Function - Revision history

Lfox at 21:20, 12 March 2025

2025-03-12T21:20:49Z

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	~~[TODO] it~~'~~s rationalistic to conceptualize functions as relationships. That~~'~~s not how people think about them. They think about functions as "things you could do~~."		A '''function''' <math>f : A \rightarrow B</math> is a potential action, which, if performed on an existent <math>a</math> of type <math>A</math>, would uniquely produce an existent <math>f(a)</math> of type <math>B</math>.

	A '''function''' is a quantitative relation between two groups<ref group="note">In this context, by group I mean what AR means by group in ITOE. I do ''not'' mean [[Group\|the standard math notion of group]]. </ref> of existents (called the "'''domain'''" and "'''codomain'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain. The notation <math>f : A \rightarrow B</math> means that <math>f</math> is a function with domain <math>A</math> and codomain <math>B</math>.		A '''function''' is a quantitative relation between two groups<ref group="note">In this context, by group I mean what AR means by group in ITOE. I do ''not'' mean [[Group\|the standard math notion of group]]. </ref> of existents (called the "'''domain'''" and "'''codomain'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain. The notation <math>f : A \rightarrow B</math> means that <math>f</math> is a function with domain <math>A</math> and codomain <math>B</math>. [TODO it's rationalistic to conceptualize functions as relationships. That's not how people think about them. They think about functions as "things you could do."]

	A '''function''' is a binary quantitative relation between two groups[TODO see 2024-07-05 comment] of existents, in which one (the "'''codomain'''") is uniquely determined by the other (the "'''domain'''"). [TODO this definition is not ''bad'', but it is a bit ''vague''. what do I mean by determined? I mean something like, can be determined.]		A '''function''' is a binary quantitative relation between two groups[TODO see 2024-07-05 comment] of existents, in which one (the "'''codomain'''") is uniquely determined by the other (the "'''domain'''"). [TODO this definition is not ''bad'', but it is a bit ''vague''. what do I mean by determined? I mean something like, can be determined.]
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	The method that an elementary schooler learns for carrying out addition (carry the 1s and so forth) is not itself a function. It is an [[algorithm]], which implements the addition function <math>+:\mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N}</math>.		The method that an elementary schooler learns for carrying out addition (carry the 1s and so forth) is not itself a function. It is an [[algorithm]], which implements the addition function <math>+:\mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N}</math>.

			Consider the following potential action. For any natural number <math>x : \mathbb{N}</math>, I can produce (using a calculator) a decimal <math>q : \mathbb{Q}</math> such that <math>q^2 \approx x</math>. This is not a function, because its result is not determined uniquely by the input.

	== The traditional concept ==		== The traditional concept ==

Lfox at 00:14, 13 February 2025

2025-02-13T00:14:41Z

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			[TODO] it's rationalistic to conceptualize functions as relationships. That's not how people think about them. They think about functions as "things you could do."

	A '''function''' is a quantitative relation between two groups<ref group="note">In this context, by group I mean what AR means by group in ITOE. I do ''not'' mean [[Group\|the standard math notion of group]]. </ref> of existents (called the "'''domain'''" and "'''codomain'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain. The notation <math>f : A \rightarrow B</math> means that <math>f</math> is a function with domain <math>A</math> and codomain <math>B</math>.		A '''function''' is a quantitative relation between two groups<ref group="note">In this context, by group I mean what AR means by group in ITOE. I do ''not'' mean [[Group\|the standard math notion of group]]. </ref> of existents (called the "'''domain'''" and "'''codomain'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain. The notation <math>f : A \rightarrow B</math> means that <math>f</math> is a function with domain <math>A</math> and codomain <math>B</math>.

Lfox at 19:26, 29 October 2024

2024-10-29T19:26:58Z

← Older revision		Revision as of 19:26, 29 October 2024
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	[TODO maybe the "problem" is just that standard mathematics is precise about functions, and I'm not.]		[TODO maybe the "problem" is just that standard mathematics is precise about functions, and I'm not.]

			== Functions are necessarily unitless ==
			consider the function describing the relationship between the side length of a square, and the area of a square:

			f(a meters) = a^2 meters^2.

			consider the function describing the relationship between an angle of a circle and the x location of the point at that angle on the circle of radius 1 meter

			f(a radians) = cos(a) meters

	== Notes ==		== Notes ==

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

Lfox at 20:44, 5 July 2024

2024-07-05T20:44:25Z

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	A '''function''' is a ~~binary~~ quantitative relation between two groups of existents (called the "'''~~codomain~~'''" and "'''~~domain~~'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain.		A '''function''' is a quantitative relation between two groups<ref group="note">In this context, by group I mean what AR means by group in ITOE. I do ''not'' mean [[Group\|the standard math notion of group]]. </ref> of existents (called the "'''domain'''" and "'''codomain'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain. The notation <math>f : A \rightarrow B</math> means that <math>f</math> is a function with domain <math>A</math> and codomain <math>B</math>.

