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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)Reply

Thanks for stating this. The article seems to be aimed at graduate level or above and is completely incomprehensible to me. Jayarava (talk) 11:08, 9 January 2026 (UTC)Reply

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

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)Reply

Recently, Farkle Griffen re-added the following paragaph to the article

Strictly speaking, this does not fully describe a function in the usual sense, as there is no way to apply a function ${\displaystyle f}$ to an argument ${\displaystyle x}$. However, this definition is sufficient to be equivalent to a system with the ability to apply functions. Thus, one may use a conservative extension (called an extension by definition) to introduce a new primitive function symbol "${\displaystyle \bullet (\bullet )}$", as in "${\displaystyle f(x)}$", denoting function application. This function symbol takes in expressons denoting a set-theoretic function, "${\displaystyle f}$", and an arguent, "${\displaystyle x}$", then "${\displaystyle f(x)}$" denotes the unique object ${\displaystyle y}$ such that ${\displaystyle (x,y)\in f}$. That is, if "${\displaystyle f}$" denotes a function, and ${\displaystyle x}$ is in its domain, ${\displaystyle f(x)=y\iff (x,y)\in f}$. If ${\displaystyle f}$ is not a function, or ${\displaystyle x}$ is not in its domain, "${\displaystyle f(x)}$" denotes the empty set.

By this edit, they restart, without any discussion on the talk page, an edit war for which we were blocked fo 24 hours.

This paragraph is highly confusing by several aspects: It uses several concepts of logic that are not supposed to be known by readers of this article. Also, it start by talking of the "usual sense" of a function without saying whether this usual sense is the one used in standard set theory or the one used when a function is a primitive notion (such as in lambda calculus). It is only when a function is a pimitive concept that "application of a function" must be defined.

However, I must agree that previous version of the section lacks explanation how functional notation and usual jargon ("⁠${\displaystyle f}$⁠ maps ⁠${\displaystyle x}$⁠ to ⁠${\displaystyle y}$⁠", ...) are defined from the formal definition. So, I'll replace Farkle Griffen's paragaph with a short paragraph explaining this.

Please, do not change or remove it without providing strong reasons here. D.Lazard (talk) 10:35, 5 September 2025 (UTC)Reply

Hi, @D.Lazard. My biggest issue with the current version is that it doesn't really offer any explanation, and, under reasonable interpretations, is simply misleading.

if ${\displaystyle x}$ and ${\displaystyle y}$ are related by ${\displaystyle R}$, this is commonly denoted ${\displaystyle f(x)=y}$ instead of ${\displaystyle (x,y)\in R}$ or ${\displaystyle xRy}$.

If the whole string "${\displaystyle f(x)=y}$" is a kind of alternative notation for ${\displaystyle xfy}$, as in "${\displaystyle \bullet (\bullet )=\bullet }$" is "one symbol", then the string "${\displaystyle f(x)}$" isn't defined on its own.

I agree that mathematical logic is fairly technical, but that's simply how it's done. And I don't think relevant, foundational content should be removed on the basis of being "too technical", especially content with several sources and in a section titled "Formal definition". – Farkle Griffen (talk) 14:20, 5 September 2025 (UTC)Reply

then the string "⁠${\displaystyle f(x)}$⁠" isn't defined on its own: Fixed. D.Lazard (talk) 10:05, 6 September 2025 (UTC)Reply

"[...] the unique element of the codomain that is related to ${\displaystyle x}$ is denoted ${\displaystyle f(x)}$."

Hmm... But how are you doing this without functions? Are you just introducing a whole new symbol for every possible input? As in "${\displaystyle f(2)}$" is just "one symbol"? If so, you still don't have function application; you cannot have ${\displaystyle f(x)}$ for a variable x, or have an expression inside f(...). Both the input and the function need to be capable of being variables, e.g. in a differential equation, where functions can be unknowns.

Please respond to the last paragraph above in my previous reply. – Farkle Griffen (talk) 13:26, 6 September 2025 (UTC)Reply

I do not understand your last sentence, since, per the manual of style, an answer must always appear after the answered post.

"doing this without functions" is nonsensical in an article about functions in mathemtics and in a section about the formal definition of functions in terms of set theory. If you are talking of functions in other frameworks than set theory, it is in § In the foundations of mathematics that this must be discussed.

