Talk:Function (mathematics)

Definition issues

[Rescued 2 recent posts into the middle of an old discussion, by restoring them here:]

While many authors ostensibly use the "labeled" definition (in which the codomain $Y$ is part of the function), I have observed that they are often inconsistent in their use of it. Is the codomain of the sine function the reals, or just the interval [-1_+1]? In the "labeled" view, either $sin:\mathbb {R} \rightarrow \mathbb {R}$ or $sin:\mathbb {R} \rightarrow [-1\_+1]$, not both.
As for surjective, in the "plain" definition one defines "$f$ is surjective on a set $Y$"; which is more conveniently stated as "$\mathop {\mathrm {Img} } (f)=Y$, where $\mathop {\mathrm {Img} } (f)$ is the image of $f$ (which is uniquely defined by the set of pairs).--Jorge Stolfi (talk) 19:07, 15 November 2023 (UTC)

A mathematics article, more than any other, should start with a definition of its topic -- not merely with statements of some interesting properties. The definition of function as set of ordered pairs is the simplest that is mathematically precise, and is perfectly valid for all fields that use that concept -- even for those who would rather define it in some indirect or more abstract way. Yes, it requires the concept of "set" -- but it is impossible to write an encyclopedia article about "function" that would be mathematically correct and understandable by a reader who does not know what a "set" is. --Jorge Stolfi (talk) 19:07, 15 November 2023 (UTC)

[ Jochen Burghardt (talk) 21:10, 15 November 2023 (UTC) ]

As I wrote in more detail on my talk page: Stolfi's attempts to install one specific definition from advanced mathematics into the lead of an article whose topic is already the subject of middle-school mathematics (in the form of graphing real functions) is a violation of WP:TECHNICAL and a way to make our articles unreadable and offputting (see recent WT:WPM thread on accessibility of mathematics articles). It is also a violation of WP:NPOV in favoring one specific definition as the only definition when there are multiple definitions possible and in use. —David Eppstein (talk) 21:15, 15 November 2023 (UTC)

Having restored the posts doesn't mean I'd agree with Stolfi. Having read the discussion at User_talk:David_Eppstein#Function_(mathematics), I'd just like to add that the 1st lead sentence is not just some "interesting property", but a pretty precise informal definition. - Jochen Burghardt (talk) 21:30, 15 November 2023 (UTC)

Thanks Jurgen for recovering my posts. While I don't have the energy to fight any more, let me refer to my comment on David's Talk page about the supposed precision of the word "associate". And then add this: Wikipedia is meant to be an encyclopedia, that is, a reference book -- not a textbook, or popularization book. Making people like mathematics (or any other science) is not one of its goals. Yes, articles should be written so as to be useful to the widest possible readership, which means they must avoid unnecessary jargon and technical concepts -- but they should be written for those readers who come here to find out what a word like "function" precisely means. Hardly anyone will come to this article looking for an "informal definition": they already have that, thank you...
All the best, --Jorge Stolfi (talk) 16:11, 18 November 2023 (UTC)

@Jorge Stolfi I recently made a change in the formal definition (HERE, application and discussion here) which was rejected. The new definition (triplet) I proposed was introduced in such a way that the old definition (binary relation) was placed in the "reduced formal definition" section, where it is mentioned in the introduction (along with citing) that both definitions are commonly used. What do you think about this change - do you think it is an improvement to the article and could be reinstated? Kamil Kielczewski (talk) 20:54, 12 March 2024 (UTC)

Several years ago I tried to edit the definition in this article, but all I achieved into a three-page discussion in this Talk page with an editor who insisted that the word "set" should be avoided because it reminds readers of high-school math which drives them away. I gave up when a third editor joined the discussion, arguing that it should be defined using category theory.
Then I swore I would never again try to edit a math article here in Wikipedia.
I should have stood by that oath.
All the best, Jorge Stolfi (talk) 10:03, 18 March 2024 (UTC)

Speaking of which, I find it somewhat curious that there is no mention in this article of type theory and the way that functions are formalized in type theory. —David Eppstein (talk) 21:22, 15 November 2023 (UTC)

Total reset needed for this article

Outsiders watching the style of this discussion would probably think about a pond of frogs. When one person replies to another, three more jump in between; others exhibit adolescent behavior in explicitly refusing even to hear any technical arguments; still others wield Wikipedia rules instead of logic. All this is only to say that a total reset is urgently needed, and that restraint is needed to avoid derailing the entire project. I will set the example in exercising restraint, and add a brief note only when logic is all too severely violated. Otherwise I just observe the proceedings...

In any case, can we at least agree on the following points?

- The article should primarily serve the readers, not some personal hidden agenda.

- The current state of the art requires (at least) two conceptually different definitions.

- The simplest of these would be the best starting point.

- Any definition should be referenced accurately (immediately verifiable, no misquotes). The reason for insisting on this last point is not some bureaucratic consideration but the hope that in the literature of the past hundred years there is at least one formulation people are prepared to quote literally, even if it does not fit their own dream definition. Functions are not rocket science but high school material. Boute (talk) 05:14, 1 March 2024 (UTC)

You should try to make the same point without the insulting preface. Comparing people to frogs and calling them children is not an exemplary "exercise in restraint". –jacobolus (t) 15:47, 1 March 2024 (UTC)

No insult, but a general observation in empathy with potential outsiders, and apparently an effective wake-up call. Restraint here obviously means making only contributions that improve the paper. Look at the pages and pages wasted on an high-school level issue. What would outsiders think? Boute (talk) 19:41, 1 March 2024 (UTC)

I'm not clear what you're hoping for. I guess you want someone to answer the complaint politely one time and thereafter just let the critic talk to themselves / shout into the void without reply? –jacobolus (t) 20:27, 1 March 2024 (UTC)

Re "high-school level issue": a big part of the problem here is that this is not (only) a high school level issue. We need to describe functions at a level that can be understood by high school students, which is a high school level issue. But we also need to describe functions with the understanding gained from research into the foundations of mathematics, in set theory (well covered in this discussion), in type theory (not so well covered), and maybe also with some discussion of constructivism (currently not covered at all), at a level far above high school mathematics. The need to treat these things at multiple levels significantly adds to the difficulty of formulating an article that anyone interested in only a single level can be satisfied with. —David Eppstein (talk) 21:35, 1 March 2024 (UTC)

That's exactly why I suggested several non-equivalent definitions. However, the discussion got stuck at high school level issues. This must be resolved first before moving to the next level. Boute (talk) 12:12, 2 March 2024 (UTC)

To substantively address your comment though: I think everyone wants the article to serve readers (in particular including high school students and early undergraduates). I agree with you that it would be nice to immediately reference one or several sources about any definition(s) here, though I don't think an exact quotation is necessary to insert inline into the text, and making basic material look like a quotations can even be kind of confusing to readers (exact quotations in footnotes can be helpful though). –jacobolus (t) 16:37, 1 March 2024 (UTC)

Please don't distort clear messages. Good intentions need constant reminders. Evidently basic material should not look like quotations, but in a Wikipedia article a definition should be traceable literally to some source, because not observing this principle is precisely the cause of the endless repetitive discussions we are witnessing here. Don't believe that the literature of the past hundred years does not contain excellent candidate definitions (examples found by opening the hidden text box at the start of this discussion page). Enough generalities, make a solid proposal and submit it for discussion. Boute (talk) 20:19, 1 March 2024 (UTC)

Is the hidden "Outline for a new version of the article" supposed to be text to go directly into the article? I personally find it to be incredibly confusing and distracting, and not written in anything like the encyclopedic "house style" I expect for Wikipedia articles. I don't think it's suitable to go into a Wikipedia article as-is, but perhaps contains some useful ideas which could be turned to productive use; the sources themselves seem fine to cite. –jacobolus (t) 20:36, 1 March 2024 (UTC)

Logical error: it was clearly stated that the outline was meant as a resource to the other editors, and that integration into an article was left to them. Boute (talk) 02:26, 2 March 2024 (UTC)

Fair enough, and thanks for answering the question. I don't personally see a clear way to do that, and even each portion is rewritten as flowing encyclopedic prose, I find the organization confusing. Maybe someone else has concrete ideas of how to write some section(s) based on that, but I don't think you can reasonably expect someone to appear to do such translation for you. –jacobolus (t) 03:37, 2 March 2024 (UTC)

Your outline cites Apostol's Calculus. Here is what Apostol says:

The word "function" was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas. It was later realized that Leibniz's idea of function was much too limited in its scope, and the meaning of the word has since undergone many stages of generalization. Today [1960s], the meaning of function is essentially this: Given two sets, say $X$ and $Y,$ a function is a correspondence which associates with each element of $X$ one and only one element of $Y.$ The set $X$ is called the domain of the function. Those elements of $Y$ associated with the elements in $X$ form a set called the range of the function. (This may be all of $Y,$ but it need not be.)

Letters of the English and Greek alphabets are often used to denote functions. The particular letters $f,$ $g,$ $h,$ $F,$ $G,$ $H,$ and $\varphi$ are frequently used for this purpose. If $f$ is a given function and $x$ is an object of its domain, the notation $f(x)$ is used to designate that object in the range which is associated to $x$ by the function $f,$ and it is called the value of $f$ at $x$ or the image of $x$ under $f.$ The symbol $f(x)$ is read as "$f$ of $x$."

