "the arrogance of modern physics." It's so ridiculous for them to say we know everything except for like dark matter and confinement and quantum gravity and times earlier than the first 10^-47 seconds of the Big Bang. Everything else is jUsT aN eNgInEeRiNg pRoBlEm.

I'm only going to cover classical physics in this essay. Quantum physics makes everything way more confusing, and I don't think it ultimately would be relevant for the key points of my argument.

== The first initial condition ==
Physics doesn't explain the Big Bang

== The nature of time ==
Time is motion. A measurement of how much time has passed is a quantitative identification of how much motion has happened. To say "it has been 30 seconds" is to say that the second hand has moved halfway across the clock.

In reality, there is no such thing as an instant in time. No one has ever observed one, and no one ever could even in principle. An instant is an ''idealization'', corresponding to an arbitrarily small amount of motion. This idealization is clearly useful for many things, but no one knows precisely what its domain of validity is.

A corollary of the above is that in reality, there is no such thing as ''the state of a physical system'' at some instant in time. Again, the concept of an instantaneous state is a useful idealization, but its domain of validity remains unclear.

Obviously, I can't paint a precise picture of what it looks like when this picture breaks down. If I could, I would be the next Isaac Newton. But to jog the reader's imagination, I will concoct a fictional story:

Suppose we have a system consisting of many parts that are moving independently. Each part alternates between going through some fixed motion which takes time <math>\tau</math> to complete, and doing nothing at all. The parts act asynchronously. Next, I want to say what the state of the system is at time <math>t= 0</math>, but I can't do that, because I'm keeping firmly in mind the fact that such a thing is an idealization.

There's no such thing as the state

Logic

2025-05-24T18:37:28Z

Lfox: /* Subjunctive quantifiers */

I think that set theory and formal logic need to be re-done rationally. This is a big project. I will list out some of my problems with formal logic below:

== Material implication ==
<math>A \implies B</math> is held to equivalent to <math>\neg A \vee B</math>, so anything follows from a falsehood. This doesn't fit natural language reasoning. For example, suppose that it is not raining outside right now, and I say "if it were raining outside right now, then communism would be true." The former statement is false, so the implication is true. Clearly that's absurd.

== Essence ==
In plain English, we have a lot of sentences like "bread nourishes," which don't fit into "for all" or "there exists." Indeed, "bread nourishes" doesn't just mean that there exists some bread that nourishes (it means way more than that), nor does it mean that ALL bread, even bread laced with arsenic, nourishes. It means that bread, ''in essence'', nourishes.

Examples of universal statements that fit this pattern:

* Lamps provide light (but not ALL of them provide light, because some lamps are broken)
* Mirrors reflect light (but not ALL mirrors reflect, because some mirrors are covered in soot)
* Leaves absorb light through photosynthesis (but not ALL leaves do, because some are dead)
* Birds have feathers (but not ALL birds have feathers because some have been plucked)

Maybe examples:

Non examples:

* Birds can fly (isn't this just a false sentence? Not true in essence, and not true for all birds.)

Conjecture(?): It appears that the above is to "for all" as statements about particulars, like "Bob is mortal," are to "there exists."

== Subjunctive quantifiers ==
In English, there is a difference between "all" and "any," and between "there exists" and "there could exist." Linguists call the former "indicative" and the latter "subjunctive."

This seems very important to me, especially the difference between "there exists" and "there could exist." The current theory of logic has a "block universe" picture built into it. Either something exists in the block universe, or it doesn't; there is no "could."

The "all" vs "any" distinction comes up when dealing with the infinite. With the doubling function <math>\mathbb{N} \rightarrow \mathbb{N}</math> sending <math>n\mapsto 2n</math>, it's not right to think about it as "all" natural numbers. You should think about it as doubling any natural number that may be presented.

There's some connection between this and the material implication. Usually, though not always, when we say "if X then Y," we have in mind that X could be true and that Y could be true.

== Terms vs. predicates ==
[TODO]

Logic

2025-05-24T17:05:25Z

Lfox: Created page with "I think that set theory and formal logic need to be re-done rationally. This is a big project. I will list out some of my problems with formal logic below: == Material implication == <math>A \implies B</math> is held to equivalent to <math>\neg A \vee B</math>, so anything follows from a falsehood. This doesn't fit natural language reasoning. For example, suppose that it is not raining outside right now, and I say "if it were raining outside right now, then communism w..."

Main Page

2025-05-24T16:29:49Z

Lfox:

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Category theory]]
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Logic]]
* [[Against models (essay)]]
* [[Category theory: abstracting mathematical construction (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
* [[Perspective Theory]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].

Main Page

2025-05-22T21:17:36Z

Lfox:

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Category theory]]
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Against models (essay)]]
* [[Category theory: abstracting mathematical construction (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
* [[Perspective Theory]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].

Category theory: abstracting mathematical construction (essay)

2025-05-20T03:27:23Z

Lfox:

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.) We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case if the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding"), or choose a concrete set-theoretic model for their category.

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to a morphism <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ ( 1,2, a) , (1,2, b ), (3,4, \alpha) , (3,4, \beta), (3,4 , a ) \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := \{ (i,j,x) \ | \ (x\in S \wedge i= 1 \wedge j =2 ) \vee (x\in T \wedge i = 3 \wedge j =4) \}. </math>♦

It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with "1", and putting the latter into a tuple with "2". The "disjoint sum" construction reverses the order, and keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a 3-tuple with "1" and "2", and putting the latter into a 3-tuple with "3" and "4".

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. In particular, if there were a different category of sets from the standard one (and I alluded to the fact that I think there ought to be a better one), then this property would still make sense. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that in answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-20T03:19:43Z

Lfox: /* Universal properties */

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.) We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case if the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding"), or choose a concrete set-theoretic model for their category.

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to a morphism <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ ( 1,2, a) , (1,2, b ), (3,4, \alpha) , (3,4, \beta), (3,4 , a ) \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := \{ (i,j,x) \ | \ (x\in S \wedge i= 1 \wedge j =2 ) \vee (x\in T \wedge i = 3 \wedge j =4) \}. </math>♦

It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with "1", and putting the latter into a tuple with "2". The "disjoint sum" construction reverses the order, and keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a 3-tuple with "1" and "2", and putting the latter into a 3-tuple with "3" and "4".

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that in answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-20T03:13:51Z

Lfox: /* Universal properties */

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.) We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case if the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding"), or choose a concrete set-theoretic model for their category.

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to a morphism <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ ( 1,2, a) , (1,2, b ), (3,4, \alpha) , (3,4, \beta), (3,4 , a ) \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := \{ (i,j,x) \ | \ (x\in S \wedge i= 1 \wedge j =2 ) \vee (x\in T \wedge i = 3 \wedge j =4) \}. </math>♦

It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with "1", and putting the latter into a tuple with "2". The "disjoint sum" construction reverses the order, and keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a 3-tuple with "1" and "2", and putting the latter into a 3-tuple with "3" and "4".

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that in answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-20T03:08:29Z

Lfox:

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.) We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case if the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding"), or choose a concrete set-theoretic model for their category.

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to a morphism <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ ( a, A) , (b, A), (\alpha ,B) , (\beta, B), (a , B) \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := \{ (x,X) \ | \ (x\in S \wedge X=S) \vee (x\in T \wedge X=T) \}. </math>♦

It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with "1", and putting the latter into a tuple with "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with <math>A</math>, and putting the latter into a tuple with <math>B</math>.

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that in answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-18T19:22:55Z

Lfox:

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.) We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case if the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding"), or choose a concrete set-theoretic model for their category.

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦
It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>.

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that in answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Main Page

2025-05-18T19:18:23Z

Lfox: /* All Pages */

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Category theory]]
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Against models (essay)]]
* [[Category theory: abstracting mathematical construction (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].

Category theory: abstracting mathematical construction (essay)

2025-05-18T19:17:47Z

Lfox: Lfox moved page Category theory to Category theory: abstracting mathematical construction (essay) without leaving a redirect

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.)

