Concept

[TODO: this page is under construction]

A concept is "a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted."

Objective Mathematics' view of concepts is heavily influenced by the book Introduction to Objectivist Epistemology (ITOE), but with a few minor additions. The purpose of this article is to summarize the parts of ITOE most relevant to math (though by no means should it be counted as a replacement of ITOE), and to explain Objective Mathematics' additions to ITOE in detail.

A concept is a mental unit. A concept refers to an unlimited number of concretes.

With the exception of proper names, there is basically a one-to-one correspondence between the concepts that a man uses and the words that he uses. There are some exceptions to that rule. For example, in my mind, I think that "dimensional analysis" is treated as a single concept, rather than as a type of analysis which is about dimensions (though the name is apt). Conversely, in my mind the different words "squadron" and "battalion" denote the exact same concept (though I am sure that they denote different concepts for people with more knowledge about militaries).

In German, the one-to-one correspondence between concepts and words is closer to being true, because they have more compound words than English (e.g "dimensional analysis" is "Dimensionsanalyse"). In Chinese, the one-to-one correspondence between concepts and characters is far from being true, because often a single Chinese character is ambiguous until combined with other characters. This fact has been recognized by Chinese, which has a word ("词语") that seems to effectively mean: a grouping of one or more characters that stands for a single concept. "词语" gets translated into English as "word."

Maybe I would say something like... the English language pushes you to form concepts in a certain way, because of what concepts it has words for. [TODO not sure]

Most pages on the Objective Mathematics wiki are concepts.

Unit

A unit is "an existent regarded as a separate member of a [set] of two or more similar members."

Examples

A shirt in a closet, regarded as one of the many shirts in a closet, is a unit.

The first foot of an extension cable, regarded as part of the whole cable, is a unit.

One gram of water, regarded as one of the many grams of water inside of a bathtub, is a unit.

One second, regarded as part of an ongoing period of time, is a unit.

Anything regarded as an example of a concept, or as a concrete instance of a concept, is in that capacity a unit.

Non-examples

Planet Earth, regarded on its own (and not as part of a set of other planets, celestial bodies, etc.) is not a unit.

A square meter of a 1.1 square meter table, considered as a part of the table, is not a unit. This is because it is not being considered as part of a set containing other square meters of table. By contrast, some 0.1 square meter piece of a 1.1 square meter table, is a unit.

Units in science

Anyone with basic scientific knowledge knows that scientists have a concept of "units" (sometimes called "dimensions") as well. For example, we say that the mathematical expression "3 meters" has units of meters, and that π is a unitless quantity. Science's concept of units is pretty much the exact same as the concept of unit described above.

Measurement

Measurement is "the identification of a quantitative relationship, by means of a standard that serves as a unit."

Some measurements are implicit, and some measurements are explicit. Implicit measurements are measurements made using consciousness only, with no external tools. Explicit measurements are measurements made using consciousness plus additional tools. Implicit measurements are extremely commonplace: whenever one makes a valid conceptual identification (e.g. "That is a book."), one has made an implicit measurement. Explicit measurements are common in modern civilization, but they are significantly less common than implicit ones: whenever one uses a measurement tool---like a measuring tape or a Michelson interferometer---which goes beyond perceptually obvious similarities, one has made an explicit measurement.

Examples

To conclude that some table is 3 meters long, is to identify a quantitative relationship, between the length of that table and the unit of 1 meter.

To find that one has received 97 nanosieverts of radiation, is to identify a quantitative relationship, between the radiation one received and the unit of 1 nanosievert.

Concept formation

Concepts are formed by a process of measurement omission.

Definition

A definition is "a statement that identifies the nature of a concept’s units."

The purpose of a definition is [...]

Rules of concept formation imply [...]

Any definition provides a genus, and some differentia. The genus tells us in what broad category of our mind we should look to find the concept. The differentia tell us how to distinguish the concept from the other concepts in that broad category.

Examples

For examples of definitions, look at the first sentence of most pages and sections on the Objective Mathematics wiki. If the first sentence contains a bolded word, I am probably defining that word.

Standard math's idea of definitions

Standard mathematics has a very different view of definitions, which is expressed in the following definition from Russel and Whitehead

A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known.

On this view, when one wants to introduce a new concept, it is necessary and sufficient to provide a definition of that concept. A concept means exactly what it was defined to mean, and nothing else. It follows that any non-basic concept is in some sense superfluous. Indeed, a couple paragraphs later Russel and Whitehead write

Thus [...] definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous.

Note also that circular definitions would be extremely problematic on this view. Upon encountering a new circularly-defined concept, the reader would encounter an infinite regress if he tried to understand what it meant. (Circular definitions are not, however, necessarily problematic according to Objective Mathematics.)

The view expressed above is held in some form by almost every mathematician, but like all bad ideas, it is not held consistently. In practice, mathematicians know very well that seeing a definition is not enough for a human mind to grasp a concept. In a math book, definitions are often supplemented with many "examples", diagrams, pictures, or "intuitive explanations." There is, however, a common view that these extra things, beyond the definitions, aren't properly a part of mathematics, and that they are more like a crutch that the human mind needs to use. Some mathematicians, epitomized by the Bourbaki group, even think that an ideal math text should throw away the "crutch" altogether.