The function idea may be illustrated schematically in many ways. For example, in Figure 1.3(a) the collections $X$ and $Y$ are thought of as sets of points and an arrow is used to suggest a "pairing" of a typical point $x$ in $X$ with the image point $f(x)$ in $Y.$ Another scheme is shown in Figure 1.3(b). Here the function $f$ is imagined to be like a machine into which objects of the collection $X$ are fed and objects of $Y$ are produced. When an object $x$ is fed into the machine, the output is the object $f(x)$.

Although the function idea places no restriction on the nature of the objects in the domain $X$ and in the range $Y,$ in elementary calculus we are primarily interested in functions whose domain and range are sets of real numbers. Such functions are called real-valued functions of a real variable, or, more briefly, real functions, and they may be illustrated geometrically by a graph in the $xy$-plane. We plot the domain $X$ on the $x$-axis, and above each point $x$ in $X$ wit plot the point $(x,y),$ where $y=f(x).$ The totality of such points $(x,y)$ is called the graph of the function.

Leaving aside definitional disagreements about the range of a function, this seems overall like a nice introduction to me, which could quite plausibly be cited here and recommended to readers as a source to read. –jacobolus (t) 20:56, 1 March 2024 (UTC)

Possibly a good start, but which edition of Apostol's Calculus are you using? The second edition (Wiley, 1967) literally states on page 53:

" A function f is a set of ordered pairs (x, y) no two of which have the same first member. "

and goes on (same page) defining domain and range as, respectively, the set of first and second members.

Did Apostol (in your assumed quotation: edition? page?) really say in one paragraph that the range need not be all of Y and later on call Y the range? Sorry, but as this stands, it cannot be plausibly cited. Boute (talk) 02:53, 2 March 2024 (UTC)

This is the 2nd edition of Apostol, from a couple pages before the part you are quoting; it's essential context for the "formal" version, considering the intended audience of ~18 year olds. –jacobolus (t) 03:38, 2 March 2024 (UTC)

An informal paragraph on p. 51 in Apostol's Calculus indeed contains this error (a typo? better not speculate), which makes it unsuitable here. A Wikipedia article should contain no logical errors, a fortiori in an introductory context because setting misconceptions straight afterwards is always more difficult. As for the intended audience of ~18 year olds, I don't know whether this is a Wikipedia rule, but it is a reasonable working hypothesis. We also must assume a minimum of interest in mathematics (otherwise, why read the article?). In Apostol's time (1960s) set theory was unknown in most high schools. Nowadays this has changed, and Apostol's formal definition on p. 53 is perfectly accessible to that audience. After all, the first paragraph you quoted (from p. 50) assumes the reader knows about sets. The formal definition has the extra advantage of also appearing in other textbooks (e.g. Flett), and is less prone to the misconceptions caused by prematurely mentioning the set Y (hence the warning by Flett). So I think your suggestion to follow Apostol's path is worth pursuing. Selectivity is important. Boute (talk) 09:05, 2 March 2024 (UTC)

In other words, we should skip the gentle explanatory prose and cut straight to something unmotivated and inscrutable, so as to maximize novice readers' confusion? No thanks. –jacobolus (t) 09:40, 2 March 2024 (UTC)

"In other words"? Logic violation alert! Motivation is essential for any definition (even for an advanced audience, perhaps more). It must be done carefully, based on a lot of experience with and thought about how various different people think about mathematics. Boute (talk) 11:43, 2 March 2024 (UTC)

Sorry, that comment of mine was more dismissive than necessary, but let me try to be precise: I don't think we can transplant Apostol's discussion directly, but for his context of a calculus book I like the way he leads off with (1) a very brief history of the word (2) a mention that a function can be thought of as a formula, a machine, or a pairing, (3) some discussion of notation setting up common letters used later in the book, (4) a comment that functions might involve any objects but in calculus are usually real numbers, (5) [not quoted above] a couple of pages of concrete examples before trying to make the "formal" definition. These features let a reader who is not too familiar with functions ease into the idea.

It seems to me that you want to skip Apostol's 3 pages of leading context and jump straight to the "formal definition" part. That seems like a big mistake to me. –jacobolus (t) 16:14, 2 March 2024 (UTC)

This looks like a good plan, which most people will endorse. Apt considerations on the importance of justifying definitions can be found in Rogaway's paper [1]. I'm sorry for not being able to contribute anything to this article for several months (as explained after Lazard's post below). The resources and references under the "green slab" at the start of this talk page may suffice for now, but if any additional resources seem useful I can be reached by email. Boute (talk) 17:24, 2 March 2024 (UTC)

Chiming in as an outsider - this article is incredibly hard to read, and really serves no one but the maths elites.

I think the first thing that should be addressed is explaining "functions" in simple terms, without assuming people know what "of a set x to a set y" even means. This reads like mathematicians trying to one-up each other, not like an encyclopedia that is useful for anyone.

Here's my basic attempt, as a math-hating programmer : "A function is an equation that transforms a value (x) into another value (y), or describes X's relation to Y. A function of x {f(x)} is an equation including X, which can be assigned a value and solved. The solution to that equation, for a given x value, is often considered Y."

It may lack nuance that people with mathematics degrees want, but should the intro really be written to appease them? They already know what functions are. 47.55.178.193 (talk) 23:51, 22 March 2024 (UTC)

If it were true "serves no one but the maths elites". It definitely does not. Believe me. It is an embarrassment of an article. The article has a really really hard time defining precisely "of a set x to a set y". In my opinion, one reason why it is hard to read, is because it does not define this as explicitly as possible. Instead, the article makes one, two, three failed attempts at definition, that are not definitions, before it finally reaches a definition that is okay. No wonder it is hard to read. Nothing more confusing than an explanation that meanders on and on and on before getting to the point and missing it. Thatwhichislearnt (talk) 11:31, 23 March 2024 (UTC)

Keep up the good work, Jacobolus. I have nothing to add today, but two things have become clear. 1) Your views are the standard, referenced views. And 2) The people pushing their own views will argue 'til the cows come home. Rick Norwood (talk) 10:59, 2 March 2024 (UTC)

Fully agree, that's why I always insist on definitions that are literally traceable to reliable sources. Boute (talk) 11:46, 2 March 2024 (UTC)

I agree also with Jacobulus. As Wikipedia is not a text book, we must follow the dominant practice in mathematics and not the pedagogical simplifications that appear in some textbooks. Moreover, the ambiguity of the term "range of a function" motivated its replacement with "image of a function" and "codomain". We must not go back to the old ambiguity, which is induced when the codomain is not included in the definition of a function, at least because this would imply the rewrite of many Wikipedia articles. However, there are two cases where one considers functions whose domain (and codomain) are not well defined. The presentation of these two cases requires to be improved.

• A subsection "Partial functions" of § Definition must be added, which should be a short summary of Partial function, and must say that in some contexts, typically calculus, "function" is commonly used for "partial function". Otherwise, reader may believe that the multiplicative inverse of the real funtion $x\mapsto x$ would not be a function.
• A section must be added on the definitions of a function that are not based on set theory. These definitions are mainly used in computability theory and in constructive mathematics, and do not include the domain and the codomain because the concept of a set is lacking. Examples of such definitions are lambda calculus and general recursive functions. Personally, I do not know how functions are defined in constructive mathematics. The addition of such a section would imply to rewrite several sections of the article.

D.Lazard (talk) 12:22, 2 March 2024 (UTC)

In Bishop's masterpiece Constructive Analysis, a function from A to B is defined as a rule assigning to each a in A a unique f(a) in B, with the extra (constructivist) condition that "the rule must afford an explicit, finite, mechanical reduction from the procedure for constructing f(a) to the procedure for constructing a." The quoted part might be streamlined with the terminology from algorithmics. Boute (talk) 12:44, 2 March 2024 (UTC)

Since another deadline is coming up and I already exceeded the time I allotted myself for Wikipedia, Il sign off now, and will not be giving "replies" or input for several months. At this time I have nothing significant to add to the collection of resources and references I compiled a year ago at the start of this talk page. I wish everyone good progress in editing this article. Boute (talk) 17:02, 2 March 2024 (UTC)

Partial functions

The above lengthy discussions revealed me that, in many contexts, "function" is used in place of "partial function". This may confuse readers. This is my reason for adding a subsection § Partial functions of the section § Definition. I have tried to explain this confusing terminology with examples. D.Lazard (talk) 15:25, 5 March 2024 (UTC)

1. I do not see how partial functions would explain or solve anything in the above two threads "Formal definition ..." (e.g., the problem of surjectivity). Please clarify this.

2. You did not provide any citations

3. The 'Partial functions' section can remain in the article. It constitutes a separate thread." Kamil Kielczewski (talk) 06:12, 6 March 2024 (UTC)

§ Partial functions is not a subsection of § Formal definition; it is a subsection of § Definition, at the same level as § Formal definition and § Multivariate functions. D.Lazard (talk) 11:31, 6 March 2024 (UTC)

Riemann Hypothesis and partial functions

The section Partial Functions which has no source, contains this paragraph:

"Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set $\mathbb {C}$ of the complex numbers. The difficulty of determining the domain of definition of a complex function is illustrated by the multiplicative inverse of the Gamma function: the determination of the domain of definition of the function $z\mapsto 1/\Gamma (z)$ is equivalent to the proof or disproof of one of the major open problems in mathematics, the Riemann hypothesis."