	A '''function''' is a binary quantitative relation between two groups[TODO see 2024-07-05 comment] of existents, in which one (the "'''codomain'''") is uniquely determined by the other (the "'''domain'''"). [TODO this definition is not ''bad'', but it is a bit ''vague''. what do I mean by determined? I mean something like, can be determined.]		A '''function''' is a binary quantitative relation between two groups[TODO see 2024-07-05 comment] of existents, in which one (the "'''codomain'''") is uniquely determined by the other (the "'''domain'''"). [TODO this definition is not ''bad'', but it is a bit ''vague''. what do I mean by determined? I mean something like, can be determined.]

Lfox at 20:36, 5 July 2024

2024-07-05T20:36:17Z

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	A '''function''' is a ~~potential process, which, if enacted, would convert units~~ of ~~one mathematical concept to units of another. [TODO I~~'~~m not sure about this definition. Like normally I would be happy to say that addition is a function. But addition is not about converting units of one math concept to units of another, it~~'~~s about~~ ''~~identifying~~'' ~~one thing based on one~~'s ''~~identification~~'' of ~~another thing~~.]		A '''function''' is a binary quantitative relation between two groups of existents (called the "'''codomain'''" and "'''domain'''" of the function), in which any existent in the domain is related to a ''unique'' existent in the codomain.

	The notation <math>f : A \rightarrow B</math> is shorthand for the following statement: <math>A</math> and <math>B</math> are two mathematical concepts, <math>f</math> is a function converting an <math>A</math> into a <math>B</math>. We say that <math>A</math> is the ''domain'' of <math>f</math>, and <math>B</math> is the ''codomain'' of <math>f</math>.		A '''function''' is a binary quantitative relation between two groups[TODO see 2024-07-05 comment] of existents, in which one (the "'''codomain'''") is uniquely determined by the other (the "'''domain'''"). [TODO this definition is not ''bad'', but it is a bit ''vague''. what do I mean by determined? I mean something like, can be determined.]

			[TODO the following is deprecated but I don't want to delete it yet] A '''function''' is a potential process, which, if enacted, would convert units of one mathematical concept to units of another. [TODO I'm not sure about this definition. Like normally I would be happy to say that addition is a function. But addition is not about converting units of one math concept to units of another, it's about ''identifying'' one thing based on one's ''identification'' of another thing. The notation <math>f : A \rightarrow B</math> is shorthand for the following statement: <math>A</math> and <math>B</math> are two mathematical concepts, <math>f</math> is a function converting an <math>A</math> into a <math>B</math>. We say that <math>A</math> is the ''domain'' of <math>f</math>, and <math>B</math> is the ''codomain'' of <math>f</math>.

	== Examples ==		== Examples ==

Lfox: /* Functions */

2024-07-02T01:55:51Z

Functions

← Older revision		Revision as of 01:55, 2 July 2024
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	A [[Graph (function)\|graph]] can describe a function: it can be thought of as providing instructions for producing outputs from inputs, or it can be thought of as describing some function which was implemented elsewhere.		A [[Graph (function)\|graph]] can describe a function: it can be thought of as providing instructions for producing outputs from inputs, or it can be thought of as describing some function which was implemented elsewhere.

			Let's suppose I have a process where I take a line segment, and I use a pencil to copy it onto transparent paper twice, resulting in a line segment of twice the length of the original. This is an example of the function <math>x \mapsto 2x</math>; it is ''not'' an example of the function <math>x \mapsto 2.00001 x</math>, even though those functions only differ at a scale much smaller than the one relevant to my pencil drawings.

	=== Non-functions ===		=== Non-functions ===

Lfox at 18:07, 23 June 2024

2024-06-23T18:07:38Z

← Older revision		Revision as of 18:07, 23 June 2024
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	A '''function''' is a potential process, which, if enacted, would convert units of one mathematical concept to units of another.		A '''function''' is a potential process, which, if enacted, would convert units of one mathematical concept to units of another. [TODO I'm not sure about this definition. Like normally I would be happy to say that addition is a function. But addition is not about converting units of one math concept to units of another, it's about ''identifying'' one thing based on one's ''identification'' of another thing.]

