Talk:Function (mathematics)

Function is Overloaded in definition and meaning, this article should be a survey

The function as relation or mapping in the single and multiple-valued and in classical functions, continuous functions, smooth functions, and about the language of functions, and domains and ranges and images and codomains, is very overloaded. This article is picking a course of opinion which does not represent its wide and varied usage, function the term. Over time, as other aspects of mathematics solidified, it's "function" the term that is most loosely thrown about, then as with regards to relations that are admitted to various sub-fields, each claiming their own definition of function has those are each distinct and different ant not compatible. This article, which could be called "mappings" instead as that's largely what it defines, does not from the outset affect to describe the development of the definition over time, nor does it very well reflect the most usual sort of arithmetic definition with which most people are familiar, or as with regards to domain and range. Mathematics is not merely differential geometry, and the definition of function is among the very most general and general throughout. So, this article should largely start explaining that "function theory" is its own sort of world, and a history and survey of "this is what is called a function historically or in these various settings", then with regards to an opinion of "this is a function today and in the most usual setting", which it is largely arguable that this article does not reflect, instead expect.

It reminds one of "graph", "chart", and "plot", about diagrams of functions, drawing a function.

Functions are modern, and Cartesian thus including the multi-valued, and not just classical functions, smooth classical continuous functions that are single-valued, and not just differential geometry's functions with neither vertical nor horizontal tangent, "functions" in mathematics are very general, and sub-fields that restrict the definition for their own purpose are presumptious that their definition is implicit, where it is not.

This article is opinionated and needs context in itself why the definition of function is so broad that it's about its own sub-field of mathematics, in matters of relation.

This article needs a brief survey of the development of the term over time, and to point to the many different intended interpretations of the term.

This article needs a thorough introduction detailing the survey of the meaning of the term "function" over time as mathematics has grown, and, specifically not removing what has become its fuller definition, in the interest of su-fields that would restrict its meaning for their own purposes in notation, where instead they should declare their own regions of syntax, because general usage does not agree.

This article has problems and hides them. 97.113.179.80 (talk) 15:03, 19 May 2024 (UTC)

Are you volunteering to track down a pile of sources and write that draft? –jacobolus (t) 21:01, 19 May 2024 (UTC)

Keeping in mind the goal of simultaneously being readable to middle-schoolers and providing pointers to current research-level mathematics... —David Eppstein (talk) 21:03, 19 May 2024 (UTC)

Why Set Theory?

As noted somewhere above, the intent was to introduce something that is usable in schools mathematics courses. Ok, nice. But why not mention it explicitly? Why not say right away (I guess it was before) that this definition is specific for set theories.

Or we could add a section dedicated to the history of the term an the notion. Leibnitz, Newton, Cauchy had no clue about functions being their graphs ("pairs of values").

To me, it's a shame to promote just one specific view of things, the school-level mathematics. It's so XX century, the century were everything was "defined" as sets. These days mathematicians must be familiar with model theory, and see clearly that functions as "sets of pairs of sets" is just a model (in sets).

Vlad Patryshev (talk) 00:38, 17 June 2024 (UTC)

I agree. The "Formal definition" section should be expanded to note that the graph definition is one possible formalization. We should concisely explain alternatives (e.g. other constructions, or an axiomatic approach) for which we can find citations. But keep in mind WP:UNDUE as the graph definition probably remains the most common or conventional one.

In fact, the existing text under "Formal definition" should be cut down. As written, it's not terrible, but it repeats essentially the same definition twice. Invoking the concept of a relation is unnecessary to explain the concept of the graph of a function, and its verbosity only distracts. I think it would be better to state just the graph definition directly, with a concise note that "This graph $R$ can also be viewed as defining a binary relation between the domain $X$ and codomain $Y$".

It might also be worth connecting the graph-based definition to the "Multi-valued functions" section further down, as the graph definition is more natural for that generalization than for the basic concept of a function. 73.223.72.200 (talk) 02:25, 28 July 2024 (UTC)

In fact, we already have an article covering the history: History of the function concept. So perhaps we only need to explain the math of a few selected alternative formalizations without regard to their history. We can then link to the history article in lieu of putting in detailed history in this overview article. 73.223.72.200 (talk) 02:38, 28 July 2024 (UTC)

The text could be improved by removing the two conditions in the middle of the text and only using a worded definition followed by the set-builder form. Roryyarr (talk) 12:14, 28 September 2024 (UTC)

Please, be more accurate: where the text should be modified and which conditions should be removed" D.Lazard (talk) 14:05, 28 September 2024 (UTC)

Paragraph 3 of the formal definition subsection is redundant as paragraph 5 presents the same information. Below is a suggested edit.