1. I think that it means the Riemann zeta function, instead of the Gamma Function.
2. Also the meaning of the phrase : "the determination of the domain of definition" of the function $z\mapsto 1/\zeta (z)$ is not clear to me and also then meaning of "equivalent to the proof or disproof of" the Riemann hypothesis.
   For example, does it mean that if we prove that the R.H. is false (by giving a counterexample), then we can (or cannot) "determine" the domain of definition of the function $z\mapsto 1/\zeta (z)$"?

--Cbigorgne (talk) 17:38, 13 March 2024 (UTC)

Thank you for pointing this. I'll fix this. for the second point, I'll simply replace "equivalent" with "more or less equivalent". I think that it is a good way for making clear without entering in technical details that it not a true equivalence. D.Lazard (talk) 19:19, 13 March 2024 (UTC)

Formal definition (new version)

It seems that the above endless discussions were motivated by the fact that it was unclear for some readers whether the domain and the codomain belong to the definition of a function; also, some readers could be confused by the unmotivated introduction of relations. Therefore, I edited section § Formal definition for clarifying these two points. From now on, I suggest to discuss the new version only. D.Lazard (talk) 11:58, 13 March 2024 (UTC)

There are two attempts at definitions at this moment, in the section Formal Definition. Here is one observation on why the first of them is not yet a well-formed definition and it is not the same as the second. The second is fine.

The first attempt, begins "A function with domain X and codomain Y", but does not defines what "with [...] codomain Y" means for the object (the function) that this predicate is applied. That attempt then continues "is a binary relation R between X and Y", which defines the object as a relation ("is a binary relation"). This leaves the predicate "with codomain Y" still undefined. There is one way to understand the "with codomain Y" from "is a binary relation R between X and Y" and that is by interpreting the relation as an "embedded relation", the relation together with the inclusion in $X\times Y$. But let's face it, how many think automatically of relations as embedded? Interpreting it this way is just delegating the question to whether one understands relations as remembering their codomain.

The second definition is good. Now, one observation. By writing it, it means the article has sided with "functions know its codomain". While I personally tend to favor that one, it is also true that that is not what everybody does. There are authors that define functions as the relation alone. Someone above was reminding of Wikipedia needing to be balanced. In that light, something should be said. Thatwhichislearnt (talk) 13:30, 13 March 2024 (UTC)

Firstly "with codomain" is not a predicate, since a predicate must be a complete sentence or a well-formed formula; it is a qualifier that makes the definition more specific. Thus the sentence means that the sets that qualify the function are the same that qualify the relation. Indeed the definition of a binary relation involves two sets explicitly. Maybe some sources define "non-embedded" relations, that is, define relations betweeen arbitrary mathematical objects; this is not defined this way in Wikipedia or any source that I know. Moreover, this may cause many errors. For example, "equality" does not refer to a "non-embedded relation" since $x^{2}-y^{2}=(x-y)(x+y)$ is true when "=" is considered as a relation between polynomials, and false when considered as a relation between expressions. D.Lazard (talk) 15:02, 13 March 2024 (UTC)

A qualifier has a corresponding predicate that is implied, in this case "x is with codomain Y". And exactly as I said, what the first attempt at definition does is pass the ball to whether relations carry codomain or not. They do not. They do not when you talk about the relation itself. They could, when the language de facto emphasizes talking about the relation embedded in a Cartesian product, like when saying "a relation over". Your example on that equation is saying nothing. The = in expressions is a symbol of a language and not a relation, until interpreted semantically. Also, while everyone abuses languages and writes that equation alone, there are implied quantifiers on the variables and restricted ranges over which those variables are quantified. To write informally one skips them, but to write formally one should add them. If one is going to think of a relation, there is no need for the embedding to know that the relations are different. The elements of one version are pairs of say real numbers, while the other are pairs of expressions. The relation alone is sufficient to tell. The only confusion in writing only that equation is that it hasn't specified any relation in particular. Thatwhichislearnt (talk) 15:34, 13 March 2024 (UTC)

Regarding "Maybe some sources define "non-embedded" relations, that is, define relations betweeen arbitrary mathematical objects; this is not defined this way in Wikipedia or any source that I know.". There is some confusion here. Here is a relation defined by comprehension $R=\{(x,y)\in A\times B\mid \phi (x,y)\}$. While the comprehension uses $A\times B$, the relation is the set $R$. The set R does not necessarily remember A or B, just like it doesn't remember multiple elements, or order, or anything that is not a property of objects of Set. When I say "non-embedded" is only to emphasize that, as opposed to using phrases like "a relation from A to B" or similar, that are implicitly (informally) talking about the triple (A,B,R). Note also, that the relation R does not change, if in the comprehension one uses larger $A',B'$ for which no new elements satisfy $\phi$. It is the same relation, it is not the same triple (A', B', R).

We are talking about a section called Formal Definition. While the section doesn't need to be written completely formally, discussing about it we need to keep in mind what is the actual formalization that the English translates to. Thatwhichislearnt (talk) 16:11, 13 March 2024 (UTC)

A relation should also be defined to "remember" A and B, and should parallel the definition of function, so that "a function is a relation ..." remains a true statement. That was one thing I was getting at with attempts above. –jacobolus (t) 18:23, 13 March 2024 (UTC)

I totally agree with consistent definitions. But again, note that that should be explicit. Not everyone does that. Of the top of my head, I don't remember anyone doing that (the triple thing) for relations. I am not doubting the existence of such sources. I will need to check sources, and the Wikipedia article(s) on relation and binary relation would need those sources too. I have only seen relations being the set of pairs/tuples. Thatwhichislearnt (talk) 19:02, 13 March 2024 (UTC)

One that I have at hand is Halmos' Naive Set theory. In pages 26 and 27 you can see him being explicit about the distinction between considering a "relation" versus "a relation from ... to ..." (see end of page 27). Halmos counts among those who define functions as just the relation (and thus the graph) and relations as only the set. Thatwhichislearnt (talk) 19:16, 13 March 2024 (UTC)

Having separate definitions for "partial function" (with underspecified domain) vs. "function" (with unspecified codomain) vs. "function from X to Y" seems excessively pedantic to me, but I suppose could be described that way, with the note that even when not specified the word "function" still usually means "(partial) function from X to Y".

As for sources with this definition of relation, I'm not sure about textbooks but above I linked some course notes: Meyer (2005) "Binary Relations", notes for MIT 6.042J/18.062J:

A binary relation, $R,$ consists of a set, $A,$ called the domain of $R,$ a set, $B,$ called the codomain of $R,$ and a subset of $A\times B$ called the graph of $R.$

–jacobolus (t) 19:59, 13 March 2024 (UTC)

I don't think one would be able to prove the "usually". I believe Wikipedia has some sort of policy/guidance of avoiding such claims regarding frequency, unless supported by a reliable source that makes such a claim. Again, count me among those who prefer having codomains, but lots of people also don't do that. I think more appropriate is just say that that "is the definition used for the rest of this article". Thatwhichislearnt (talk) 20:06, 13 March 2024 (UTC)

@Thatwhichislearnt I completely agree with you - both definitions (of triples - with codomain) and of the set of pairs (without codomain) - due to their ubiquity, should be described (which I did in in reverted change ). Kamil Kielczewski (talk) 19:07, 13 March 2024 (UTC)

I reverted your edit because you not provide appropriate citations and it is a "Personal invention". In article you wrote: "A function is formed by three sets" - You merged Bourbaki's version (HERE - which you arbitrarily withdrew witch comment "Here is a clear consensus on the talk page against this change" - without asking all discussion participants) with the old version of the definition. And I assume that this is the reason why you couldn't find appropriate citations.

In the edit you arbitrarily withdrew - there was a source (even several) (here) describing two commonly used definitions: Bourbaki's (triple) and the set of pairs (which have significant structural differences) - you created a third one by combining these two. Kamil Kielczewski (talk) 18:09, 13 March 2024 (UTC)

@David Eppstein I see that you have reverted (19:01, 13 March 2024) what I did (17:57, 13 March 2024) with the comment "tendentious" - please explain how this relates to the issues I have highlighted, namely, lack of sources and "personal invention" - unless you are in some kind of conspiracy with D.Lazard. Kamil Kielczewski (talk) 22:17, 13 March 2024 (UTC)

Please refrain from wildly speculative and unlikely ad hominem conspiracy theories. The explanation is (1) "tendentious": you keep repeating the same fixed idea of there being only one possible correct way of defining functions, and responding to any attempt to bring you into a wider view by repeating the same thing again. WP:IDIDNTHEARTHAT. (2) "technical": this is supposed to be an article that can, to the extent possible, be read and understood by high schoolers. Formulating things as pedantically as possible using sets doesn't achieve that. —David Eppstein (talk) 00:34, 14 March 2024 (UTC)

@David Eppstein You did not answer the asked question: what about the lack of citations and "personal invention" in the changes made by D. Lazard ?