	The notation <math>f : A \rightarrow B</math> is shorthand for the following statement: <math>A</math> and <math>B</math> are two mathematical concepts, <math>f</math> is a function converting an <math>A</math> into a <math>B</math>. We say that <math>A</math> is the ''domain'' of <math>f</math>, and <math>B</math> is the ''codomain'' of <math>f</math>.		The notation <math>f : A \rightarrow B</math> is shorthand for the following statement: <math>A</math> and <math>B</math> are two mathematical concepts, <math>f</math> is a function converting an <math>A</math> into a <math>B</math>. We say that <math>A</math> is the ''domain'' of <math>f</math>, and <math>B</math> is the ''codomain'' of <math>f</math>.

Lfox: /* Functions */

2024-04-21T01:27:00Z

Functions

← Older revision		Revision as of 01:27, 21 April 2024
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	A table describing several [[trigonometric functions]] is shown in figure (2).		A table describing several [[trigonometric functions]] is shown in figure (2).

	A [[Graph (function)\|graph]] can describe a function: it ~~provides~~ instructions for producing outputs from inputs.		A [[Graph (function)\|graph]] can describe a function: it can be thought of as providing instructions for producing outputs from inputs, or it can be thought of as describing some function which was implemented elsewhere.

	~~The method that an elementary schooler learns for carrying out addition (carry the 1s and so forth) is a~~ function.

	=== Non-functions ===		=== Non-functions ===
	A coffee machine is not a function. It carries out a process; it takes some inputs (energy, water, coffee grounds, paper filter) and converts them into coffee. But the inputs and outputs are not mathematical notions.		A coffee machine is not a function. It carries out a process; it takes some inputs (energy, water, coffee grounds, paper filter) and converts them into coffee. But the inputs and outputs are not mathematical notions.

			The method that an elementary schooler learns for carrying out addition (carry the 1s and so forth) is not itself a function. It is an [[algorithm]], which implements the addition function <math>+:\mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N}</math>.

	== The traditional concept ==		== The traditional concept ==

Lfox at 01:23, 21 April 2024

2024-04-21T01:23:13Z

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	A '''function''' is a ''process~~'' that converts~~ units of one mathematical ~~[[Concept#Notion\|notion]] into~~ units of another.		A '''function''' is a potential process, which, if enacted, would convert units of one mathematical concept to units of another.

	The notation <math>f : A \rightarrow B</math> is shorthand for the following statement: <math>A</math> and <math>B</math> are ~~some~~ mathematical ~~notions~~, <math>f</math> is a function converting an <math>A</math> into a <math>B</math>.		The notation <math>f : A \rightarrow B</math> is shorthand for the following statement: <math>A</math> and <math>B</math> are two mathematical concepts, <math>f</math> is a function converting an <math>A</math> into a <math>B</math>. We say that <math>A</math> is the ''domain'' of <math>f</math>, and <math>B</math> is the ''codomain'' of <math>f</math>.

	== Examples ==		== Examples ==
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	\|0		\|0
	\|}		\|}
	That table describes the input and output of an XOR gate. A XOR gate (another example of something that performs a function) can be seen in figure (1).		That table describes the input and output of an XOR gate. A XOR gate (another example of something that performs a function) can be seen in figure (1).

	The following lines of C++ code ~~perform~~ a function:		The following lines of C++ code implement a function:
	[[File:Bernegger Manuale 136.jpg\|thumb\|Figure (2): A 1619 book of mathematical tables by Matthias Bernegger, showing some of the sine, tangent, and secant functions. ]]		[[File:Bernegger Manuale 136.jpg\|thumb\|Figure (2): A 1619 book of mathematical tables by Matthias Bernegger, showing some of the sine, tangent, and secant functions. ]]
	string reverseString(const string &s) {		string reverseString(const string &s) {
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	return returnStr;		return returnStr;
	}		}
	A [[Sequences\|sequence]] is a function.		A [[Sequences\|sequence]] of fractions is a function <math>\mathbb{N}\rightarrow\mathbb{Q}</math>.

	A table describing several [[trigonometric functions]] is shown in figure (2).		A table describing several [[trigonometric functions]] is shown in figure (2).
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	A [[Graph (function)\|graph]] can describe a function: it provides instructions for producing outputs from inputs.		A [[Graph (function)\|graph]] can describe a function: it provides instructions for producing outputs from inputs.

	The method that an elementary schooler learns for carrying out ~~long division~~ is a function.		The method that an elementary schooler learns for carrying out addition (carry the 1s and so forth) is a function.