For an example of a standard mathematics definition, consider its definition of a group:

Definition. A group $(G,\mu ,i,e)$ is a set $G$, a map $\mu :G\times G\rightarrow G$, a bijection $i:G\rightarrow G$, and an element $e\in G$, satisfying the following axioms

1. Associativity: $\mu (x,\mu (y,z))=\mu (\mu (x,y),z)$ for all $x,y,z\in G$.
2. Identity: $\mu (e,x)=x$ for all $x\in G$.
3. Inverse: $\mu (x,i(x))=e=\mu (i(x),x)$ for all $x\in G$.

In standard math, a group is nothing less and nothing more than what these definitions say it is. A group has exactly those properties which follow deductively from its definition, and no other properties. In fact, any definition of any term is considered to be all there is to the term.

In contrast, the properties of objects in real life are identified rather than deduced from axioms. Consequently, there is always the possibility that a real-life object has properties which have not yet been identified---that some totally novel property of the object could be discovered. However, anything that is discovered about (the standard math notion of) a group, though it may have been previously unknown, is not novel in the same way: it is necessarily just a deductive consequence of the group axioms.

For the concepts of standard mathematics, there are no borderline cases or grey areas. Given some mathematical structure (in standard mathematics), either it is a group, or it isn't a group. In real life, though, there are borderline cases. There are objects whose measurements don't clearly fall inside the range of a given concept, but also don't clearly fall outside the range of that concept. For example, is a feminine hermaphrodite a woman? I imagine that there is not always a clear answer to that question.

AR describes what is wrong with this idea of definitions:[TODO this seems out of place]

A definition is not a description; it implies, but does not mention all the characteristics of a concept’s units. If a definition were to list all the characteristics, it would defeat its own purpose: it would provide an indiscriminate, undifferentiated and, in effect, pre-conceptual conglomeration of characteristics which would not serve to distinguish the units from all other existents, nor the concept from all other concepts.

Specification [TODO]

A specification of a concept is a description of the measurements which were omitted in forming it. A specification can have differing levels of precision, depending on the context. A specification is closer to standard mathematics' idea of definitions than is a concept-definition. Below I will describe some contexts, mathematical and otherwise, where specifications are needed.

An example is that if one is programming computer, then one has to specify exactly what all one's "concepts" (i.e. classes or types) mean. Computers are dumb, so there is no room to be vague, or to omit non-essential properties. To define a class in, say, C++, one must precisely specify all of its attributes, down to the most trivial details.

Another example of a context in which one needs a "definition" which includes more details than what the concept-definition requires (but fewer details than what a computer program requires) is: legal definitions. To identify the nature of the units of the concept "man," it suffices to say that man is a rational animal. But to legally define "man," one has to specify many more things, like at what---if any---level of brain damage (and by what standard of brain damage) shall someone cease to be counted as a man, at which precise time shall a fetus be counted as having begun its life as a man, etc.

Lastly, and most importantly for Objective Mathematics, there are many situations where precision seems to require us to specify more than just those characteristics of a concept's units which are essential. For an example of such a situation, consider the concept of a prime number. The definition of prime number is: a natural number which cannot be factored nontrivially into natural numbers. That definition states the essential properties of primes. However, from that definition, it is not clear how one should deal with the borderline case of 1: does $1=1\times 1$ count as a "nontrivial" factorization or not? In some antiquated definitions of primes, 1 is counted as a prime, and indeed, I think that there is not an essential reason why it is wrong to count 1 as a prime. It is, however, generally more convenient to count 1 as a non-prime. For example, if 1 were to be counted as a prime, then the fundamental theorem of arithmetic

Fundamental Theorem of Arithmetic. Any natural number admits a unique decomposition into prime factors.

would be more complicated to state, for the decomposition into primes would not be not unique. If one wants to use a theorem like the above, then one's concept of prime must agree with that of the person who stated the theorem, even down to some minute and trivial details like whether 1 is a prime. So we see that beyond just the "definition" of a prime, some extra specification of details can be useful in math.[TODO I am skeptical of this example. Use continuity or something.]

Notion

A notion is a compound concept. A notion is like a concept, in the sense that it subsumes and includes an unlimited number of concretes. A notion is unlike a concept, in the sense that it might be made of other concepts.

[TODO maybe I should rename this to "phrase"....]

Examples:

• 3 is a concept, but 16654 is a notion. In general, any number is a notion.
• "Fox" is a concept, but "the quick brown fox" is a notion. In general, any noun phrase is a notion.
• "To jump" is a concept, but "to jump off the gangplank with one's hands tied behind one's back" is a notion.

If a concept does not refer to any concretes, then it was formed invalidly. By contrast, one can use valid concepts to validly form a notion that does not refer to any concretes in reality. For example, "the talking purple platypus" does not refer to anything in reality, but it nonetheless makes sense, and "the," "talking," "purple," and "platypus" are all valid concepts. For another example, $9^{9^{9^{9^{9^{9^{9^{9}}}}}}}$is too large to refer to anything in reality, but it is clear what it means, and 9 and exponentiation are both valid concepts.

I reserve the right to be sloppy about the distinction between notion and concept, in situations where I think it doesn't matter.

Proposition

A proposition is a basic identification of a fact of reality. It is "the form in which we make conceptual identifications."