The sentence "technical: ..." explains nothing - it is irrelevant to the question. Moreover, D. Lazard's formal definition is more complicated than the Bourbaki definition (reverted here) because it uses the redundant concept of a binary relation, so the argument about better accessibility for high school students is misguided. But this is a separate issue - and it has nothing to do with the question I asked and which you are avoiding answering.

P.S.1: "there being only one possible correct way of defining functions" I'll add immediately that you are seriously mistaken - because in this change, I explicitly added a separate "Reduced function definition" section with an appropriate introduction and preserved the old definition. So, there are two, not one as you claim.

P.S.2 Accusing formal mathematical definitions of being too pedantic is absurd. It's like faulting poetry for having multiple meanings. (not to mention that D. Lazard in his innovation also tried to use precise language - but I see that you are biased and fail to notice that.)

P.S.3: Your attempt to dodge the answer and responsibility by changing the subject is transparent. Such eristic tricks are contrary to substantive discussion. Kamil Kielczewski (talk) 06:47, 14 March 2024 (UTC)

@D.Lazard, besides the fact that you didn't provide citations and your proposal is a "personal invention" (as I demonstrated in the comment at 18:09, 13 March 2024 (UTC).) I've also noticed that it contains an internal contradiction. Specifically, at one point you define a function as a binary relation, that is, a set of pairs ""a binary relation between two sets X and Y is a subset of the set of all ordered pairs (...) A function... is a binary relation,"" and at another point, you state that ""A function is formed by three sets."" Therefore, there is a contradiction because on one hand, "a function is a binary relation so it is a set of pairs," and on the other hand, "a function is three sets X, Y, R." - contradicition! Kamil Kielczewski (talk) 08:04, 14 March 2024 (UTC)

Even though he is doing the reverse (introducing functions and relations that do not know codomains), you can see how Halmos still addresses the semantics of "relation from X to Y" and "relation in X" versus just "relation" in pages 26 and 27 of his Naive Set Theory. Perhaps it can be seen online in google books here, for those who don't have the book. He is still using (almost only) plain English, yet being formally precise. Note how to do give meaning to those two, the inclusion needs to be mentioned: "If R is a relation included in ..." Whether wanting to introduce functions with codomain or not, I think it is worth imitating the level of precision in Halmos' (which is again, is written in pretty much plain English). Thatwhichislearnt (talk) 19:58, 13 March 2024 (UTC)

Please don't revert without discussing first. If you don't understand how to write a definition, just recuse yourself from the article. The current imitates masters of exposition, like Halmos. There is nothing undefined there. Thatwhichislearnt (talk) 12:12, 14 March 2024 (UTC)

Note that is it not true that X and Y are undefined. They has been mentioned before. Compare with what you had written, in which "with domain X and codomain Y" was indeed undefined. Not X, not the Y, the "with" was undefined. Thatwhichislearnt (talk) 12:15, 14 March 2024 (UTC)

Whether X and Y are undefined

This section is to keep the discussion focused. User Lazard questioned whether X and Y are undefined in this version. I claim that they have been defined in the paragraph above the definition, while this version does leave the "with" in "with domain X and codomain Y" undefined (specially for the codomain Y, the domain can be strictly speaking recovered). There is essentially no difference in content and intension between the two versions, only this point. For comparison, refer to Halmos' Naive Set Theory, pages 26 and 27 (end of page 27) for how the "with" is given meaning. Thatwhichislearnt (talk) 12:33, 14 March 2024 (UTC)

I have only one question: why don't we use two simple, well-known definitions: Bourbaki's and the set of pairs for which there is abundant literature (and which are simple)? Instead, we're trying to forge our own new (complicated) definition based on D. Lazard's personal invention? Why "reinvent the wheel"? Kamil Kielczewski (talk) 14:10, 14 March 2024 (UTC)

The shield of keeping this Wikipedia simple. And that is fine. The definition can be done in any language. It can be done as close to English as any wants. It is better not to digress. Focus, in this section, on discussing whether X and Y are undefined (any more or any less than they are in the other version). And whether the other version has "with" still undefined. Thatwhichislearnt (talk) 14:16, 14 March 2024 (UTC)

And let me make it even simpler. When you define a new set, say by (restricted) comprehension $F=\{x\in A\mid \phi (x)\}$, the set $A$ needs to be known to be a set from before, it is assumed to have been defined before. The $F$ is the one being defined in the comprehension. Likewise, when defining a function (which in set theory foundations, like everything, is a set) we have previously defined sets X and Y and from them one defines f. There, is never a question whether X and Y are defined. One could have had the same unfounded concern in this inadequate version. All the discussion(s) above, are not regarding X and Y themselves, but whether the meaning of "f with codomain Y" has been defined or not. That inadequately written version does not define "f with codomain Y" (in the first of its purported "formal definitions"). Thatwhichislearnt (talk) 13:08, 14 March 2024 (UTC)

A definition must be mathematically consistent. If a definition implies X and X without saying what they are, it cannot be consistent. More precisely, in a clause "something is defined as some_description", the only free variables that could appear must appear in both "something" and "some_description". As your edit does not follow this basic principle, it make your definition nonsensical, and cannot be accepted in Wikipedia. D.Lazard (talk) 13:37, 14 March 2024 (UTC)

In $\{x\in A\mid \phi (x)\}$ the $A$ is not a free variable. Likewise, the X and the Y in the definition are not. They are defined, they are some sets, that (you) in the previous paragraph(s) have named. If your "objection" were valid, your version would have exactly the same problem. The objections about your version are not even that. Have you already understood that the "with" is what is undefined? Thatwhichislearnt (talk) 13:46, 14 March 2024 (UTC)

And stop reverting when you have trouble understanding set theory. This is so basic that it should make you ashamed. Thatwhichislearnt (talk) 13:50, 14 March 2024 (UTC)

The onus is on you to get consensus support for your changes. Making personal attacks is not going to help you do that. MrOllie (talk) 14:06, 14 March 2024 (UTC)

Wait, but where was the consensus when he added his version, that does still leave undefined what nearly the entire talk page is dedicated to discuss. My edit, nearly the exact same content he added. He raised a new "concern" that even his edit would have, if it were at all founded. Thatwhichislearnt (talk) 14:11, 14 March 2024 (UTC)

None of that is a reason for you to edit war and make personal attacks. MrOllie (talk) 14:15, 14 March 2024 (UTC)

Ok. Sufficient digression. Would you like to add to the discussion in this section? Thatwhichislearnt (talk) 14:18, 14 March 2024 (UTC)

I broadly agree with D.Lazard's criticism of your change. Definitions should be as self-contained and self-consistent as possible. I also find the added paragraph (Viewing the relation...) to be ungrammatical and confusing, it doesn't add anything to the section and should be omitted. MrOllie (talk) 14:39, 14 March 2024 (UTC)

Do you think the "with" in "with codomain Y" has been defined in that version? What that paragraph in green adds is giving meaning to precisely that. Whether ungrammatical or not, while I disagree, I am perfectly happy with the grammar being fixed. Compare with authors that take the care of defining "with codomain Y", like Halmos A large volume of discussion in the talk page, throughout the years has been spent on the definition(s) in the article not properly defining the meaning of the phrase "with codomain Y". That last edit, still leaves it undefined. Thatwhichislearnt (talk) 14:50, 14 March 2024 (UTC)

@MrOllie In my comment at 18:09, 13 March 2024 (UTC), I pointed out serious objections to D. Lazard's changes (lack of literature and "personal invention") - so I also do not understand what this consensus is based on since he introduces any changes he wants - and ignores all voices of opposition.He also prevents improvements to the article that are based on literature (e.g. here) - by providing arguments that are irrelevant or contradict the said literature. Kamil Kielczewski (talk) 14:21, 14 March 2024 (UTC)

Making more personal attacks in a response to a warning about personal attacks is an unusual tactic. Kindly stop. MrOllie (talk) 14:25, 14 March 2024 (UTC)

What I wrote above was not a personal attack - only opposition to the actions of D. Lazard - it's not the same thing. And it's not unfounded what I write - I have mentioned some arguments. Kamil Kielczewski (talk) 14:33, 14 March 2024 (UTC)

Keep your 'opposition' civil and without personal comments. You are only harming your own position. MrOllie (talk) 14:35, 14 March 2024 (UTC)

In this version, I added to the formula to address your concern (which would exist also in the previous, but well). While also addressing the lack of definition of "with". Happy now? Any new concern? Thatwhichislearnt (talk) 14:38, 14 March 2024 (UTC)

@Thatwhichislearnt: I am only speaking for myself, but my assessment of the situation is that you are dragging other contributors into spending disproportionate amounts of time and energy solving a non-existent problem. I find @D.Lazard's version of the article better-written than yours and perfectly suitable for this kind of article; and despite your explanations above, I fail to see what is the exact problem you are trying to address (and apparently I am not the only one here).

Please try to convince us that there is a legitimate problem with the current version of the article — taking into account the fact that this is a Wikipedia article, not a foundational treaty on set theory. If you bring up legitimate points, I (and I am sure D.Lazard as well) will be happy to think about how they could be best fixed in a broad-audience article. If no-one share your concern that there is something really wrong that needs to be fixed in the article, consider the possibility that maybe you are splitting-hair and that is not worth the time investment.