"Both the input and the function need to be capable of being variables": No; a variable is a symbol that represents (that is, is a name for) a mathematical object. In ⁠${\displaystyle f(x)}$⁠, there are two variables, ⁠${\displaystyle f}$⁠ and ⁠${\displaystyle x}$⁠, and the mathematical objects that they represent are defined in the preceding text. It is not a new symbol ("one symbol") that is introduced, this is a syntactic rule for forming an expression. As usual, when a variable appears in an expression, it can be substituted with any expression that can be evaluated to an acceptable value of the variable. D.Lazard (talk) 14:43, 6 September 2025 (UTC)Reply

""doing this without functions" is nonsensical in an article about functions in mathemtics"

This is a bit rude; you could simply ask for clarification. I am not talking about other foundations; I am talking about set theory. This is supported by all three sources I included in my edit.

By default, the language of ZFC only has variables, predicates, and logical connectives. There are no constant symbols, functions, or operations to create larger terms than an individual symbol. If you are asserting ${\displaystyle f(x)}$ is an expression that contains two individual symbols, then you are attempting to introduce an operation, which cannot exist in the initial language of ZFC. – Farkle Griffen (talk) 15:08, 6 September 2025 (UTC)Reply

To reply to the superscripted initial line, I was asking you to respond to this paragraph:

"I agree that mathematical logic is fairly technical, but that's simply how it's done. And I don't think relevant, foundational content should be removed on the basis of being "too technical", especially content with several sources and in a section titled "Formal definition"." – Farkle Griffen (talk) 15:22, 6 September 2025 (UTC)Reply

@D.Lazard? – Farkle Griffen (talk) 15:19, 7 September 2025 (UTC)Reply

Sorry, I am unable to decipher what do you mean with "how are you doing this without functions". I understand "doing this" as "defining functions", and, with this interpretation, the question is indeed nonsensical.

If you are asserting ⁠${\displaystyle f(x)}$⁠ is an expression that contains two individual symbols, then you are attempting to introduce an operation, which cannot exist in the initial language of ZFC: In other words, you assert that the integers ⁠${\displaystyle \mathbb {Z} }$⁠ and addition ⁠${\displaystyle a+b}$⁠ cannot exist in ZFC. If you means something else, you must be clearer.

However, as this page is not for discusssing editor's posts, but for discussing the article, I give below more reasons for removing the paragraph quoted above:

• "Strictly speaking, this does not fully describe a function in the usual sense": fixed by my edit.
• "there is no way to apply a function to an argument": wrong, as the application of a function to an argument is defined as the unique element related by the function to the argument.
• "one may use a conservative extension (called an extension by definition) to introduce a new primitive function symbol": In other words you say that you do not work in ZFC, but in an extension of ZFC. This is wrong, since ZFC allows the use of new symbols and new notations for abbreviating complicated logical expressions.
• "primitive function symbol": Apparently, nobody but you consider functional notation as a primitive of ZFC

Apparently, your paragaph is an attempt for defining function application. Function application is a binary operation that takes a function and an element of its domain to povide an element of its codomain. It is defined this way in the linked article, where the definition uses the fact that the set of all functions from ⁠${\displaystyle D}$⁠ to ⁠${\displaystyle R}$⁠ form a set denoted ⁠${\displaystyle R^{D}}$⁠. This definition does not requires any extension of set theory.

My opinion is that the view of function application as a binary operation deserves to be mentioned in the article, but this must be the object of a separate section. D.Lazard (talk) 16:28, 7 September 2025 (UTC)Reply

"with this interpretation, the question is indeed nonsensical."

You are assuming an interpretation that is nonsensical. That does not mean what I said is nonsensical, only that your interpretation is. Again, you can simply ask for clarification.

"In other words, you assert that the integers⁠ and addition ⁠⁠cannot exist in ZFC."

Not in the usual ZFC; or not that notation, at least. The symbols ${\displaystyle \mathbb {Z} ,0,1,-1,...,+}$ are not part of the language of ZFC. The logic of the integers can be modeled by ZFC, or a conservative extension can include them to get them explicitly.

"since ZFC allows the use of new symbols and new notations for abbreviating complicated logical expressions."

Strictly speaking, this is not correct. Introducing new symbols via a definition requires you to do an extension by definition. That is what that article is about.