We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that, even though ''prima facie'' it may look like the objects of a category are like "entities" and the morphisms are like "actions," I don't think that's precisely what categories are conceptualizing. Another reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case of the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding").

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦
It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>.

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-18T19:08:30Z

Lfox:

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.)

We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that, even though ''prima facie'' it may look like the objects of a category are like "entities" and the morphisms are like "actions," I don't think that's precisely what categories are conceptualizing. Another reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case of the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding").

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors ''on groupoids''.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦
It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>.

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|217x217px]]

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. You don't even have to know that <math>X </math>, <math>S </math>, and <math>T </math> are sets, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are functions. For the statement to make sense, we just have to know that <math>X </math>, <math>S </math>, and <math>T </math> are objects in some category, and that <math>i_S </math>, <math>\lambda_T </math>, etc. are morphisms in that category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but one could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!).

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

''This'' is the sense in which the disjoint sum and the disjoint union are really "the same" construction! They satisfy the same universal property, and therefore are isomorphic in a canonical way.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a canonical/unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-18T18:56:57Z

Lfox: /* Universal properties */

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.)

We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that, even though ''prima facie'' it may look like the objects of a category are like "entities" and the morphisms are like "actions," I don't think that's precisely what categories are conceptualizing. Another reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case of the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding").

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors on groupoids.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties."

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. Disjoint union and union are different concepts. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction.

Given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

It is possible to get at the exact same idea in a way which is differs in mere technical details. We could instead have defined the disjoint union as<math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally,

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦
It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>.

We see that there is something arbitrary about the details of these constructions. What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there is always an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. These data possess the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram:

[[File:Screenshot 2025-05-18 at 2.51.41 PM.png|center|The universal property of the coproduct.|300px]]

Ex

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. To make the statement about <math>S + T </math> rather than <math>S \sqcup T </math>, all we had to do was change the words. We didn't have to dig into the details of the way that these two sets are constructed. In other words, all that this universal property cares about is the properties of these things ''qua'' elements of some category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!). [TODO]

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

File:Screenshot 2025-05-18 at 2.51.41 PM.png

2025-05-18T18:52:23Z

Lfox:

Universal property of the coproduct.

Category theory: abstracting mathematical construction (essay)

2025-05-18T18:42:16Z

Lfox:

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category <math>\text{Cat}</math> whose objects are categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a ''structure'' on <math>E </math>. (A structure is an order, or a sigma algebra, or an involution, or a multiplication operation, or something like that. I won't try to define precisely what a "structure" is (see Bourbaki for that), but I claim that everything in standard mathematics can be thought of as some structure on some set.)

We then get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is ''equal to'' <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math>. </ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One reason why is that, even though ''prima facie'' it may look like the objects of a category are like "entities" and the morphisms are like "actions," I don't think that's precisely what categories are conceptualizing. Another reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory, which wouldn't be the case of the latter was truly foundational (and hence integral to human thought). It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain below. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding").

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( Y ,Z)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(Y),F(Z))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I will only give one example, one which I think demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A set-theoretic construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>? Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting<math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get
<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.
Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's define <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear that if <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, sending <math>(s,i) \mapsto (\psi(s), i)</math> which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets. (Exercise: verify this last statement.)

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we were to change <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Technical caveat: it is the essence of functors on groupoids.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>. But I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference: The resulting category would have been essentially the same. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n: The resulting computer program will be essentially the same.)

'''Functors are a way of talking about constructions, that abstracts away the arbitrary choices that went into the construction.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choice. Here we also see why category theory is difficult: saying how you would handle ''any possibility'' is much more difficult than just saying how you would do the construction in one specific case.

== Universal properties ==
Besides functors, there is another way in which categories abstract away from arbitrary choices, called "universal properties." [TODO explain set theoretic union?]

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given ''two'' sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>This should be contrasted with the ''union'' of <math>A</math> and <math>B</math>, <math>A \cup B := \{ a, b, \alpha, \beta \} </math>. The disjoint union keeps the <math>a \in A</math> separate from the <math>a \in B</math>, and the union does not. In standard set theory, the union is taken to be an axiomatic notion, and the disjoint union is taken to be a construction. And of course, given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally, '''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set

<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

However, we can get at the exact same idea in a way which is technically different. We could instead have defined the disjoint union as <math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally
'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦
It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>. We see that there is something arbitrary about the details of these constructions.

What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there always exists an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. This data possesses the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram [TODO].

Ex

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. To make the statement about <math>S + T </math> rather than <math>S \sqcup T </math>, all we had to do was change the words. We didn't have to dig into the details of the way that these two sets are constructed. In other words, all that this universal property cares about is the properties of these things ''qua'' elements of some category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!). [TODO]

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Conclusion (general relativity and beyond) ==
This theme of ''canonical'' math constructions, i.e. those which can be performed without making any choices, is very powerful. I believe that the ideas of category theory will have implications that go far beyond the ridiculously abstract fields (algebraic geometry, homotopy theory) in which they are presently employed. I will conclude with just one tantalizing example:

The laws of classical mechanics are formulated with respect to coordinate systems. A coordinate system is a ''choice'', which could have been otherwise: the 0˚ longitude line could have been just as easily defined to run through Paris as it was defined to run through Greenwich. If your coordinate system is changed drastically, then the form of the laws of classical mechanics change drastically in turn. For example, if your coordinate system is rotating, then there is a Coriolis force.

One might ask the question: How do we formulate the laws of physics in a way so that they look the same, regardless of what coordinate system we are using? I suspect that answering that question, you will arrive at something like General Relativity.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-16T21:15:54Z

Lfox: /* Functors */

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category of categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a structure (I won't define what a "structure" is; see Bourbaki for that. But think about it as an additional set. [TODO look up]). Then we get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is equal to <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math></ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One big reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory. It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain later. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding").

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( X ,Y)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(X),F(Y))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and which respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I won't give the standard examples here. I will only give one example, which demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction."<ref group="note">One might ask what I mean by a "construction." What exactly are set-theoretic constructions in reality? A construction is a method of forming one set, given some other sets. To "form a set" is to make the identification that some existents are a set; it is to consider the existents as belonging together, as part of a single collection; it is to take the unit perspective on some existents. </ref> For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>. Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting <math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.

Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's call <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear (exercise!) that <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets.

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we changed <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Or at least, it is the essence of functors on groupoids.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>, but I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n.)

'''Functors are a way of talking about constructions, without making any arbitrary choices.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choices. (Here we also see why category theory is difficult: to define a functor, you have to say how you would handle any possible choice, which is much more difficult than just saying how you would do the construction in some specific case).

== Universal properties ==
There is another way in which categories handle arbitrary choices. [TODO explain set theoretic union?]

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given two sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>And of course, given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

However, we can get at the exact same idea in a way which is technically different. We could instead have defined the disjoint union as <math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦

It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>. We see that there is something arbitrary about the details of these constructions.

What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there always exists an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. This data possesses the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram [TODO].