Best, Malparti (talk) 15:04, 14 March 2024 (UTC)

@Malparti In my comment at 18:09, 13 March 2024 (UTC), I pointed out serious objections to D. Lazard's changes (lack of literature and "personal invention"). I also found contradiction - details in my comment: 08:04, 14 March 2024 . Kamil Kielczewski (talk) 15:09, 14 March 2024 (UTC)

Look, I am not dragging anything. If you look from the very beginning of the talk page to the bottom, and likely much older version, the topic of whether the definition properly determines the codomain keeps bringing people other than me, discussing that. The version that you such praise, still has the definition malformed, because the predicate "[f] is (or has the property) with codomain Y" is not defined in it (I am talking about the first of the two attempts at definition). I personally disagree that Lazard does a good job at mathematical exposition, but I am happy with any style as long as when the literal translation into formal set theory corresponds to a well-formed definition. One can write as plain English as anyone wants, while still maintaining a proper correspondence to formally written set theory. For example, Halmos pages 26 and 27 are pretty plain English. See them addressing exactly the meaning of "with domain X and codomain Y" and "from X to Y" in the last paragraph of page 27. Thatwhichislearnt (talk) 15:21, 14 March 2024 (UTC)

I just went to the history of the talk page, clicked "older 500" a few times, clicked on a version, and here, 2012 and people discussing that the definition does or does not define codomain. I would bet that there are earlier. Is it me dragging? The only way resolve that, is by being explicit on that meaning. It doesn't have to be complicated, it doesn't have to be abstract, or formal, just explicit. Of course, translatable into formal language, for correctness. Thatwhichislearnt (talk) 15:51, 14 March 2024 (UTC)

A la Halmos "with codomain Y" while close to previous version

Ok, here is a version that address "with codomain Y" still being undefined in the current version.

Given sets X and Y, a function can be defined to be a binary relation R between X and Y that satisfies the two following conditions:

• For every $x$ in $X$ there exists $y$ in $Y$ such that $(x,y)\in R.$
• $(x,y)\in R$ and $(x,z)\in R$ imply $y=z.$

Viewing the relation R together with the inclusion into $X\times Y$ allows to speak of the function with domain X and codomain Y or say that it is a function from X to Y. Some authors do this implicitly, some explicitly, and some view functions as only the relation itself.

Without referring explicitly to the concept of a relation, but using more notation (including set-builder notation), a function with domain X and codomain Y is formed by three sets, the domain $X,$ the codomain $Y,$ and the graph $R$ that satisfy the three following conditions:

• $R\subset \{(x,y)\mid x\in X,y\in Y\}$
• $\forall x\in X,\exists y\in Y,\left(x,y\right)\in R\qquad$
• $(x,y)\in R\land (x,z)\in R\implies y=z\qquad$

The first paragraph introduces the function. A careful reader will note that this is the function viewed as only the set of pairs. The paragraph "Viewing the relation R ..." then defines the meaning of "with codomain Y". Well, also "with domain X", but a careful reader can note that already the first paragraph determines X from the set of pairs, if not the phrase "with domain X" itself. This second paragraph is imitating Halmos' Naive Set Theory last paragraph of page 27, which gives meaning to "with codomain Y" while still mostly in English and without much formal language. The concern of whether that second paragraph has some grammatical problems ... I am happy for it to be written in any other form.

The concern of whether X and Y are undefined. Sorry, but mathematically makes no sense. Please suggest a wording that you think "solves" that. I am happy with different wording and style. An exercise that everyone should do is to think about how does the English translates into a completely formal definition. The "Given X and Y", as is customary, translates to the $\forall A$ in the axiom of comprehension. Note that there is no antagonism between being precise and using simple wording. A simple, plain English wording can be assessed to be precise, when you can take the translation of each of its syntagmas into set theory and you get a valid formula. Thatwhichislearnt (talk) 15:07, 14 March 2024 (UTC)

I still do not understand what is the problem with the current version, and why this version would be an improvement... In fact, to me this version seems worse for several reasons:

• I do not find it to be well-written: (1) "can be defined to be" is heavy; (2) it feels very awkward for me to introduce X and Y and then talk about "a function" instead of "a function from X to Y" (as is done, e.g, in Halmos's Naive Set Theory). A bit like comparing "Let r be a positive real number. A circle is [...]" instead of "Let r be a positive real number. A circle with radius r is [...]"
• I find the paragraph "Viewing the relation R [...]" unnecessarily complicated and confusing.

For these reasons, I prefer the current version over the proposed new one. Since I still don't understand exactly what problem that new version is supposed to address (I guess it might have to do with whether the codomain is included in the definition of the function or not? If so, then to me that discussion is a waste of time and I'm not willing to invest any more time into it), I don't see a reason to change what is currently written. Malparti (talk) 15:48, 14 March 2024 (UTC)

• Happy with any other wording of "can be defined".
• The "Given X and Y", is the $\forall A$ in the Axiom_schema_of_specification#Statement. See how the $\forall A$ in it is before the $\exists B$, which is the set that the axiom effective defines. That is the same structure. Again, happy to move it, or reword it. There are many styles of writing the $\forall A$, including the customary omission, which in the very first version I used, but there was the unfounded claim that X and Y are left as free variables.
• Note that Halmos defines "a function from X to Y" after having defined "relation" and having defined "relation from X to Y" (page 30). Regarding the previous point, I quote his page 30 "If X and Y are sets, a function from (or on) X to (on into) Y ...".
• Finally, the version that you like is nearly the same as this. The only difference is that it still leaves undefined (in the first of the two version of definitions that it claims equivalent) the meaning of "with codomain Y". A subject that has kept bringing people to discuss since as early as 2012, and likely before. Leaving it it still under the rug, undefined, is just going to keep being (unnecessarily) a point of contention.
• There is also the issue with the current version of the article regarding neutrality. The current version of the article (tries to) present functions as having codomain as part of the concept. That is not the only case in current use. Big guns of set theory, of functional analysis (and many introductory books that did it without paying attention), like to use the definition that does not carry the codomain. Per neutrality, the article should also make as little bit of effort in presenting that that position also exists.

Thatwhichislearnt (talk) 16:09, 14 March 2024 (UTC)

Formal definition - why not use the literature?

It seems that the current definition is an attempt to "reinvent the wheel" - let's note that:

• In the literature there are two common definitions (more info here p. 1158) of a function: Bourbaki triple and as a set of ordered pairs (sometimes using a binary relation, though it is not necessary). This second definition is effectively Bourbaki's definition reduced only to the function's graph.
• Literary definitions are well-thought-out, simple, direct, clear, short and unambiguous - we can introduce both of these popular definitions directly (especially since there is a relationship between them)
• The above literary definitions usually do not have the inaccuracies that the above endless discussions (for many years!) are about.
• The "Formal definition" section clearly informs the reader that formalism will appear here, so the reader knows what to expect from this section: a good, well-thought-out, clear, concise formal definition (and preferably simple).
• In order not to distract the reader the "Formal definition" section doesn't even need to contain any comments on the definition (or history information) - these can be placed further on
• Currently, the "Formal Definition" section is becoming something of a "monster" - it is long, complicated (e.g. too many quantifiers - it can be simplified), contains contradictions (details in the comment at 08:04, 14 March 2024 (UTC)) and is basically a personal invention (details in the comment at 18:09, 13 March 2024 (UTC)) that consists of combining Bourbaki's definition and the set of ordered pairs into one.
• The current "Formal definition" section is definitely not easy to understand for a high school student
• Because the current definition is a patchwork of two popular definitions, it will be difficult to find references in the literature - perhaps some will be found, but certainly not as abundant as for the two above common definitions.

Can someone explain to me why we don't use simple classic short literary definitions but invent our own?

What is blocking us, where is the problem?

PS: For example, my withdrawn edition introduces Bourbaki's definition with references to literature, and places the old definition based on a binary relation in the appropriate context. Kamil Kielczewski (talk) 16:39, 14 March 2024 (UTC)

Bourbaki wrote in French. I have not their book under hands but, as far as I remember, they do not define fonctions (functions), but applications (maps). The two terms are more or less synonyms in English but not in French ("a linear function between vector spaces" seems very odd when literally translated in French). Moreover, Bourbaki's preface says that their aim was to establish a strong basis for analysis, and that the chapters on analysis was not yet written (it is still not written). Since this article must cover functions in analysis as well as functions in set theory and algebra, Bourbaki's and Halmos' references must be considered with care, since both references are focused on the algebraic and set theoretic concept (Halmos' book is entitled Naive set theory). D.Lazard (talk) 18:45, 14 March 2024 (UTC)

Bourbaki's exposition setting everything up is extremely convoluted and painful to decode, because they want to define everything entirely symbolically with utmost precision. They usually use Roman letters for sets, but sometimes bold italic letters instead for a reason I don't quite understand. For someone who doesn't already know exactly what it's supposed to be saying, just reading the text takes repeatedly solving tricky logic puzzles to work out the meaning. I would expect deciphering the relevant definitions to take at least a half hour for a newcomer, maybe hours. Here's a link into the thick of it.