There is no other way to get the function application notation to work as intended. This is all fairly standard, though Levy and Mendelson from the sources I added go into more detail if you need more convincing. – Farkle Griffen (talk) 16:57, 7 September 2025 (UTC)Reply

@D.Lazard? – Farkle Griffen (talk) 18:10, 8 September 2025 (UTC)Reply

@Farkle Griffen Can you explain a bit more about why you want to make the definition more complicated and technically detailed? To me personally, this seems like the wrong direction to move – if anything, we should be trying to make the early sections of this article about such a centrally important topic more accessible. If you think additional technical material is essential to mention somewhere, perhaps it can be moved further toward the bottom of the page, put into a footnote, or moved into a secondary article. –jacobolus (t) 18:43, 8 September 2025 (UTC)Reply

"[...] if anything, we should be trying to make the early sections of this article about such a centrally important topic more accessible."

I agree with this... The only issue is that what I'm trying to add is specific to set theory, and the only place that makes sense is right after the set-theoretic definition.

"Can you explain a bit more about why you want to make the definition more complicated and technically detailed?"

The motivation is kinda the opposite. Calling the set-theoretic definition the "formal definition" in the first section is confusing for less advanced readers, since it's not clear how to apply that function (and as it turns out, it's not super easy to get that). Honestly, I've never really come across the set-theoretic definition outside of introductory set theory books. Most authors try to immediately move to the usual notation. To your point, I wonder if the best choice is to move the set-theoretic definition to the bottom too.

I'll go ahead and make a restructuring edit so it is clear what I'm advocating for. Feel free to revert me or edit it if you disagree or think it needs more discussion. – Farkle Griffen (talk) 19:05, 8 September 2025 (UTC)Reply

If you think it's likely to be controversial, maybe this can be workshopped on the talk page instead of done as a revert war on the article? –jacobolus (t) 00:01, 9 September 2025 (UTC)Reply

More like, consensus through the talk page and edits. I don't think any administrator would call this a revert war if we're actively talking and I'm inviting you to do so.

If you disagree with my edit and don't want to revert me for procedural reasons (I guess?) I can self-revert if you want.

In any case, do you have an opinion? – Farkle Griffen (talk) 01:09, 9 September 2025 (UTC)Reply

I don't have a particularly strong feeling about it. Just as a general matter, if you think something is more likely to be reverted than left by other editors, it can be less trouble to discuss first. YMMV. –jacobolus (t) 01:43, 9 September 2025 (UTC)Reply

I think I would only disagree with this in the case of complicated edits. Restructuring involving a lot of section moves seems like a case of "show don't tell" for getting a point across. – Farkle Griffen (talk) 02:48, 9 September 2025 (UTC)Reply

If someone filibusters a talk page and, whenever everyone else gets too tired of responding to the same points, keeps reinstating the same previously-reverted content, then I would call it an edit war. —David Eppstein (talk) 02:12, 9 September 2025 (UTC)Reply

@David Eppstein I mean, I agree, but I don't understand how that relates to the comment you're replying to. The edit mentioned is not reinstating an earlier edit. Unless this is passive-agressively hinting at some other discussion unrelated to this one, I have no idea what this reply is saying. – Farkle Griffen (talk) 02:41, 9 September 2025 (UTC)Reply

This seems a bit harsh. Farkle Griffen has only made 5 or 6 not-outrageously-long comments here, and the "everyone" seems to just be D.Lazard. You might be mixing the discussion up with one on some other page / some past discussion here at Function (mathematics) involving different editors? –jacobolus (t) 02:43, 9 September 2025 (UTC)Reply

Unless if I missed some points, this restructuring consists mainly of:

• Renaming section "Formal definition" as "In set theory"
• Moving this section, § Partial functions and § Multivariate functions near to the bottom of the article
• Making section "Formal definition" a subsection of § In foundation of mathematics
• Making the two other moved sections as subsections of a new section § Generalizations

I stongly disagree with these changes:

• The formal definition given in the article appear in most elementary textbooks of algebra and is commonly taught to college students. So, it belongs to the beginning of the aticle, and more specifically to section § Definition
• Partial functions are soon encountered by many college students (for example, the logaithm and the tangent function are partial function of a real variable). As partial functions are often simply called "functions", they must be defined in section § Definition for avoiding reader confusion. Also, partial functions are not generalizations of functions; they are functions whose domain is not fully specified.
• I agree that multivalued functions are generalizations of functions. However, they are encountered soon in complex analysis (the complex square root and the complex logarithm are multivalued functions) and are also often simply called functions. So, they must be introduced soon in the aticle. However, the place of this section deserves to be discuted, and, before a consensus, it must be kept at the previous place.