=== Formal definition ===

File:Injection keine Injektion 2a.svg

Diagram of a function

File:Injection keine Injektion 1.svg

Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair. Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair therein

The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler. However, it cannot be formalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory. This set-theoretic definition is based on the fact that a function establishes a relation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets Template:Math and Template:Math is a subset of the set of all ordered pairs $(x,y)$ such that $x\in X$ and $y\in Y.$ The set of all these pairs is called the Cartesian product of Template:Math and Template:Math and denoted $X\times Y.$ Thus, the above definition may be formalized as follows.

A function is formed by three sets, the domain $X,$ the codomain $Y,$ and the graph $R$ that satisfy the three following conditions.

• $R\subseteq \{(x,y)\mid x\in X,y\in Y\}$
• $\mathrm{\forall }x\in X,\mathrm{\exists }y\in Y,(x,y)\in R$
• $(x,y)\in R\wedge (x,z)\in R⟹y=z$

Roryyarr (talk) 11:07, 29 September 2024 (UTC)

I have edited the article for clarifying that the definitions are the same. However the two versions must been kept, because most people do not like to read a big succession of formulas without prose, while other prefer to not refer to relation theory or need help for formalizing prose. D.Lazard (talk) 11:44, 29 September 2024 (UTC)

Thank you D.Lazard Roryyarr (talk) 14:09, 29 September 2024 (UTC)

The current definition remains unacceptable, logically and conceptually

About once a year I present corrections to this article and am disappointed to see that the indicated defects remain. Ample references showing how to do better are in the bibliography I added to the article a couple years ago, but now they appear hidden. Anyway, the current definition still states:

In mathematics, a function from a set Template:Mvar to a set Template:Mvar assigns to each element of Template:Mvar exactly one element of Template:Mvar. The set Template:Mvar is called the domain of the function and the set Template:Mvar is called the codomain of the function.

It is logically unacceptable because it makes both the domain and the codomain ill-defined. Indeed, let $f$ be a function obeying the above definition, claiming the domain to be $X$ and the codomain to be $Y$. Let now $X^{\prime }$ be a proper subset of $X$ and let $Y^{\prime }$ be a proper superset of $Y$. Then, by the above definition, the same functrion $f$ assigns to each element of $X^{\prime }$ exactly one element of $Y^{\prime }$, so its domain is $X^{\prime }$ and its codomain is $Y^{\prime }$, different from $X$ and $Y$ respectively.

The idea of making a codomain an extra attribute of a function is an unnecessary complication and also extremely limiting. One example: normally, the composition $g\circ f$ is defined for any functions $f$ and $g$; its domain is the set of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$, no further conditions. Requiring (as with most definitions using codomains) that the codomain of $f$ equals the domain of $g$ makes composition unusable even for very basic purposes such as the chain rule in calculus.

The only way to do fully justice to the universality of the function concept is simply defining a function as being fully specified by its domain and by a unique value for every domain element, nothing more. Any basic article about functions that makes things more complicated detracts from its value to the readers. The simpler defination does not prevent defining a labeleled function as a different concept that indeed has a codomain, but that should be presented only later as a variant.

To conclude, any editors who insist on attaching a codomain to a function in the basic definition of a Wikipedia article should present very serious arguments why they think that is necessary or even useful. Boute (talk) 10:04, 8 June 2025 (UTC)

The argument for the current definition is simply that this is the one given in most reliable sources. This is the strongest possible argument since the role of an encyclopedy is not to provide new definitions. Moreover, if you remove the codomain from the definition you get "In mathematics, a function from a set X assigns to each element of X exactly one element." Most people would be confused and would ask the question: what is the nature of assigned elements? The answer is: there are taken from some set Y. So, we come back to the current definition. D.Lazard (talk) 12:23, 8 June 2025 (UTC)

The current definition is not given in any reliable source, certainly not in Halmos as claimed. Whether or not you are a codomain-fan, the current definition is logically incorrect, as I have proven. No one is proposing any new definitions, there are plenty of acceptable ones around. Cutting the definition in half to remove the codomain would not solve the logical defect. Boute (talk) 16:08, 13 June 2025 (UTC)

For context, previous discussions: Archive 6 § Codomain and mathematics, Archive 9 § Suggestions ..., Archive 10 § Slowly getting there ... and § A Modest Proposal, Archive 11 § Letting logic prevail ..., Archive 12 § Correcting shortcomings ..., Archive 14 § Towards a coherent article ..., Archive 15 § Total reset needed .... Maybe you can summarize the arguments made by other editors at those discussions and what you think has changed in the intervening time. You may also want to ping past participants who may be interested in discussion. From a skim it looks like in past discussions people did some careful literature surveying and found that your proposed version was not as well supported by sources. I might be mischaracterizing though; I only did the briefest skim. –jacobolus (t) 17:03, 8 June 2025 (UTC)