Ex

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. To make the statement about <math>S + T </math> rather than <math>S \sqcup T </math>, all we had to do was change the words. We didn't have to dig into the details of the way that these two sets are constructed. In other words, all that this universal property cares about is the properties of these things ''qua'' elements of some category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!). [TODO]

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-16T21:01:44Z

Lfox:

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists<ref group="note">In the literature this is called a "locally small" category.</ref> of the following data:

# A class<ref group="note">A class is just another word for a set. In conventional mathematics, we aren't allowed to use the word "set" here, because for many categories it would give rise to a Russell's paradox. I think that Russell's paradox only arises in situations where we aren't being careful about our what math refers to in reality, but that discussion would take us too far afield.</ref> <math>\text{ob}(\mathcal{C})</math>, called the "objects" of the category
# For any two objects <math>X,Y \in \text{ob}(\mathcal{C})</math>, a set <math>\text{mor}_{\mathcal{C}}(X,Y)</math>, called the "morphisms from <math>X </math> to <math>Y </math>." We take <math>f : X \rightarrow Y</math> to be a statement which means the exact same thing as the statement <math>f \in \text{mor}_{\mathcal{C}}(X,Y)</math>.
# For any two morphisms <math>f : X \rightarrow Y</math> and <math>g : Y \rightarrow Z</math> , a function <math display="block">\circ : \text{mor}_{\mathcal{C}}(Y,Z) \times \text{mor}_{\mathcal{C}}(X,Y) \rightarrow \text{mor}_{\mathcal{C}}(X,Z), </math> which we call "composition." That is, we can compose <math>f </math> and <math>g </math> to get an element <math>g \circ f : X \rightarrow Z </math>,
subject to the following conditions:

# For any object <math>X \in \text{ob}(\mathcal{C}) </math>, there exists an "identity" morphism <math>1_X : X \rightarrow X </math>, satisfying <math>1_X \circ f = f </math> for any <math>f : Y \rightarrow X </math> and <math>g \circ 1_X = g </math> for any <math>g : X \rightarrow Z </math> .
# Composition is associative: <math>(f \circ g) \circ h = f \circ (g \circ h) </math>.

♦

One example of a category---''the'' example, in fact---is <math>\text{Set} </math>, the category of sets. The objects of <math>\text{Set} </math> are sets, the morphisms of <math>\text{Set} </math> are functions, and composition of morphisms of <math>\text{Set} </math> is composition of functions: <math>(g \circ f)(x) : = g(f(x)). </math> The identity morphism of a set <math>X</math> is the function <math>X \rightarrow X </math> which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call <math>\mathbb{I} </math>. The objects <math>\text{ob}(\mathbb{I}) </math> are a set consisting of 2 elements (for concreteness we could take <math>\text{ob}(\mathbb{I}) := \{ a,b \} </math> where <math>a = \emptyset </math> and <math>b = \{ \emptyset \} </math>).<ref group="note">I think sets contain actual existents. One type of existent is indeed a set which is empty. However, I don't think such things should play any sort of foundational role in a proper theory of sets. Another place where I disagree with the mainstream is that I don't think there is a unique empty set. The place where any set exists is within the mind of an individual; a set is one of his forms of awareness. The empty set is a man's awareness of ''nothing'', i.e. of some existent being ''other than'' what context suggests. Thus there is one empty set in Bob's mind when he identifies that his pantry is empty, another empty set in Mary's mind when she identifies that she doesn't have any meetings today, etc. </ref> The morphisms of <math>\mathbb{I} </math> are the identity elements, and nothing else. More formally, we have<math>\text{mor}_{\mathbb{I}}(a,a) = \{ * \} </math>, <math>\text{mor}_{\mathbb{I}}(a,b) = \emptyset </math>, and <math>\text{mor}_{\mathbb{I}}(b,b) = \{ * \} </math>.

Another example of a category is the category <math>\text{Grp} </math> of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category <math>\text{Top} </math> of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ''ad infinitum''. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, algebraic varieties, the points of the sphere, the open subsets of the plane, the natural numbers, compact Hausdorff totally disconnected spaces, etc etc. There is even a category of categories themselves. In fact, ''every'' mathematical object forms a category,<ref group="note">Indeed, suppose that some "''thing"'' in mathematics is defined as <math>(E,\Sigma) </math>, where <math>E </math> is a set and <math>\Sigma </math> is a structure (I won't define what a "structure" is; see Bourbaki for that. But think about it as an additional set. [TODO look up]). Then we get a category <math>\text{Thng} </math> of "things" for free. The objects of <math>\text{Thng} </math> are "thing"s, and a morphism <math>(E,\Sigma) \rightarrow (E', \Sigma') </math> is an isomorphism <math>\varphi : E \rightarrow E' </math> of sets, such that: the structure <math>\varphi_*\Sigma </math> living over <math>E' </math> that you get when you transport <math>\Sigma </math> by <math>\varphi </math>, is equal to <math>\Sigma' </math>, or in symbols <math>\varphi_*\Sigma = \Sigma' </math></ref> though perhaps not in a useful way. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has reminded me that category theory can be taken as an alternative foundation to set theory (see [https://ncatlab.org/nlab/show/ETCC here] for Lawvere's attempt at that). I have barely studied Lawvere's ideas, but I am skeptical that it is useful as a foundation. One big reason why is that when it comes to actually proving things, set theory is much ''easier'' than category theory. It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically. I don't think this is just my own ineptitude; I think there is a deep reason why category theory is difficult, which I will explain later. It's not just me, either: When working mathematicians want to understand something category theoretic, they often do things that allow them to think about it in a set theoretic manner, e.g. they embed their category into the category of sets (see "Yoneda embedding").

== Functors ==
I mentioned earlier that there is a category <math>\text{Cat}</math> of categories. Clearly the objects of <math>\text{Cat}</math> are ... categories, but what are the morphisms of <math>\text{Cat}</math>? They are called ''functors'', and they are defined as follows.

'''Definition.''' A ''functor'' <math>F : \mathcal{C} \rightarrow \mathcal{D}</math> is a function which sends any <math>X \in \text{ob}(\mathcal{C})</math> to an object <math>F(X) \in \text{ob}(\mathcal{D})</math>, along with a function which sends any <math>\varphi \in \text{mor}_{\mathcal{C}}( X ,Y)</math> to an object <math>F(\varphi) \in \text{mor}_{\mathcal{D}}(F(X),F(Y))</math>. This assignment respects composition (meaning <math>F(\varphi \circ \psi) = F(\varphi) \circ F(\psi)</math> for any morphisms <math>\varphi, \psi</math>), and which respects identity (meaning <math>F(\text{Id}_{X} ) = \text{Id}_{F(X)} </math> for any object <math>X</math>). ♦

If you want examples of functors, you can find many of them in a category theory textbook. I won't give the standard examples here. I will only give one example, which demonstrates the essence and purpose of functors.

Let's suppose that we are mathematicians who wish to do some set-theoretic "construction." For example, one construction we could do is we could take any set <math>A</math>, and we could replace it with a set <math>A\sqcup A</math>, where <math>A\sqcup A</math> is a set which has exactly two copies of every element of <math>A</math>. For example, if <math>A = \{a, b\}</math> then we might have <math>A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\}</math>.

Now, let's suppose that we change <math>A</math> in a "trivial" way. For example, instead of considering <math>A = \{a, b\}</math>, what if we considered <math>A' = \{a', b'\}</math>. Since <math>A</math> and <math>A'</math> are basically the same, nothing important would change. Instead of getting <math display="block">A\sqcup A = \{(a,1), (b,1), (a,2), (b,2)\},</math>we would get<math display="block">A' \sqcup A' = \{(a',1), (b',1), (a',2), (b',2)\},</math>which, again, is basically the same thing.

Now, why am I entitled to say that <math>A</math> and <math>A'</math> are "basically the same," and that replacing <math>A</math> with <math>A'</math> was trivial? Because all I did was I took the contents of <math>A</math>, and I added a symbol ' to both of them! That's not interesting. More formally, I think that <math>A</math> and <math>A'</math> are "the same" because I have in mind a function <math>\varphi : A \rightarrow A' </math>, which sends <math>a \mapsto a'</math> and <math>b \mapsto b'</math>, and this function is an ''isomorphism;'' it has an inverse function <math>A' \rightarrow A </math> which sends <math>a' \mapsto a</math> and <math>b' \mapsto b</math> . Likewise, I think that <math>A \sqcup A</math> and <math>A' \sqcup A'</math> are "basically the same" because there is an isomorphism <math>A \sqcup A \rightarrow A' \sqcup A'</math> sending <math>(a, 1) \mapsto (a', 1)</math>, <math>(b, 2) \mapsto (b', 2)</math>, etc.