	=== Non-functions ===		=== Non-functions ===

Lfox: /* The traditional concept */

2024-02-03T20:39:10Z

The traditional concept

← Older revision		Revision as of 20:39, 3 February 2024
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	In standard mathematics, a function always converts elements of a set <math>X</math> into elements of a set <math>Y</math>. The sets could be [[Sets#Infinite sets\|infinite]], so the function doesn't have to describe a real process. In fact, though standard mathematicians usually think about functions ''as if'' they were processes, standard mathematics technically doesn't define functions as processes at all. Rather, it defines a function from <math>X</math> to <math>Y</math> as a subset of <math>X \times Y</math> satisfying certain conditions.<ref group="note">Namely, for each <math>x \in X</math>, the subset of <math>X \times Y</math> must contain one and only one tuple whose first argument is <math>x</math>.</ref> This sharp disconnect between formal mathematical jargon, and the intuition applicable to real life, is part of what Objective Mathematics seeks to avoid. [TODO rewrite last sentence]		In standard mathematics, a function always converts elements of a set <math>X</math> into elements of a set <math>Y</math>. The sets could be [[Sets#Infinite sets\|infinite]], so the function doesn't have to describe a real process. In fact, though standard mathematicians usually think about functions ''as if'' they were processes, standard mathematics technically doesn't define functions as processes at all. Rather, it defines a function from <math>X</math> to <math>Y</math> as a subset of <math>X \times Y</math> satisfying certain conditions.<ref group="note">Namely, for each <math>x \in X</math>, the subset of <math>X \times Y</math> must contain one and only one tuple whose first argument is <math>x</math>.</ref> This sharp disconnect between formal mathematical jargon, and the intuition applicable to real life, is part of what Objective Mathematics seeks to avoid. [TODO rewrite last sentence]

	Standard mathematicians usually prefer not to consider things like "the set of all strings containing less than ''n'' characters," where ''n'' is the number of bytes of free RAM available on some computer. Instead, standard mathematicians usually prefer to consider things like "the set of all strings." The reason for this is clear enough: the latter is much more elegant and simple than the former. The problem with the latter, though, is that it is completely disconnected from reality. There is no such thing as a function from the set of all strings to the set of all strings, since everything that exists is delimited and finite. For example, in the [[Function#Examples\|examples section §]], Objective Mathematics made the identification that <math>r : \text{String} \rightarrow \text{String} </math>, where <math>r</math> is the <code>reverseString</code> function and <math>\text{String}</math> denotes the ''[[concept]]'' of a string,<ref group="note">C++, being a practical language, basically says the same thing as Objective Mathematics; it's just a bit more specific. It says that the [[Type theory\|type]] of <code>reverseString</code> is that of a (C++) function from <code>std::string</code> to <code>std::string</code>. </ref> but standard mathematics ''cannot'' identify <code>reverseString</code> as a function from "the set of all strings" to "the set of all strings", since there are limits on how big its input and output could be.		Standard mathematicians usually prefer not to consider things like "the set of all strings containing less than ''n'' characters," where ''n'' is the number of bytes of free RAM available on some computer. Instead, standard mathematicians usually prefer to consider things like "the set of all strings." The reason for this is clear enough: the latter is much more elegant and simple than the former. The problem with the latter, though, is that it is completely disconnected from reality. There is no such thing as a function from the set of all strings to the set of all strings, since everything that exists is delimited and finite. For example, in the [[Function#Examples\|examples section §]], Objective Mathematics made the identification that <math>r : \text{String} \rightarrow \text{String} </math>, where <math>r</math> is the <code>reverseString</code> function and <math>\text{String}</math> denotes the ''[[concept]]'' of a string,<ref group="note">C++, being a practical language, basically says the same thing as Objective Mathematics; it's just a bit more specific. It says that the [[Type theory\|type]] of <code>reverseString</code> is that of a (C++) function from <code>std::string</code> to <code>std::string</code>. </ref> but standard mathematics cannot identify <code>reverseString</code> as a function from "the set of all strings" to "the set of all strings", since there are limits on how big its input and output could be. In other words, there are some strings in "the set of all strings" that are so large that <code>reverseString</code> would not be able to reverse them.

	In conclusion, standard mathematicians are forced to make an impossible choice. Do we work with nice, clean mathematical objects and be disconnected from reality, or do we keep our connection with reality and suffer through nasty, non-conceptual messes?		In conclusion, standard mathematicians are forced to make an impossible choice. Do we work with nice, clean mathematical objects and be disconnected from reality, or do we keep our connection with reality and suffer through nasty, non-conceptual messes?

	[TODO I think this is a bad example, and I also am not sure if it belongs in this article.]		[TODO I think this is a bad example, and I also am not sure if it belongs in this article.]

			[TODO maybe the "problem" is just that standard mathematics is precise about functions, and I'm not.]

	== Notes ==		== Notes ==

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