In English translation, Bourbaki define a graph to be a set of ordered pairs, use the term relation to mean more or less any meaningful statement about some list of terms, and the relation $R\{x,y\}$ is said to have a graph $\mathrm {G}$ if $(\forall x)(\forall y)R\iff {}\!$$((x,y)\in \mathrm {G} ).$ The projections (1 and 2) of the graph are the set of first/second elements of the pairs, also respectively called domain and range (what we call image here).

They then define a correspondence to be a triple $\Gamma =(\mathrm {G} ,\mathrm {A} ,\mathrm {B} ),$ with source $\mathrm {A}$ and target $\mathrm {B}$ where the domain of the graph is a subset of the source and the range of the graph is a subset of the target.

A graph is called a functional graph if each element in the domain only appears as the first element of one pair, and a correspondence is called a function if its graph is a functional graph and if furthermore its source is the domain of the graph.

Then a mapping of $\mathrm {A}$ into $\mathrm {B}$ is defined as a function whose source (the domain of its graph) is $\mathrm {A}$ and whose target is $\mathrm {B}$.

In other mathematical literature, any of "graph", "functional graph", "correspondence", "function", "mapping of $\mathrm {A}$ into $\mathrm {B}$" might sometimes be called a "function", depending on the context. –jacobolus (t) 19:34, 14 March 2024 (UTC)

There is english translation of Bourbaki book "Theory of Sets", Springer, 1970 - page 81.

But it is not necessary to accept Bourbaki's definition in its original form (because he used quite specific formalism to express it), only in the contemporary one provided by other authors (bearing in mind only that the idea of the triplet comes from him).

If it turns out in the future that Bourbaki's definition is insufficient (which I think is, however, unlikely.), there is nothing to prevent supplementing or modifying the article.

Perhaps Bourbaki intended to use their definition only for analysis, but over time it turned out that the definition of the triplet is surprisingly general despite its simplicity. And maybe that's why it became widespread. Kamil Kielczewski (talk) 20:11, 14 March 2024 (UTC)

• And the current section Formal definition, is not neutral, in the sense that there is no explicit mention on the definition of function as its graph (even though the first attempt at definition in it, that is what it unintentionally defines). The article on binary relations#Definition does a better job at neutrality. Unfortunately the article binary relations does exactly the same mistake of starting trying to define "binary relation R over sets X and Y" and then says it is a subset of $X\times Y$. Hilarious. How to carry the information of "over X and Y" using an object from the most forgetful of categories? Magic! Thatwhichislearnt (talk) 20:19, 14 March 2024 (UTC)

In Finitary_relation#Definitions they did a better job. It doesn't have the issues in this article and the issues in binary relation. The owners of this article should probably imitate what is done in Finitary relations. There, there is a sentence, that like the one in this article forgets what is trying to define, but acknowledges it by saying "is given by", instead of "is". Then it issues a correction acknowledging that it is a correction with "may be more formally defined". Thatwhichislearnt (talk) 21:12, 14 March 2024 (UTC)

I think we could clear this up with a definition that includes the following information (but rewritten for clarity):

• A binary relation R between X and Y consists of a set X called the source or domain, a set Y called the target or codomain, and a set of ordered pairs {(x, y)} such that x is in X and y is in Y, called the graph, which is a subset of the Cartesian product X × Y. A binary relation and its graph are routinely conflated, and some sources define the term binary relation to mean a graph; the symbol R can be used interchangeably to refer to either one, with meaning inferred from context.
• A function f between X and Y is a binary relation for which each element x in the domain is the first entry of exactly one ordered pair in the graph. As a result, each x in X unambiguously indicates a specific corresponding y in Y, called the image of x under f and denoted f(x). The set of every y in Y which is included as the second entry of some pair in the graph is called the image of the function. As for binary relations, a function and its graph are routinely conflated, and some sources define the term function to mean a graph satisfying the same constraints, but without a defined codomain. The domain of a function can be inferred from its graph.
• A partial function f between X and Y is a binary relation for which each element x appearing as the first entry of an ordered pair in the graph appears in only one pair, however not every element of X need appear in the graph. Sometimes a partial function is just called a function; this can be convenient because it saves the trouble of defining a more restricted domain where the function is defined.
• A multivalued function f between X and Y drops the condition that each x in X can only have a single image under f.
• A multivariate function f between (X₁, ..., X_n) and Y is a function for which each element of the domain is itself an n-tuple (x₁, ..., x_n).

I would skip the business about ordered triples of sets, which is a bit of set-theory bookkeeping not necessary for describing the concept. –jacobolus (t) 22:20, 14 March 2024 (UTC)

I agree with bullet (1), in general. In bullet (2) I think "relation" should not drop its surname "over X and Y". After all, that is the full name of that defined in bullet (1). If dropped, then there is the uncertainty for the reader: "Are we doing the conflation that that bullet (1) was saying, or not?". Bullet (3) is awkward, because it is using a concept that is muddied. A solution can be to start fresh, and define it in the style of the beginning of bullet (1): "A set X, a set Y, and a set of ordered pairs ...". In bullet (5), I don't think that that language is common, of referring to the "first part of the between" (what would be the domain) using a symbol for a tuple of sets. It is not what multivariate functions are. Anyway, multivariate functions are not important in the section of Formal Definition. It is just a label for some functions. Every n-variate function is also uni-variate, and k-variate for every k in between. But let's ignore bullets 3,4, and 5. The discussion is regarding the section Formal Definition. Thatwhichislearnt (talk) 23:19, 14 March 2024 (UTC)

I don't personally think we need to have "Formal definition" as a section heading. I would just have one top-level "definition" section, try to keep it moderately concise, and not divide it into subsections. I think it's worth including partial functions and multivalued functions there because they are sometimes called "functions" and it's helpful to readers to clearly describe them; multivariate functions can plausibly be set-theoretically formalized in multiple distinct ways some of which might not make them straight-forwardly "functions" per se. I'm not an expert on what the literature says about this though. (My bullet points were not intended to be wiki-ready definitions of any of these, and anything going into the article should be written carefully). –jacobolus (t) 23:31, 14 March 2024 (UTC)

I believe that the 'Formal definition' section is important and has its practical significance. For instance, individuals who are familiar with mathematical formalism and wish to recall or refine the meaning, can quickly find the necessary information in this section. This section also allows in a concise way to keep the remaining elements of the article in check, which should not contradict this section. Probably that's why many mathematical articles contain this section. 85.221.133.173 (talk) 07:53, 15 March 2024 (UTC)

I just think the article would be more readable if we left § Definition as a consolidated section without internal subsections, and tried to keep it as concise as practical. We should also try to shorten the following sections about notation if we can. This article spends way too long on technical minutiae near the top in a way which is distracting and unhelpful to less-technical readers, and not really very legible for anyone. –jacobolus (t) 08:01, 15 March 2024 (UTC)

I would skip the business about ordered triples of sets - I think we should not overlook definitions that are commonly accepted in the literature. 85.221.133.173 (talk) 08:02, 15 March 2024 (UTC)

There is a nice discussion about this general topic discussing the motivation and considerations involved with various definitions in:

Goldblatt, Robert (1984). "2. What Categories Are". Topoi: The Categorial Analysis of Logic (Rev. ed.). Amsterdam: North-Holland. pp. 17–36.

... Although the modified [triple (X, Y, R)] definition does tidy things up a little it still presents a function as being basically a set of some kind – a fixed, static object. It fails to convey the “operational” or “transitional” aspect of the concept. One talks of “applying” a function to an argument, of a function “acting” on a domain. There is a definite impression of action, even of motion, as evidenced by the use of the arrow symbol, the source–target terminology, and commonly used synonyms for “function” like “transformation” and “ mapping” . The impression is analogous to that of a physical force acting on an object to move it somewhere, or replace it by another object. Indeed in geometry, transformations (rotations, reflections, dilations etc.) are functions that quite literally describe motion, while in applied mathematics forces are actually modelled as functions. This dynamical quality that we have been describing is an essential part of the meaning of the word “function” as it is used in mathematics. The “ordered-pairs” definition does not convey this. It is a formal set-theoretic model of the intuitive idea of a function, a model that captures an aspect of the idea, but not its full significance.

–jacobolus (t) 23:14, 14 March 2024 (UTC)

That author is about to introduce diagrams (with its arrows), commutative diagrams, and categories. That is his whole point in that paragraph. The better way he has in mind, is to look at the entire category. That is the "dynamical quality" that he wants to bring up, the functions, the arrows, not in isolation, but living in a category (See after he defines pullback). That is beyond the scope of this article. Thatwhichislearnt (talk) 10:53, 15 March 2024 (UTC)

The quoted paragrapg does not talk at all about categories. I suppose that its aim is to motivate category theory. The last sentence of this quotation seems fundamental for understanding functions. I had something like this in mind when I wrote the first paragraph of section § Formal definition. D.Lazard (talk) 12:24, 15 March 2024 (UTC)

Just like if you take the first word alone, it doesn't talk about functions at all. Read the book. And if it is not interesting for a career of implementing other's algorithms, read at least until after he introduces pullback. The entire point of the author in that paragraph is "functions' dynamic nature is exposed when viewed within the (concrete) category where they live". Thatwhichislearnt (talk) 12:49, 15 March 2024 (UTC)

What a curious exegesis! D.Lazard (talk) 15:40, 16 March 2024 (UTC)

Yeah, things look like magic, when you know know how they work. They fly over your head. Thatwhichislearnt (talk) 19:33, 16 March 2024 (UTC)

Please avoid personal attacks. —David Eppstein (talk) 21:11, 16 March 2024 (UTC)

To be clear, I would recommend reading the text before the part I quoted, which I think does a good job of setting the context. I just didn't want to quote several pages of exposition into a comment here.