It is clear that <math>A</math> and <math>A'</math> didn't play any special role in our construction; we could have done the exact same thing for any set <math>S</math>. Thus, for an arbitrary set <math>S</math>, let's call <math>F(S) := S \sqcup S</math>. Furthermore, let's call the latter isomorphism of the above paragraph <math>F(\varphi) : F(A) \rightarrow F(A')</math>. It's also clear (exercise!) that <math>\psi : S \rightarrow T</math> is any isomorphism of sets, then we get an isomorphism <math>F(S) \rightarrow F(T) </math>, which we shall call <math>F(\psi)</math>. Thus our construction defines a ''functor'' <math>F : \text{Iso}(\text{Set}) \rightarrow \text{Iso}(\text{Set})</math>, where <math>\text{Iso}(\text{Set})</math> is that category whose objects are sets and whose morphisms are isomorphisms of sets.

Let's take a step back and observe what happened. We have a set-theoretic construction <math>S \mapsto S \sqcup S</math>. This set-theoretic construction is robust against arbitrary choices, in the sense that if we were to relabel all the elements of <math>S</math> (that is, if we changed <math>S</math> to an isomorphic set <math>S'</math>), then there is an obvious way to relabel all the elements of <math>S \sqcup S</math>. This robustness means precisely that our construction defines a functor.

''This'', in my view, is the essence of functors.<ref group="note">Or at least, it is the essence of functors on groupoids.</ref> Functors represent mathematical ''constructions'', such that when you make an unimportant change to the input of the construction, it induces an unimportant change on the output of the construction.

As any mathematician (or computer programmer) knows, whenever you are constructing something (or programming something), there are tons of details which are "arbitrary," in the sense that they that could have been other than the way they are. We even faced this issue in the previous section, when I defined the category <math>\mathbb{I}</math>. To define the objects of <math>\mathbb{I}</math>, I needed a set with two elements, and I chose <math>\text{ob}(\mathbb{I}) := \{ \emptyset , \{ \emptyset\} \} </math>, but I could just as well have chosen <math>\text{ob}(\mathbb{I}) := \{\{ \emptyset \} , \{ \{ \emptyset \} \} \} </math>, or <math>\text{ob}(\mathbb{I}) := \{1, 2 \} </math>, or <math>\text{ob}(\mathbb{I}) := \{*, \star \} </math>, or anything else, and it wouldn't have made a big difference. (Likewise, when programming a computer, it doesn't matter if we begin our for-loops at 0 and end at n-1, or begin them at 1 and end them at n.)

'''Functors are a way of talking about constructions, without making any arbitrary choices.''' Functors gain their "independence" from arbitrary choices because to build one, you have to say how it would handle ''any possible'' arbitrary choices. (Here we also see why category theory is difficult: to define a functor, you have to say how you would handle any possible choice, which is much more difficult than just saying how you would do the construction in some specific case).

== Universal properties ==
There is another way in which categories handle arbitrary choices. [TODO explain set theoretic union?]

The construction (or as we now know, the ''functor'') <math>S \mapsto S \sqcup S</math> of the previous section suggests a more general construction. Given two sets <math>A := \{ a, b \} </math> and <math>B := \{ \alpha, \beta, a \}</math>, we can construct their ''disjoint union'' <math display="block">A \sqcup B := \{ (a, 1), (b,1), (\alpha, 2), (\beta, 2), (a, 2) \}. </math>And of course, given any set <math>S </math> and any set <math>T </math>, we can construct a set <math>S \sqcup T </math> whose elements are the elements of <math>S </math> and the elements of <math>T </math>. More formally,

'''Definition.''' The ''disjoint union'' of a set <math>S </math> and a set <math>T </math> is the set<math display="block">S \sqcup T := \{ (x,i)\ | \ (x\in S \wedge i=1) \vee (x\in T \wedge i=2) \}. </math>♦

However, we can get at the exact same idea in a way which is technically different. We could instead have defined the disjoint union as <math display="block">A + B := \{ a, b, \{ \alpha \}, \{ \beta \}, \{ a \} \}, </math>and generally

'''Definition.''' The ''disjoint sum'' of a set <math>S </math> and a set <math>T </math> is the set <math display="block">S + T := S \cup \{ \{ t\}\ |\ t\in T \}. </math>♦

It is clear that the thing I'm calling "disjoint union" and the thing I'm calling "disjoint sum" are not different in an important way. Both of them capture the idea that, given two sets, there exists a set which has all the elements of the first one and all the elements of the second one, and which keeps the elements separate. The "disjoint union" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by putting the former into a tuple with a "1", and putting the latter into a tuple with a "2". The "disjoint sum" construction keeps <math>a \in A </math> separated from <math>a \in B </math> by doing nothing to the former, and putting the latter into a set <math>\{ a \} </math>. We see that there is something arbitrary about the details of these constructions.

What is the common essence that ''disjoint union'' and ''disjoint sum'' share, which we are trying to capture with our formal definitions? Is it even possible to talk about such a thing mathematically? The answer to the latter question is yes(!!), and the answer to the former question is that these two constructions possess the same ''universal property''. I will not try to define this concept yet, but rather I will proceed inductively by studying the disjoint union and disjoint sum in greater depth.

The first thing to note about the disjoint union is that there always exists an "inclusion" function <math>i_S : S \rightarrow S \sqcup T </math> sending <math>s \mapsto (s, 1) </math>, and an "inclusion" function <math>i_T : T \rightarrow S \sqcup T </math> sending <math>t \mapsto (t, 2) </math>. This data possesses the following property:

'''Proposition 1 (universal property of the coproduct).''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S \sqcup T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

This proposition is logically somewhat complex (<math>\forall \exists \forall </math>), but the reader will find that it is very easy to prove once he gets a grip on what it is saying.

The more standard and succinct (but less clear) way to state the above proposition would be to say "for any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a ''unique'' function <math>\chi : S \sqcup T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>." But rather than writing out that long sentence every time, a category theorist would draw the following diagram [TODO].

Ex

The universal property of the coproduct is also satisfied by the disjoint sum. That is,

'''Proposition 2.''' For any set <math>X </math>, and any functions <math>\lambda_S : S \rightarrow X </math>, <math>\lambda_T : T \rightarrow X </math>, there exists a function <math>\chi : S + T \rightarrow X </math> such that <math>\chi \circ i_S = \lambda_S </math> and <math>\chi \circ i_T = \lambda_T </math>, and <math>\chi </math> is the ''unique'' function with this property. More formally, uniqueness means that if there is any function <math>f : S + T \rightarrow X </math> such that <math>f \circ i_S = \lambda_S </math> and <math>f \circ i_T = \lambda_T </math>, then necessarily <math>f = \chi </math>. ♦

Note that this "universal property of the coproduct" is entirely category theoretic: to state the property, you don't have to know any of the details about what's inside <math>X </math>, <math>S </math>, or <math>T </math>, and you don't have to know anything about what the functions <math>i_S </math>, <math>\lambda_T </math>, etc. are doing. To make the statement about <math>S + T </math> rather than <math>S \sqcup T </math>, all we had to do was change the words. We didn't have to dig into the details of the way that these two sets are constructed. In other words, all that this universal property cares about is the properties of these things ''qua'' elements of some category.

A consequence of this generality is that we could have formulated the exact same universal property in ''any other category'', whether it be groups, topological spaces, natural numbers, or anything else. There's no guarantee that any object of a given category will actually possess this property, but could state it nonetheless. Another consequence of this generality is the following:

'''Proposition 3.''' Let <math>S </math> and <math>T </math> be two objects of ''any''(!) category, and suppose that <math>S \rightarrow Y \leftarrow T </math> and <math>S \rightarrow Z \leftarrow T </math> both satisfy the universal properties of the coproduct. Then <math>Y </math> and <math>Z </math> are ''uniquely isomorphic''(!). [TODO]

'''Corollary of proposition 3.''' The disjoint sum and disjoint union of <math>S </math> and <math>T </math> are uniquely isomorphic.