I am doubtful that covering all that old ground would be helpful. It is a matter of logic, not a political campaign. Did you not find the package of 20-odd sources I provided a couple of years ago? Boute (talk) 16:16, 13 June 2025 (UTC)

Template:Ping As far as I remember I followed your last-year discussion. As of today, you would have a point if we would define just Template:Tq. However, (again: if I remember right) as a result of your discussion, we now define Template:Tq. Imo, this is the best compromise between the advocates of labeled functions and those of unlabeled functions. I wouldn't object against adding a section near the end that discusses these variants. - Jochen Burghardt (talk) 17:36, 8 June 2025 (UTC)

My criticism about the current definition was certainly about "function from X to Y" (the definiendum), not just "function". In fact, my proof shows that, by the current definition, a "function from X to Y" is also a function from X' to Y' for any subset X' of X and any superset Y' of Y. That would cause no logical problem, but would result in a very uncommon definition. Before talking about repairs, and the question of whether we should make a codomain an attribute, it should be clear to everyone why the current definition is logically flawed Boute (talk) 16:37, 13 June 2025 (UTC)

It's difficult to tell exactly what your objection is. The opening sentence does implicitly require that X and Y are part of the data that describe a particular function. It's maybe not immediately obvious from the way it's worded, but it's made so later on when a more formal definition is given. Composition of functions with dissimilar (co)domains is often just a minor notational abuse to avoid having to explicitly write out when codomains are extended, etc. So what's the problem? 35.139.154.158 (talk) 17:05, 13 June 2025 (UTC)

There are many problems, compounded by collapsing 3 definitions into one: (i) function from X to Y, (ii) domain of that function, (iii) codomain of that function. Let us disentangle.

Part (i) is implicitly quantified by "for any sets X and Y", as per common style conventions in mathematics. The opening phrase states the definiendum. In the full definition part (i) is acceptable logically, but does not lead to a very common definition (although it is nearly equivalent to Def. 2.1 in Bartle's areal analysis text). Part (ii) and (iii) are ill-defined since, by part (i), the very same function is also a function from X' to Y' (witness sets introduced in the proof). This ill-definedness is a fatal logical flaw.

As for composition of functions with a codomain f / domain g mismatch, proper definitional design means not requiring notational abuse or patching. Any necessity for abuse ("minor" being in the eye of the beholder) reveals poor design, unworthy of the elegance of mathematics. Not attaching a codomain is issue-free. Composition is just one example of the numerous issues with codomains.

A personal aside: since around 1976 I have been waiting in vain for a single well-founded argument in favor of attaching a codomain to a function. Answers have always been either based on a misunderstanding or evasive, full of red herrings etc. Here is an opportunity for giving the first sound argument ad rem!

However, since so many editors insist on codomains for unspecified reasons (fetishism?) instead of offering raders the simpler and more universal variant, they should at least ensure a logically flawless definition of the codomain variant, despite its unnecessary complications. Such flawless definitions are found in most category theory textbooks and many set theory textbooks (e.g., Bourbaki, Dasgupta): the triplet formulation with $⟨f,X,Y⟩$. This "works" because the definiendum cannot make X and Y attributes of the function; that must be done in the definiens. Boute (talk) 09:23, 14 June 2025 (UTC)

If I unsterdand well your argument, you are arguing that the identity function on the real numbers is the "very same function" as the inclusion of the positive real numbers into the real numbers, and that the identity function of the positive real numbers is the "very same function" as the inclusion of the positive real numbers into the real numbers.

Also it seems that you misunderstand the given definition by misplacing the implicit quantifiers. The definition must be interpreted as a "Template:Tmath, a function from Template:Tmath to Template:Tmath is ...". In other words, we have a parametric definition, and there is nothing flawed here. D.Lazard (talk) 13:03, 14 June 2025 (UTC)

It seems you did not understand my argument at all, and leave it up to you to find the fallacy in obtaining your conclusion about the identity function.

Also, I did not misplace any quantifiers, in my text I placed them at the same spot as you did, and also observed that there is nothing wrong until that point, just an unusual definition. The flaw is in part (ii) and (iii), clearly and undeniably. Please try to read more carefully what people write and avoid red herrings that have plagued this article from the start. Boute (talk) 19:22, 14 June 2025 (UTC)

I regret my impatience when writing the preceding remark. It is just frustrating to see that the single most important concept in mathematics (the function concept) cannot find proper coverage in this article. Before signing off until 2026, I need to clarify that, by "unusual definition", I do not mean the formulation of that definition, but the concept of function covered by that definition. In other words, although "part (i)" contains no logical contradiction, it does not cover the common concept of function. 213.132.130.11 (talk) 06:20, 15 June 2025 (UTC)