In any event, whether or not they use categories (or draw commutative diagrams), technical authors working with the concept of a 'function' don't really think of functions as static collections of sets, and a formal definition in those terms is, as Goldblatt describes, a "model" of the concept, but not really the thing itself. It is difficult to convey, especially to newcomers, that mathematical definitions are in many ways contingent and arbitrary, in the sense that many possible definitions could be chosen which describe the same structural relationships people care about, but with different incidental details of the construction, which people generally "black-box" and then ignore. The same phenomenon occurs all over mathematics: some choice of definition is necessary to make rigorous proofs, but the specific choice often doesn't really match the essence of the thing being defined. –jacobolus (t) 16:03, 16 March 2024 (UTC)

I recommend reading the book (which is about Topoi), and perhaps after, follow up with the book "Homotopy Type Theory: Univalent Foundations of Mathematics" to see how the former can be used as models of a different way to develop foundations of mathematics. Just taking out the meme "the ordered pairs definition does not convey the dynamic quality of functions" is meaningless if you don't say what **exactly** is it that you are missing, and what is the replacement that you think does a better job. By the way, it has little to do with the black-box image of input coming in and output coming out. The ordered pairs tell you everything you need to know in that regard. They tell you what can come in, and what comes out. What he is getting at has to do with what happens with the black-box when you plug things at the entrance and/or exit. That all one needs to know about it can be learned in this way. This is why he mentions it again after defining pullback. So, yes, you can drop the set of pairs, but keep the category (and more drop the category and keep all equivalent categories).

Inserting that meme in this article would be completely meaningless and unjustified, withing the context of this article. Or is the article going to start talking about categories, toposes and HoTT? Thatwhichislearnt (talk) 18:33, 16 March 2024 (UTC)

HoTT, maybe not, but it does briefly mention type theory, and the treatment of functions as primitive objects in that theory rather than something derived from sets. I think that inclusion is appropriate. —David Eppstein (talk) 19:12, 16 March 2024 (UTC)

Indeed, very appropriate. In an article that has only one of the set-theory foundations definition badly written, is missing the other, and doesn't have the language to even begin articulating what exactly are the mathematical features that one cannot express with the definition. And who knows what exactly will be written, if written by those who didn't even realize what the author was referring to. Thatwhichislearnt (talk) 19:53, 16 March 2024 (UTC)

The ordered pairs tell you everything you need to know in that regard: How you define continuous functions, differentiable functions and analytic functions in terms of ordered pairs? It is certainly possible since the two points of view (the dynamic one and the static one) are equivalent, but it seems impossible to do that in a comprehensible manner. We must not forget that a large part of the readers of this article is more interested in continuous functions (21,574 page views last 30 days) than in binary relations (7,9222 page views). D.Lazard (talk) 21:32, 16 March 2024 (UTC)

How you define continuous functions, differentiable functions and analytic functions in terms of ordered pairs To talk about continuity, additional concepts/structures must be introduced - open sets/neighborhoods in the domain and codomain (thus, topology). The mere definition of a function (regardless of whether as a triple or just its graph) is not enough - but this does not mean there is a problem with the definition of the function. Simply, the concept of function continuity is inseparably linked with the concept of topology as well as the concept of a function - and both of these independent concepts must be used to discuss continuity. (PS: It should be noted that often in the literature, a given topological space is not defined explicitly. For example, in the set of real numbers, open sets are defined using balls with certain properties - which, as it turns out, implicitly specify the topology on R) Kamil Kielczewski (talk) 22:18, 16 March 2024 (UTC)

Re "regardless of whether as a triple or just its graph": this reminds me of the famous quote about having "both kinds of music": country and western. —David Eppstein (talk) 23:07, 16 March 2024 (UTC)

Wow, just wow. Beginning from continuity, differentiability, and analyticity none of them being properties of the function alone. Did you even think about your question for one second? Thatwhichislearnt (talk) 13:04, 18 March 2024 (UTC)

But functions have been introduced for dealing with continuity and derivative. Until the 19th century, all functions were continuous and differentiable. Presently most readers of this article are more interested in continuous functions than in relations see above statistic). So the beginning of this article must be useful for the majority of its readers. This article must not define continuity and differentiability (they belong to other article), but its beginning must be compatible with these definitions. This is not the case of the definition by ordered pairs and relations. The only usge of the definition through pairs and relations is to provides a formal version of the intuitive definition trough assignment, process, mappings, etc. You must know that most users of functions do not know other definition than the intuitive one, and this is generally sufficient for them. D.Lazard (talk) 13:53, 18 March 2024 (UTC)

Here's another paper to look at, Leinster, Tom (2014). "Rethinking Set Theory". The American Mathematical Monthly. 121 (5): 403–415. JSTOR 10.4169/amer.math.monthly.121.05.403.

In this take, 'functions' are a primitive concept with certain axioms applying to them, and 'elements' of sets can be defined as a higher level concept, in terms of functions.

The root of the problem is that in the framework of ZFC, the elements of a set are always sets too. Thus, given a set X, it always makes sense in ZFC to ask what the elements of the elements of X are. Now, a typical set in ordinary mathematics is $\mathbb {R}$. But ask a randomly-chosen mathematician, ‘what are the elements of π?’, and they will probably assume they misheard you, or tell you that your question makes no sense. If forced to answer, they might reply that real numbers have no elements. [...] The traditional approach to set theory involves not only ZFC, but also a collection of methods for encoding mathematical objects of many different types (real numbers, differential operators, random variables, the Riemann zeta function, ...) as sets. This is similar to the way in which computer software encodes data of many types (text, sound, images, ...) as binary sequences. In both cases, even the designers would agree that the encoding methods are somewhat arbitrary. So, one might object, no one is claiming that questions like ‘what are the elements of π?’ have meaningful answers. [...] The bare facts are that in ZFC, it is always valid to ask of a set ‘what are the elements of its elements?’, and in ordinary mathematical practice, it is not. Perhaps it is misleading to use the same word, ‘set’, for both purposes. [...] § The working mathematician’s vocabulary includes terms such as set, function, element, subset, and equivalence relation. Any axiomatization of sets will choose some of these concepts as primitive and derive the others. The traditional choice is sets and elements. We use sets and functions. –jacobolus (t) 23:34, 16 March 2024 (UTC)

The root of the problem is that in the framework of ZFC, the elements of a set are always sets too The source of what problem?

It seems that the ZFC axioms constitute the most popular foundation of mathematics, while other systems represent exotic niches for specialists. Kamil Kielczewski (talk) 00:03, 17 March 2024 (UTC)

All you are doing is introducing noise, and noise that is unfounded withing the context of this article, and in particular the section being discussed. Even the section the brings up type theory raises an objection and instantly resolves it within the framework of set theory. Again, what is being discussed is the section Formal Definition, and the issue that the two definitions in use are not mentioned, only one is mentioned (badly in the first attempt). What that section should do is four things: (1) mention the definition without codomain, explicitly noting that there are no codomains in it (2) mention the definition with codomains, explicitly mentioning that there are codomains in it (3) mention that both definitions are in use in literature (4) mention which of the two is the one that this article will use, at least predominantly and unless otherwise stated.

A random click at the history of this talk page showed me that as far back as 2012, this article has had an influx of people complaining that the article does a poor job regarding the definition. It will continue, as long as there all that ambiguity. And it is not that it is a difficult task. There are plenty of award-winning authors that present these points in clear plain English, that can be imitated. Thatwhichislearnt (talk) 13:17, 18 March 2024 (UTC)

For given so much emphasis on functions without codomain, you must provide examples of common functions without a codomain. All examples that I know have a codomain provided implicitly by the context or by the definition of the specific function. For example, the square function on the reals is defined as assigning to each real number the product of this number by itself. As the product of two real numbers is always a real number, the codomain of this square function is implicitly the set of the real numbers. If you cannot provide an example where the codomain is not defined, at least implicitly, this mean that functions without a codomain do not deserve to appear at the beginning of the article. D.Lazard (talk) 14:20, 18 March 2024 (UTC)

Open Halmos', open Apostol's, open Carson's. There are [Talk:Function_(mathematics)#A._Functions:_the_plain_variant nearly 20 different references listed above]. Those are the ones that do it as a deliberate choice. In addition, there are myriads of Calculus books (plus this Wikipedia article's Formal Definition's first attempt) that do it by mistake, because they don't pay attention, they define a concept that doesn't have codomain, even though later talk about them and define surjectivity. The discussion here is the section Formal Definition, not the beginning of the article. Thatwhichislearnt (talk) 14:54, 18 March 2024 (UTC)

And your request for examples is entirely non-sensical. Within the context of a book that chooses to define functions as the set of ordered pairs, all functions are examples of functions without codomain. All of them. This doesn't mean that one cannot talk about codomains, or the idea of surjectivity. Instead of talking about surjectivity (as a property of the function), they talk about "surjectivity onto Y". Mathematically, no idea fails to be representable, it is only a different formulation that some authors choose. So many other properties: continuity, differentiability, analyticity, are not properties of the function itself anyway. Thatwhichislearnt (talk) 15:19, 18 March 2024 (UTC)