As an exercise, the reader should give a more precise statement of proposition 3 (i.e. state precisely what "uniquely isomorphic" means), and prove it.

I can now give a definition of a universal property. A universal property is a property possessed by an object of a category, which characterizes it up to a unique isomorphism. Besides the coproduct, there are many other familiar sets and set-theoretic constructions that possess universal properties, for example, the product of two sets <math>S \times T = \{ (s,t) \ |\ s\in S, t\in T \} </math>, a set containing a single element <math>\{ * \} </math>, a coproduct of ''three'' sets <math>S \sqcup T \sqcup R </math>, and anything else that you can dream up. For an example of the flavor of universal products in other categories, I shall note that in many geometrical categories (where the objects are shapes / spaces of some sort), the (transverse) intersection of two objects satisfies a universal property (the universal property of the pullback), and the gluing together of two objects satisfies a universal property (the universal property of the pushforward).

In conclusion, with universal properties we observe the same phenomenon that we observed with functors. There are some arbitrary details in our math constructions, but this concept helps us talk about our math constructions in a way which guarantees that those arbitrary details won't matter. If the only properties of <math>A \sqcup B </math> that we ever use are those that follow from its universal property, then it is completely indistinguishable from <math>A + B </math>. Of course, just like for functors, this robustness against arbitrary choices comes at the cost of making category theory ''more difficult'' than set theory: instead of just writing down a set and chugging along, we have to learn and automatize a complicated <math>\forall \exists \forall </math> statement.

== Notes ==

<references group="note" />

Category theory: abstracting mathematical construction (essay)

2025-05-15T02:20:56Z

Lfox:

Main Page

2025-05-15T02:12:16Z

Lfox: /* All Pages */

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Category theory]]
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Against models (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].

Category theory: abstracting mathematical construction (essay)

2025-05-15T02:11:48Z

Lfox:

Category theory: abstracting mathematical construction (essay)

2025-05-15T00:49:19Z

Lfox:

Category theory: abstracting mathematical construction (essay)

2025-05-14T23:35:45Z

Lfox: Created page with "In real life, and in particular, in mathematics, there are ''entities'' and ''actions''. We can consider types of entities. For example, let us consider two such types <math>X</math> and <math>Y</math>, where <math>X = \text{apples}</math> and <math>Y = \text{pie}</math>. There is a specific apple sitting in my fridge, There's an action of baking apple pie too broad Definition of a category '''Definition.''' A ''category'' <math>\mathcal{C}</math> consists o..."

In real life, and in particular, in mathematics, there are ''entities'' and ''actions''.

We can consider types of entities.

For example, let us consider two such types <math>X</math> and <math>Y</math>, where <math>X = \text{apples}</math> and <math>Y = \text{pie}</math>.

There is a specific apple sitting in my fridge,

There's an action of baking apple pie

too broad

Definition of a category

'''Definition.''' A ''category'' <math>\mathcal{C}</math> consists of the following data:

# A class

== Notes ==

<references group="note" />

Set

2025-05-13T03:25:47Z

Lfox: /* Functions */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
This essay was prompted by Ray's suggestion that I put my problem with the diagonal argument into writing. My problem is not really with the diagonal argument per se, it is with the more basic concept of sets. Rather than saying something negative about set theory as it stands today, I will offer something positive. Starting from scratch, I will sketch what I think a rational theory of sets would look like.

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups of the units of concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We may now see the subtlety with the idea of "the set of all points on O." Those points on O which are ''physical'' must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So regardless of which of those we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>. A function <math>f : X \rightarrow Y</math> is ''something you could do''.

A "problem" with this way of thinking about functions is that sometimes indeed you are doing something with <math>x</math>, and producing a new thing <math>f(x)</math>, but sometimes you haven't actually produced a new thing, you're just thinking about <math>x</math> in a different way. For example, all men are mortal, so there should be a function from the group of all men to the group of all mortal beings. It is arguable, however, that these functions represent something you could do, but it's something you do in consciousness---they represent a change of perspective. You aren't doing anything to <math>x</math> itself, you're just looking at <math>x</math> differently.

For an uncontroversial example of a function, it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Another type of function, one without a clear mathematization, is a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life, e.g. given a line segment on a piece of paper, pull out an amount of string equal to that length, then use the string to extend that line segment by drawing a new line segment of equal length. Mainstream math models the above process as the function <math>\mathbb{R} \rightarrow \mathbb{R}</math> which sends <math>x \mapsto 2x</math>, but that's not quite right, because <math>\mathbb{R}</math> is a number (or at least, something akin to a number), and not the length itself. It is very common that mainstream math thinks of <math>\mathbb{R}</math> as "lengths as they actually are in reality." It conflates length with measurement of length.

Is a function <math>\mathbb{R} \rightarrow \mathbb{R}</math> something you could do? It's not clear. What even are the elements of the group <math>\mathbb{R}</math>? The elements of <math>\mathbb{R}</math> are equivalence classes of Cauchy sequences, so those are two things we need to understand: equivalence classes, and Cauchy sequences.

Cauchy sequences are functions <math>\mathbb{N} \rightarrow \mathbb{Q}</math>. Is a Cauchy sequence something you could do? Some of them are, like <math>\{ 1/n \}_{n=1}^\infty </math>, but some of them aren't, like <math>\{ 1/h_n \}_{n=1}^\infty</math> where <math>h_n = </math> "the number of steps the <math>n</math>th Turing machine takes to halt." And some of them are very bizarre, like <math>\{ a_n \}_{n=1}^\infty </math> where <math>a_n = </math> "the number of stars in the Milky Way of mass <math>\leq n</math> kilograms."

Is it legitimate to talk about Cauchy sequences that are ''not'' something you could do? The answer of mainstream mathematics is a resounding "yes." My answer is a tentative "no": if it's not something which exists in reality, and it's not something which potentially could exist in reality, then your sequence doesn't refer to anything, and so it's meaningless. It's like talking about flying purple cats. The reason why my answer is ''tentative'' "no" is that sometimes we can learn things from impossible hypotheticals; sometimes they reveal things about the nature of our legitimate concepts. I don't want to police what sort of Cauchy sequences people can or cannot talk about, I just want to reorient math in the direction of reality.

What is an equivalence class? Well that's actually quite simple. First let's consider fractions: 1/3 and 2/6 are the same ''fraction'', but they are different ''expressions''. A fraction is an equivalence class of expressions <math>p/q</math> involving two integers <math>p,q\in \mathbb{Z}</math> (where <math>q \neq 0</math>), and where the equivalence relation is that <math>p/q = r/s</math> if <math>ps = qr</math>. The concept of equivalence classes extends far beyond mathematics, to any concept which is a special case of another concept. The two copies of ''Conformal Field Theory'' (some physics textbook) in my office are different ''copies'', but they are the same ''book''. Now, given two Cauchy sequences <math>\{ 1/n \}_{n=1}^\infty </math> and <math>\{ 1/n^2 \}_{n=1}^\infty </math>, which mind you are two things---two processes---actually out there in reality, we say they are different qua ''sequence'', but they are the same qua ''real number''. Generally, given two Cauchy sequences <math>\{ a_n \}_{n=1}^\infty </math>and <math>\{ b_n \}_{n=1}^\infty </math>, we say regard them as equivalent if for any <math>\epsilon \in Q, </math> such that <math>\epsilon > 0</math>, there exists an <math>N_\epsilon \in \mathbb{N}</math> such that whenever <math>n \geq N</math>, <math>| a_n - b_n| < \epsilon </math>.

So what real thing does <math>\mathbb{R}</math> refer to? It refers to a particular kind of potential action <math>\mathbb{N} \rightarrow \mathbb{Q}</math> (namely one that is "Cauchy"), from a particular perspective (namely the perspective from which two Cauchy sequences are the same if they are going to the same place).