So, since no other editor support fully these changes, which are not clearly motivated, I'll restore the previous version. D.Lazard (talk) 13:47, 9 September 2025 (UTC)Reply

The points are fair and the revert is invited. This is getting a bit off topic from the main discussion topic. I think I'll be willing to make a new discussion for a better organization for this article eventually, but my motivations are tied up in other articles at the moment.

In the meantime, @D.Lazard, can you respond to my last reply about the paragraph in question? – Farkle Griffen (talk) 18:07, 9 September 2025 (UTC)Reply

I will not respond to your last post about the opening paragraph: I gave the reasons for which I think that this paragraph is not suited for Wikipedia. You answered by essentially saying that I am wrong and that I do not understand basic mathematics. I have nothing to add to my reasons, and it is to other editors to arbitrate between us. D.Lazard (talk) 19:42, 9 September 2025 (UTC)Reply

@D.Lazard, I apologise if that's how it came off, that certainly wasn't my intention. However, mathematical logic is not basic math. It's an unfortunate state of affairs that mathematicians tend to just hand-wave away foundations, but that seems like the status quo.

This doesn't need to be a black and white discussion. We can discuss how to phrase the paragraph, rather than "me vs you". I think any formulation that explains, roughly "function application is a binary function symbol: ${\displaystyle \bullet (\bullet )}$." would be good enough for me. Unless you're not willing to discuss at all? – Farkle Griffen (talk) 20:34, 9 September 2025 (UTC)Reply

This seems like a pedantic and confusing addition. The notation with the dots might not be understood by a significant proportion of readers, and the term "binary function symbol" is obscure and a search for «"binary function symbol" "function application"» turns up mostly non-relevant uses and uses in specialized papers about symbolic logic and lambda calculus (where it doesn't refer to the common function notation, but to some specialized new symbol used as a binary operation, so that function application is explicitly written like ⁠${\displaystyle f\star x}$⁠ or the like instead of ⁠${\displaystyle f(x)}$⁠). I think discussion along those lines can probably be saved for more specialized articles about those topics. The current text doesn't seem especially ambiguous or misleading. To quote:

The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter f. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in $${\displaystyle f(x),\quad \sin(3),\quad {\text{or}}\quad f(x^{2}+1).}$$

The argument between the parentheses may be a variable, often x, that represents an arbitrary element of the domain of the function, a specific element of the domain (3 in the above example), or an expression that can be evaluated to an element of the domain (${\displaystyle x^{2}+1}$ in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let ${\displaystyle f(x)=\sin(x^{2}+1)}$".

When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin(x).

–jacobolus (t) 22:39, 9 September 2025 (UTC)Reply

[...] turns up mostly non-relevant uses and uses in specialized papers about symbolic logic and lambda calculus

Symbolic logic, specifically first order logic, is indeed the correct context. It is the context which ZFC is defined.

Maybe I'm just suffering from being on the other side of it, because I honestly don't understand what's confusing. But okay, I agree it's pedantic, but there has to be some way to to make it less confusing, right?

Specifically, the audience I am trying to write this for is readers trying to understand foundations. If a reader learns that "technically, ZFC does not have functions" and wants to learn how to get back the usual notation, the first place they're likely to check is Function (mathematics) § Formal definition. The misleading bit with what you quoted is that it gives no hint that it is not completely formal, or that there is a canonical, formal way to do it.

I think discussion along those lines can probably be saved for more specialized articles about those topics

I mean yeah, I agree, and that's fine. It is already partially there at Function application § Set theory. I'm not asking that this article does a deep-dive into formal logic, just that it actually points to the specialized articles for interested readers.

Are you willing to at least let me workshop it? – Farkle Griffen (talk) 23:44, 9 September 2025 (UTC)Reply

Side remarks: you wrote "first-order logic, is indeed the correct context. It is the context which ZFC is defined". The lead of the linked article as well as Higher-order logic#Quantification scope imply that ZFC is not a first-order logic.