And not even going to the literature, the article Finitary_relation#Definitions does an okay job regarding the definition. While Binary_relation has the same problem as this (in the first sentence of their "definition"), except that over there they do use consistently the full name of the object that they intend to be talking about throughout the whole article. Thatwhichislearnt (talk) 14:21, 18 March 2024 (UTC)

@Jacobolus of course, the information that in some systems not based on ZFC, functions can be considered as primitive concepts, can be placed in a separate section titled e.g. "Additional Information" (as a curiosity for someone who aspires to join the narrow group of specialists...). So, outside the "Formal Definition" section that we discuss in this discussion thread. Kamil Kielczewski (talk) 23:41, 18 March 2024 (UTC)

Formal definition - the structure of a triple

Based on the statement by @Jacobolus I would skip the business about ordered triples of sets, which is a bit of set-theory bookkeeping not necessary for describing the concept from the thread "Formal definition - why not use the literature", I would like to address the aspect of defining a function consisting of domain X, codomain Y, and graph G - in literature, one can encounter two main approaches:

• the classic Bourbaki: a function is an ordered triple f=(X,Y,G)
• modified: function f is "something" containing X, Y, and G

At point 2, I wrote "modified" because it seems that before Bourbaki, the literature did not consider functions as three elements X, Y, G, and it was he who introduced the use of this. Later, some other authors allowed themselves to bypass the structure of the ordered triple - nevertheless, I have not encountered a justification anywhere (but this may be because I have not come across the right sources - so if someone has an article/book from before Bourbaki where X, Y, G are used, please quote it).

An ordered triple defined as an e.g. appropriate Kuratowski pair, for example, f=(X,Y,G)= ((X,Y),G)= {...} allows us to treat the function as a set - here are the advantages of such a construction:

• the function becomes an ordinary set, a well-known primitive concept (we do not multiply entities without necessity - Occam's Razor)
• the definition of the function becomes precise and unambiguous
• completeness - the triple is a sufficient structure to capture the concept of function - no need for more complex structures
• the set f can be directly applied with the well-known and developed apparatus/formalism known from set theory
• ease of working with functions at the definition level (we can use well-known operations on sets)

I would like to emphasize that the introduction of the triple structure by Bourbaki is not merely "bookkeeping" within set theory, but is a well-thought-out structure. A similar approach is applied in the formal definitions of other important mathematical concepts such as:: group, topological space, graph, etc.

However, with authors who did not use the triple structure, but define the function as "something" a separate entity, having domain, codomain, and graph, I have not found an explanation.

So, I pose this question: What advantages does the introduction of functions as entities, distinct from sets, bring? What does it contribute?

PS1: If it turns out that it does not contribute much, I would suggest using the structure of the triple in the article first (i.e., using the Bourbaki definition) and only secondly adding information that some authors use a definition in which the function is not a set but a separate entity containing X,Y,G (with citations). In this way, we communicate to the reader that there are different definitions based on the three elements: X,Y,G.

PS2: I assume that in the "Formal definition" section we will use Zermelo–Fraenkel set theory because it is the most popular, so the above considerations are only within this system. Though of course in another section of the article one can go beyond that. Kamil Kielczewski (talk) 06:52, 20 March 2024 (UTC)

The short answer to your question is WP:TECHNICAL. The long answer, is you have been going on and on and on about this and we have answered it over and over and over and it is time to WP:DROPTHESTICK. —David Eppstein (talk) 17:03, 23 March 2024 (UTC)

Good luck with that. Work badly done will keep bringing people complaining about it. As early as 2012 one can find people complaining about the same problem. The reason is as simple: A student finds a book where functions have codomains, and a book where they don't. Comes to Wikipedia to check which is which, and finds an article written so badly that does not clarify that both are used in the literature and it doesn't clearly state either one of the definitions.

And your "short answer" is disingenuous. Just a moment ago you were saying that claiming that "a set theory definition doesn't convey the dynamical nature of functions" is a phrase that belongs in this article. This claim is demonstrably false, when taken within the context and needs of this article, and extremely technical to properly justify. So much that even those who brough up the quote didn't have the tiniest idea what the quote was really talking about. Thatwhichislearnt (talk) 14:50, 24 March 2024 (UTC)

@David Eppstein I do not see an answer to the question asked either in the WP:TECHNICAL or in the comments. If you want to prove otherwise, please indicate the relevant quote and explain.

As for the definition of the tripe f=(X,Y,G), it is commonly used in literature (what has been noticed e.g. here p. 1158), so omitting it in the article would be a serious mistake. Doubts about this seem to have been mostly dispelled in the previous thread. In this thread, I just wanted to more broadly explain certain misunderstandings that arose in the course of that discussion. Kamil Kielczewski (talk) 06:52, 20 March 2024 (UTC)

Function evaluation

A mini edit war has started about the paragraph on function evaluation. This paragraph was confusely written, and I rewrote it boldy for clarifiation. Nevertheless, some problems remain, for which I am not sure of the best solution.

• Although very sketchy, this paragraph is too long for the lead. I am not sure of the part of it that belong to the lead and of the best place for details.
• Function evaluation redirects here. This deserves more than two lines in the lead. Maybe a specific article?
• Very often, "function evaluation" refers more properly to "expression evaluation". The confusion between these two concepts is the cause of many errors by beginners in the use of computer algebra systems. Expression evaluation is a redirect to Expression (mathematics)#evaluation that I created recently. Nevertheless, the treatment of evaluation (mathematics) remains insufficient and confusing in Wikipedia.
• Although very common, the vertical-bar notation for evaluation is not clearly defined in Wikipedia. I added recently a description of it at Expression evaluation, but this is still insufficient. This vertical bar notation, is a particular case of a restriction, if one consider $x=a$ as an abbreviation of $\{x|x=a\}.$ Maybe, a description of the vertical-bar notation for evaluation could be added at Restriction.

In summary, fixing this paragraph of the lead requires more work and more discussion, for which the tag {{cn}} is of no help. D.Lazard (talk) 17:39, 24 March 2024 (UTC)

The lead section of this article seems like an inappropriate place for the only discussion of this "vertical bar notation" which is not used anywhere else in the article. It's confusing, distracting, and only tangentially relevant. It should be put somewhere else in the article or in another article. Readers aren't going to look for it here, and readers who come across it here aren't going to find it valuable in context. –jacobolus (t) 17:43, 24 March 2024 (UTC)

One problem with this vertical bar notation is that it's hard to find clear sources describing its history/use. My guess is that it evolved out of the related notation for the bounds of a definite integral, as found in introductory calculus books/courses. There's a discussion of the earliest use (under "bar notation") in https://jeff560.tripod.com/calculus.html –jacobolus (t) 18:20, 24 March 2024 (UTC)

The use of "often" and "commonly" are unsourced and unnecessary. The second sentence has not one, but two parenthetical clauses. Yikes! The third sentence, unnecessarily restricts itself to talk about function evaluation to the case when the expression "depends on x". This, "depends on x", is ambiguous as an intuitive notion and it hasn't been defined in the article as a formal concept. Thatwhichislearnt (talk) 13:35, 25 March 2024 (UTC)

I agree that {{cn}} is not sufficient to elaborate on the vertical bar and evaluation, however, I consider(ed) is necessary - I apologize for my part of the "mini edit war" (I didn't intend to fight). I also agree that the vertical bar needn't be adressed in the lead, or even shouldn't be adressed there. However, it should be explained somewhere (not sure whether inside this article). And I, too, have seen the notation only in computations of definite integral [and in the recent versions of this article's lead].

Defining the vertical bar based on restriction appears to be one possibility, which also nicely motivates the notation. Another possibility would be to define it based on substitution (logic), i.e. $e\vert _{x=a}$ means the substitution application $e\{x\mapsto a\}$, or the evaluation of it. I'm not sure about an explanation of "expression evaluation". Maybe it should be based on (item "Functions" at) First-order_logic#Evaluation_of_truth_values, which defines evaluation of an expression in terms of evaluation of its constituent functions. (First-order logic doesn't care about how to obtain a function's result value for given inputs. If a function is defined by an expression in turn, its constituent functions (usually $+,-,*,/,...$) need to be evaluated, which leads to the standard arithmetical algorithms. However, a function may well be non-computable, and it may be impossible to obtain its result value. — In computer science, term rewriting systems are capable of defining every computable function. A subclass of them, the canonical term rewrite systems, guarantee that every function defined by them can be evaluated in finitely many steps to a unique normal form, i.e. a term that can't be rewritten any further; such terms are considered "values" in this setting. Arithmetical algorithms, for example, can be written as term rewriting systems that are canonical when applied to variable-free inputs, cf. "ground confluent".) - Jochen Burghardt (talk) 20:33, 25 March 2024 (UTC)

"Overriding (mathematics)" listed at Redirects for discussion

The redirect Overriding (mathematics) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 26 § Overriding (mathematics) until a consensus is reached. Tea2min (talk) 11:59, 26 April 2024 (UTC)

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)