Now, how about functions <math>\mathbb{R} \rightarrow \mathbb{R}</math>? Well it's just something that takes a real number and sends it to a real number. If you think of the real number as a sequence <math>\{ a_n \}_{n=1}^\infty </math>, that's fine, but the function needs to be defined for <math>\{ a_n \}_{n=1}^\infty </math> qua real number rather than <math>\{ a_n \}_{n=1}^\infty </math> qua sequence. Again, this point is logical and not mathematical. We could consider a function from the group of all books to the group of all strings, e.g. which sends a book to its author; such a function can be defined on ''copies'' of books (like maybe your way of implementing it is: purchase a copy of the book and look at who the author of the copy is), but it must necessarily depend on the properties of the copy ''qua book'' rather than its properties ''qua copy''. For example, it can't depend on the physical location of the book.

So, in conclusion, the answer to the question "Is a function <math>\mathbb{R} \rightarrow \mathbb{R}</math> something you could do?" is yes. But what sort of processes does it refer to in real life? Is it really practical?

== Group formation rules [analogue of ZF] ==
A group doesn't have to be specified in a completely unambiguous way. In real life, there is no such thing.

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

I actually don't like the term "form a set." It's vague. Really what we're doing is we are ''identifying'' sets; we are pointing to some things in reality, and saying "that's a set."

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there need not be a practical way to check that all the members of a set are the same.

Quotients / identity are actually what I have to think the most about.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Set

2025-04-17T14:20:21Z

Lfox: /* Functions */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
This essay was prompted by Ray's suggestion that I put my problem with the diagonal argument into writing. My problem is not really with the diagonal argument per se, it is with the more basic concept of sets. Rather than saying something negative about set theory as it stands today, I will offer something positive. Starting from scratch, I will sketch what I think a rational theory of sets would look like.

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups of the units of concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We may now see the subtlety with the idea of "the set of all points on O." Those points on O which are ''physical'' must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So regardless of which of those we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>.

so note that uh if

A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]

So it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Something much more confusing is like, a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life. It's not clear that it has a math description, but mainstream math models it as a function <math>\mathbb{R} \rightarrow \mathbb{R}</math>.

A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.

Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

A problem with functions (at least in the way that I think about them) is that sometimes indeed you are doing something with <math>x</math>, and producing a new thing <math>f(x)</math>, but sometimes you haven't actually produced a new thing, you're just thinking about <math>x</math> in a different way.

== Group formation rules [analogue of ZF] ==

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

I actually don't like the term "form a set." It's vague. Really what we're doing is we are ''identifying'' sets; we are pointing to some things in reality, and saying "that's a set."

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there need not be a practical way to check that all the members of a set are the same.

Quotients / identity are actually what I have to think the most about.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Set

2025-04-03T15:17:38Z

Lfox: /* My email to Ray 03/27/25 */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
This essay was prompted by Ray's suggestion that I put my problem with the diagonal argument into writing. My problem is not really with the diagonal argument per se, it is with the more basic concept of sets. Rather than saying something negative about set theory as it stands today, I will offer something positive. Starting from scratch, I will sketch what I think a rational theory of sets would look like.

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups of the units of concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We may now see the subtlety with the idea of "the set of all points on O." Those points on O which are ''physical'' must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So regardless of which of those we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>.

so note that uh if

A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]

So it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Something much more confusing is like, a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life. It's not clear that it has a math description, but mainstream math models it as a function <math>\mathbb{R} \rightarrow \mathbb{R}</math>.

A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.

Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

== Group formation rules [analogue of ZF] ==

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

I actually don't like the term "form a set." It's vague. Really what we're doing is we are ''identifying'' sets; we are pointing to some things in reality, and saying "that's a set."

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there need not be a practical way to check that all the members of a set are the same.

Quotients / identity are actually what I have to think the most about.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Set

2025-03-29T16:20:30Z

Lfox: /* Axiom schema of specification */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
Ray asked me to put my problem with the diagonal argument into writing, and I think this is a good idea.

My problem is not really with the diagonal argument per se, it is with the more basic concept of infinite sets.

I want to go on a quest, where I justify set theory from the ground up. This is inspired by [TODO]

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups coming from concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We now see the subtlety with the idea of "the set of all points on O." As for those points on O which are ''physical'', they must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So whatever it is that we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>.

so note that uh if

A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]

So it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Something much more confusing is like, a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life. It's not clear that it has a math description, but mainstream math models it as a function <math>\mathbb{R} \rightarrow \mathbb{R}</math>.

A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.

Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

== Group formation rules [analogue of ZF] ==

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

I actually don't like the term "form a set." It's vague. Really what we're doing is we are ''identifying'' sets; we are pointing to some things in reality, and saying "that's a set."

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there need not be a practical way to check that all the members of a set are the same.

Quotients / identity are actually what I have to think the most about.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Set

2025-03-29T15:17:01Z

Lfox: /* Functions */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
Ray asked me to put my problem with the diagonal argument into writing, and I think this is a good idea.

My problem is not really with the diagonal argument per se, it is with the more basic concept of infinite sets.

I want to go on a quest, where I justify set theory from the ground up. This is inspired by [TODO]

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups coming from concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We now see the subtlety with the idea of "the set of all points on O." As for those points on O which are ''physical'', they must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So whatever it is that we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>.

so note that uh if

A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]

So it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Something much more confusing is like, a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life. It's not clear that it has a math description, but mainstream math models it as a function <math>\mathbb{R} \rightarrow \mathbb{R}</math>.

A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.

Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

== Group formation rules [analogue of ZF] ==

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there's no practical way to check that all the members of a set are the same.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Set

2025-03-29T05:52:34Z

Lfox: /* Group membership */

"'''Set'''" is a concept referring to an irreducible primary, and therefore can only be defined ostensively. A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar [[Existent|existents]], considered together as a whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about ''the purpose of sets''.

Sets are sometimes called "groups" (e.g. in ITOE), but Objective Mathematics reserves that terminology for [[Group|a different concept]].

== Examples ==
A set of plates.

[TODO more]

== Empty set ==
''An'' empty set (not "''the''" empty set; see below) is one way of viewing the concept of '''nothing'''. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains<ref>Rand, Ayn. ''Introduction to Objectivist Epistemology''. Penguin, 1990.</ref><blockquote>["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present. You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form. </blockquote>

=== Examples ===
If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

=== The traditional concept ===
In standard mathematics, there is supposed to be a single object called "''the''" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "''the'' empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

== Relations among sets ==
In this section, I will describe some relations<ref group="note">In this context, standard mathematics would use the phrase "operations on sets" or "constructions in the category of sets," rather than "relations among sets." </ref> among sets. That is, I will describe ways in which some sets (possibly along with [[functions]] between them) can be used to ''identify''<ref group="note">In this context, standard mathematics uses the word "construct" instead of the word "identify."</ref> other sets. This list is non-exhaustive.

=== Disjoint union ===
Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''disjoint union''' of <math>A</math> and <math>B</math>, denoted as <math>A \sqcup B</math>, is following set <math display="block">A \sqcup B := \{ a_1, a_2,\cdots, a_n, b_1, b_2, \cdots, b_m \}. </math>[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined
<math display="block">A \sqcup B := \{ (1,a_1), (2,a_2), \cdots, (n, a_n), (1, b_1), (2, b_2), \cdots, (m, b_m) \}</math>category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

==== Examples ====
I have two piles of books on my table. Equivalently, I could say that I have identified two sets <math>P_1 := \{b_1, b_2, b_3\}</math> and <math>P_2 := \{a_1, a_2 \} </math> of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of ''two piles of books on my table'', I could think of ''all the books on my table''. Equivalently, I could say that I have identified the set of all the books on my table,<math display="block">P_1 \sqcup P_2 = \{ b_1, b_2, b_3, a_1, a_2\}.</math>

=== Cartesian product ===

Given two sets <math>A = \{ a_1, \cdots, a_n\}</math> and <math>B = \{ b_1, \cdots, b_m \}</math>, the '''cartesian product''' of <math>A</math> and <math>B</math>, denoted as <math>A \times B</math>, is following set of pairs

==== <math display="block">A \times B := \{ (a_i, b_j) : 1 \leq i \leq n, 1 \leq j \leq m \}. </math>Examples ====
The power socket on my wall has two outlets; in other words, I've identified a set <math>O := \{o_1, o_2\}</math> of outlets. An outlet has three holes; in other words I've identified (abstractly) a set <math>H := \{h_1, h_2, h_3\} </math>. Now, I can consider the set of all the holes in the power socket on my wall. It is <math display="block">O \times H = \{ (o_1, h_1), (o_1, h_2), (o_1, h_3), (o_2, h_1), (o_2, h_2), (o_3, h_3) \}. </math>

=== Subset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''subset''' of <math>A</math> is some of the units of <math>A</math>, say <math>a_{i_1}, \cdots, a_{i_k} \in A </math>, considered as a single set <math>B= \{ a_{i_1},\cdots, a_{i_k} \} </math>. We denote this by <math>B \subset A</math>.