Also, the sentence "technically, ZFC does not have functions" is highly confusing, since it is true if interpreted ZFC does not have functions among its primitives and it is wrong if interpreted as "ZFC does not allows a technical definition of functions". D.Lazard (talk) 00:57, 10 September 2025 (UTC)Reply

@D.Lazard, I can see how those would be very confusing, however ZFC is always first-order. The first line of Zermelo–Fraenkel set theory#Formal language says explicitly "Formally, ZFC is a one-sorted theory in first-order logic." as well as any book on axiomatic set theory. There are second- and higher-order set theories, but they are not ZFC.

Higher-order logic#Quantification scope definitely needs to be rewritten for clarity. This is confusing because in ZFC the "individuals" are also called "sets". However, the "sets" in that article refer to sets external to the theory, not internal ones. It should say something more like "Second-order logic allows for the quantification over all subsets of the domain of discourse. Third-order logic allows for quantification over all sets of subsets of the domain of discourse, and so on." – Farkle Griffen (talk) 02:07, 10 September 2025 (UTC)Reply

@Jacobolus? – Farkle Griffen (talk) 22:52, 10 September 2025 (UTC)Reply

Personally I'd look for definitions of function found in textbooks aimed at undergraduate students (e.g. on intro analysis, abstract algebra, discrete math, or "first course in pure mathematics" kinds of courses), rather than looking for definitions from niche monographs aimed at professional logicians or whatever. YMMV. –jacobolus (t) 00:08, 11 September 2025 (UTC)Reply

We have an article History of the function concept which might be more appropriate for a deep dive into the many possible alternatives. –jacobolus (t) 00:10, 11 September 2025 (UTC)Reply

@Jacobolus, you're right, I should have been clearer: the first place on Wikipedia. But I digress. Can you respond to the last bit of my response there? To quote:

"I'm not asking that this article does a deep-dive into formal logic, just that it actually points to the specialized articles for interested readers. Are you willing to at least let me workshop it?"

I'm a bit confused by:

"[...] into the many possible alternatives"

I agree there are alternative foundations, but I don't understand how it applies to the conversation. Assuming § Formal definition is based in ZFC, what I wrote is the canonical way to introduce function application. – Farkle Griffen (talk) 00:34, 11 September 2025 (UTC)Reply

I think "formal definition" near the top here here should be a formal definition as found in a highly cited source aimed at early undergraduates (as compared to an "informal" description aimed at an audience of 12–16 year old schoolchildren). If something more elaborate or a deeper discussion than that is required, it could be put in a footnote, further down the page, or relegated to some other article. –jacobolus (t) 01:04, 11 September 2025 (UTC)Reply

@David Eppstein, I'm using this post since it seems related enough.

What exactly is the goal of this revert? If the goal is to undo the typo, then why not just fix the typo? If the goal is to make sure the notation ${\displaystyle f(x)}$ is mentioned, that notation is already introduced in section § Notation. I don't understand your edit summary. The current formulation in a section titled "Formal definition" seems very misleading. – Farkle Griffen (talk) 23:40, 10 September 2025 (UTC)Reply

The goal was simply to undo a change that did not appear to me to be an improvement. It was not part of any broader agenda (such as how formalist we should be or whether function objects come equipped with domain and codomain sets) and I think my edit summary stated the same thing clearly enough. Supplying the missing verb would only have made sense if I were sure which verb you intended and if I thought that the fixed change would be an improvement, neither true in this case. —David Eppstein (talk) 23:45, 10 September 2025 (UTC)Reply

"[...] if I thought that the fixed change would be an improvement, neither true in this case."

Can you explain this a bit more? I still don't think I understand. Lets say I fix the typo. Your reply seems to imply you would still revert me. Can you explain why? – Farkle Griffen (talk) 23:51, 10 September 2025 (UTC)Reply

Ok, since you demand a longer and more in-depth explanation than the edit itself and my gut reaction to it, then the answer is, more or less, WP:USEPROSE. Your edit took a three-sentence paragraph with lots of concrete nouns and replaced it by "If $$$ is a function defined by a functional relation $$$, and $$$ or $$$, and one says that $$$ maps $$$ to $$$, or $$$ is the image by $$$ of $$$." Where are all the words? It's too formula-heavy to be readable. Now maybe what was there before had some of the same issues but your edit made it worse. I don't think it's reasonable to demand that I spend a lot of time and effort defending my choice to undo edits that in my opinion are not improvements. Put more of your energy into crafting your content better and less on arguing about it, please. —David Eppstein (talk) 00:04, 11 September 2025 (UTC)Reply

Okay, I was sloppy and should have been more careful. Point taken.