=== Powerset ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, the '''powerset''' of <math>A</math>, denoted <math>2^A</math>, is the set consisting of all subsets of <math>A</math>, <math display="block">2^A := \{ B : B \subseteq A \}.</math>The reason for the notation is that the powerset may equivalently be defined as the set of all functions <math>A \rightarrow 2</math> where <math>2</math> denotes the set <math>2:= \{0,1\}</math>.

=== Partition ===
Given a set <math>A = \{a_1, \cdots, a_n\}</math>, a '''partition''' of <math>A</math> is a division of <math>A</math> into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is <math>A</math>.

A partition of <math>A</math> is equivalent to an '''equivalence relation''' of <math>A</math>. [TODO explain]

== Infinite sets ==
Objective Mathematics says that infinite sets are an invalid [[Concept#Notion|notion]]. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== The traditional concept ===
Standard mathematics says that a set <math>X</math> is ''infinite'' if there exist [[functions]] <math>X \rightarrow X</math> which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment<ref>Ewald, William, and Wilfried Sieg. “David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933.” ''Springer EBooks'', 2013, <nowiki>https://doi.org/10.1007/978-3-540-69444-1</nowiki>. </ref> about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number <math>n : \mathbb{N}</math>. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, ''even though the hotel is already full''. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is <math>n</math>, he should move to room number <math>n+1</math>. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.
=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

== My email to Ray 03/27/25 ==
Ray asked me to put my problem with the diagonal argument into writing, and I think this is a good idea.

My problem is not really with the diagonal argument per se, it is with the more basic concept of infinite sets.

I want to go on a quest, where I justify set theory from the ground up. This is inspired by [TODO]

=== What are sets(="groups")? ===
The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

# "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
# "Continuous" groups, like the group of all points on the south-facing wall of my room.
# Groups coming from concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as ''infinite sets''. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have ''some'' quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

=== How many points are there on my wall? ===
In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We now see the subtlety with the idea of "the set of all points on O." As for those points on O which are ''physical'', they must necessarily have a finite size (otherwise we couldn't know about them in the first place), and so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely things on which one is focusing, rather than physical objects in reality, they do not actually exist ''until someone focuses on them''. And a man can only ever focus on finitely many points of O.

So whatever it is that we mean by "point," we see that there are only finitely many points on my wall.

=== How many natural numbers are there? ===
I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:<blockquote>A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified. </blockquote>I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is ''identification''. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

=== Group membership ===
As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it ''is'' always meaningful to ask "is this thing a member of that group?"

To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

=== Functions ===
A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]

So it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

Something much more confusing is like, a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life. It's not clear that it has a math description, but mainstream math models it as a function <math>\mathbb{R} \rightarrow \mathbb{R}</math>.

A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.

Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

== Group formation rules [analogue of ZF] ==

=== Axiom schema of specification ===
yeah it's fine to "form a group" according to a predicate. BUT

1) you don't ''need'' a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

=== Axiom of extensionality ===
This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there's no practical way to check that all the members of a set are the same.

=== Axiom of choice ===
Bullshit.

=== Axiom of infinity ===
Bullshit.

=== Powerset axiom ===
I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

=== Other axioms ===
????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set <math>X</math> is ''infinite'' if there exists a function <math>f : X \rightarrow X</math> which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.

== Notes ==
<references group="note" />

== References ==

Set

2025-03-29T04:02:16Z

Lfox:

Set

2025-03-27T05:42:39Z

Lfox: /* How many natural numbers are there? */

Set

2025-03-27T05:15:36Z

Lfox: /* My email to Ray 03/27/25 */

Set

2025-03-27T05:10:09Z

Lfox:

Multiplication

2025-03-12T22:33:05Z

Lfox:

'''Multiplication''' is [TODO]

== Examples ==
My thesis is that multiplication always identifies quantities resulting from a unit-conversion (i.e. a change in unit-perspective) of some sort.

=== Examples which obviously come from unit-conversion ===
Suppose you wish to find the number of squares in a rectangular grid. You measure that there are <math>n</math> squares per column, and that there are <math>m</math> columns. In total, therefore, there are <math>nm</math> squares.

Each pack of hot dog buns contains 6 buns, so if I buy 10 packs then I will have enough buns for 6*10 = 60 hot dogs.

=== Examples which don't obviously come from unit-conversion ===
The volume of a disk of radius r = 2 meters is approximately 3.14 * 2 * 2 = 12.56 square meters. It could be argued, however, that this ''does'' come from unit-conversion: the derivation consists of finding the area <math>2\pi r \times dr</math> of small annuli (which is done by unit conversion and approximation), then adding their areas together.

A for loop which runs <math>n</math> times, inside of a for-loop which runs <math>m</math> times, will take approximately <math>nm</math> steps to run. Again, I think this ''does'' come from a unit-conversion.

A coin flip has 2 possible outcomes, and so a series of 3 coin flips has 2 * 2 * 2 = 8 possible outcomes.

Multiplication

2025-03-12T22:24:53Z

Lfox:

Number

2025-03-12T21:54:09Z

Lfox: /* Counting */

A '''number''' is an identification of a [[quantity]], by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>

The three concepts of ''number'', ''quantity'', and ''numeral'' are closely related. They conceptualize the same thing, but from three different perspectives. Number is epistemological. Quantity is metaphysical. Numeral is symbolic. Note that the latter two are referred to in the definition of number. Those concepts are more basic; quantity is an irreducible primary, and numerals are some symbols that man created. Number is an integration of the two.

This is also basically the definition of "numeral" and "counting."

Numbers work by identifying a quantity of things with a quantity of symbols. That is, the quantity of these dots {*,*} is ''the same as'' the quantity of these symbols {"1", "2"}.

== Definitions of specific numbers, like "2" ==
[TODO unify this page with the natural number page.]

The definition of the ''noun'' "2" is: the numeral succeeding "1" in the counting algorithm.

The definition of the ''adjective'' "2" is: the quantity of numerals up to and including "2," in the counting algorithm.

[TODO maybe I should delete "in the counting algorithm...." ]

I use "two" and "2" interchangeably.

The two definitions above can be generalized to arbitrary natural numbers. Indeed, we define "3" as the numeral succeeding "2" in the counting algorithm, "844" as the numeral succeeding "843" in the counting algorithm, etc.

== Basic arithmetic operations ==
[TODO unify with the several other articles, especially [[multitude]].]

[TODO quantity assumes the unit perspective...]

=== Succession ===
The '''successor''' to a numeral is the numeral which comes after it in the fixed sequence of numerals.

Successorship is the [[function]] computing successor. There is a very simple [[algorithm]] computing successorship, which is learned by schoolchildren. [TODO copy paste the part where I wrote it?]

=== Counting ===
'''Counting''' is the process which produces a number by means of repeated application of the successorship function.

=== Addition ===
The '''sum''' or '''disjoint union''' of two quantities of existents is the quantity of those existents considered together as a single group. (See also the "disjoint union" thing.)

'''Addition''' is the function which takes two numbers as input, and outputs the number identifying the sum of their corresponding quantities.

=== Subtraction ===
The '''difference''' of two quantities is the quantity which would have to be added to the lesser in order to make it equal to the greater. (Note: this sense of the word is used derivatively. It should not be confused with that primary and axiomatic concept of "difference," which stands in opposition to the axiomatic concept of "similarity.")

'''Subtraction''' is the function which takes two numbers as input, and produces the number identifying the difference between their corresponding quantities.

=== Multiplication ===
see page [[multiplication]].

Before we can give a definition of multiplication, we must say a bit more about how numbers work.

In the identification of a quantity as a number (in the "numbering" of a quantity), a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.

[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later. OK it's later now. Ratios ARE numbers. And ratios actually do come with units. Like I am driving at 30 MPH, that's expressing a ratio but it has units. Even if I take something with the same units, like the ratio of two measurements of length, maybe I shouldn't omit that fact from the notation. Like maybe we should write 3.14 m/m instead of just 3.14. The former retains some information about where it came from, the latter doesn't; it's just a numeral and that's it.]

'''Multiplication''' (of A and B) is an algorithm which produces the number identifying a quantity in units of ''v'', where B identifies the number of ''v''s making up one ''u'', and ''A'' is the quantity of ''u''s.

=== Division ===
[TODO]

== Is a fraction a number? ==
Yes.

To identify a quantity as <math>p/q</math> is to say that it consists of <math>p</math> pieces of quantity <math>1/q</math>. It is not fundamentally different from identifying a quantity as <math>p</math>; it's just that the unit is different. In the fraction, the unit is being regarded as a <math>q</math>th of another unit.

== Is a negative a number? ==
Yes.

We can consider a sequence that goes backwards; 1, 0, -1, -2, -3, and so forth. Then we can identify a quantity like -5 by putting it in correspondence with a position in that sequence.

== Is <math>\pi</math> a number? ==
No.

[[π]] is the [[ratio]] between the circumference and the diameter of a [[circle]]. Unlike standard mathematics, by "circle," OM means real, finite, ''physical'' circles, with thickness and height and bumps and other non-uniformities. Observe the following facts about OM's concept of π, which are not true of the standard concept:

# Since π is the result of a continuous measurement, it is not a specific value, but is rather a range of possible values. For example, one circle might have a circumference and diameter whose ratio is found to be 3.1 ± 0.1.
# π is not the same range for every circle. For one circle, we might measure π = 3.1 ± 0.1; for another we might measure π = 3.15 ± 0.02; for yet another, we might measure π = 3.14159 ± 0.00005.

OK I'm failing to differentiate between two things. On the one hand, there's the actual ratio of the actual circumference and diameter of the actual circle. On the other hand, there's the thing we measure....

To treat something as a circle is to say like "all these ways in which I might measure its circumference, they're the same."

== Is <math>\sqrt{2}</math> a number? ==
No, I don't think so.

First of all we must settle on a definition of <math>\sqrt{2}</math>. There are two potential definitions that I can think of

# (The usual definition) <math>\sqrt{2}</math> is the number that squares to 2.
# <math>\sqrt{2}</math> is the side length of a square of area 2.

If definition 2 is correct, then <math>\sqrt{2}</math> is not a number, for basically the same reason that π is not a number. It seems that <math>\sqrt{2}</math> defines a range of numbers (and one which, to boot, differs between concrete situations) rather than a specific number.

If definition 1 is correct, then what is the fixed sequence of symbols which we are using to identify <math>\sqrt{2}</math>? It's certainly not fractions. I don't think can be anything else, either.

== Is a number just "an identification of a quantity"? ==
No.

My sofa has some length. In saying that, I have identified a quantity, but I haven't given a number.

== Is an element of the cyclic group <math>\mathbb{Z}_n</math> a number? ==
Yes, I think so.

For example, an amount of time could be identified by saying something like: after that much time has passed, it will be 4:00.

We are identifying the amount of time by drawing a comparison between it and a "sequence" of fixed symbols appearing on a clock. This sequence is <math>\mathbb{Z}_{12}</math>, or <math>\mathbb{Z}_{12} \times \mathbb{Z}_{60}</math>, or <math>\mathbb{Z}_{12} \times \mathbb{Z}_{60} \times \mathbb{Z}_{60}</math>, depending on the clock.

== References ==

Multiplication

2025-03-12T21:53:44Z

Lfox: Created page with "'''Multiplication''' is [TODO]"

'''Multiplication''' is [TODO]

Main Page

2025-03-12T21:53:24Z

Lfox:

Welcome to the [[Objective Mathematics]] wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math [[concept]], and thinking about the concrete, [[perceptual]] data to which it ultimately refers. If a math concept is ''not'' reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects [[Platonism]]: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects [[Intuitionism]]: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects [[Formalism]] and [[Logicism]]: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is ''the science of measurement''. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is ''foundational'', meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an ''idealization'' of that [real-world thing]," or "this [collection of mathematical ideas] is a ''model'' of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it ''is'' a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects ''other than'' by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is <blockquote>an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand. </blockquote>I have the exact same opinion about standard mathematics.

== Recommended reading order ==
[TODO]

== All Pages ==
Math pages:
* [[Coordinates]]
* [[Derivative]]
* [[Sequences|Sequence]]
* [[Sets|Set]]
* [[Continuity]]
* [[Number]]
* [[Multiplication]]
* [[Natural numbers|Natural number]] (obsolete. See [[multitude]] instead)
* [[Multitude]]
* [[Magnitude]]
* [[Integers|Integer]]
* [[Fractions|Fraction]] (obsolete. See [[magnitude]] instead)
* [[Radicals|Radical]]
* [[Real number]]
* [[Imaginary numbers|Imaginary number]]
* [[Vector space]]
* [[Triangulations|Triangulation]]
* [[Function]]
* [[Polynomial]]
* [[Circle]]
* [[Line]]
* [[Point]]
* [[Curve]]
* [[Surface]]
* [[Solid]]
* [[π]]
* [[e]]
* [[Limit]]
* [[Nill]]
* [[Induction]]
* [[Ordering]]
* [[Symmetry]]
* [[Probability]]
* [[Group]]
Physics pages:

* [[Entropy]]
* [[Newton's laws]]
* [[Uncertainty]]
* [[Perturbation theory]]
* [[Velocity]]
* Notes on "[[On the Electrodynamics of Moving Bodies]]" by Albert Einstein
* Notes on "[[On the Law of Distribution of Energy in the Normal Spectrum]]" by Max Planck
* Notes on "[[On a Heuristic Viewpoint Concerning the Production and Transformation of Light]]" by Albert Einstein

Computer Science pages:

* [[Algorithm]]
* [[Type]]
* [[Information]]
* [[Computation]]
* [[P]]

Philosophy pages:

* [[Against models (essay)]]
* [[Existent]]
* [[Concept]]
* [[Entity]]
* [[Unit]]
* [[Identity]]
* [[Definition]]
* [[Platonism]]
* [[Intuitionism]]
* [[Formalism]]
* [[Induction]]
* [[Objective Mathematics]]
* [[Zeno's Paradox]]
* [[Possible|Counterfactuals]]
* [[Occam's razor]]
Essays for Harry Binswanger's Philosophy of Mathematics course:

* [[Limits (essay)]] [TODO this is old, replace]
* [[The Limits of Limits]]
Other essays:

* [[Coordinate invariance: a manifesto]]

== Notation ==
Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of [[Type Theory]]. It is easiest to demonstrate what is meant by this through examples:

* <math>q = \frac{2}{3}</math> is a fraction, and I denote this fact---this identification---by writing <math>q:\mathbb{Q}</math>.
* Any integer is a fraction, and I denote this fact by writing <math>\mathbb{Z} : \mathbb{Q}</math>.

== Donate ==
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== Contact ==
If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

== Legal ==
All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].