I am honestly trying to discuss. My point in moving to the talk page is not to argue to reinstate my edit, but to understand your issues with it so my next edit can be an improvement. I apologize if that is not how it comes off. I really am trying here.

TBH, the whole paragraph as phrased seems off-topic in the section § Formal definition, and is mostly repeated in the section § Notation. The current paragraph seems redundant at best, and outright wrong at worst. My motivation is just to make it less misleading. Do you have an objection to simply deleting that paragraph? I can't accidentally introduce typos if I do that. – Farkle Griffen (talk) 00:16, 11 September 2025 (UTC)Reply

Yes, I object to the removal of an explanatory paragraph. It doesn't have to be that specific explanatory paragraph. But the rest of that section merely puts together some abstract mathematical objects. The paragraph is needed to explain how those objects match our less-formal intuitions and notations about what a function is supposed to do (namely, that it takes inputs and produces outputs, rather than just sitting there inertly as a set of pairs of things). —David Eppstein (talk) 01:48, 11 September 2025 (UTC)Reply

@David Eppstein, Okay, that makes sense. The paragraph reads a little better now after the most recent edit, which, given it still stands, you don't oppose?

Taking in jacobolus's opinion above, would either of you object to adding just one sentence at the end of that paragraph? Roughly:

"A more formal definition of function application can be given in terms of first-order logic (cf. Function application § Set theory)."

Satisfying jacobolus's request (hopefully?), all information is delegated to the main article, and this article would merely point there. No possibly confusing notation is added or explanation removed, hopefully satisfying your conditions. And the interested reader would have clear indication of where to find it. Does anyone oppose this? – Farkle Griffen (talk) 17:58, 11 September 2025 (UTC)Reply

Is the formalization of syntactic sugar important to direct readers to? Why? For the formalization of functions themselves, and the syntactic sugar that makes the formalization understandable, I can see the value. But why do readers care about formalization of syntax? —David Eppstein (talk) 18:08, 11 September 2025 (UTC)Reply

You're probably right that ~90% of readers don't care about foundations. But the last ~10% is who I'm writing for. It's kind of hard to explain concisely, but the basic gist of it is, it's not just syntax or syntactic sugar. What you're getting from first-order logic isn't just a "notation" for functions, but real, bona fide function application. And trying to formalize it any other way can result in genuine foundational issues. – Farkle Griffen (talk) 18:51, 11 September 2025 (UTC)Reply

I think you missed my point. The readers who care about foundations of functions can find them in the section already existing here. The readers who care about foundations of syntax can read other articles, like formal language; the details about formalization of the syntactical sugar used for functional notation are not really specific to functions, not really about functions per se, and I think not relevant to most readers here. —David Eppstein (talk) 20:12, 11 September 2025 (UTC)Reply

If I did, I don't think I understand your point still, because I feel like my response just above mostly responds to this. In slightly stronger language (without the intention of being rude) :

The definition is not about syntax. If you treat ZFC as a set theory, and not some uninterpreted formal language, then the definition not just a notation but literally function application. It is how to actually take the set-theoretic definition of "function" and introduce real function application. Without it, as you said, "functions" are just static sets of ordered pairs.

It is not about the foundations of syntax. It is exclusive to set theory, and to this specific set-theoretic definition of "function". – Farkle Griffen (talk) 20:47, 11 September 2025 (UTC)Reply

Who does not treat Zermelo–Fraenkel set theory (ZFC) as an uninterpreted formal language that is not a set theory? Otherwise, I cannot understand this post (what does mean "literaly" here? what is a real function application?, etc.) D.Lazard (talk) 21:05, 11 September 2025 (UTC)Reply

In mathematical logic, you work with formal theories (such as ZFC), which don't necessarily have to be given an interpretation. You would just work with the symbols and transformation rules. In this sense, none of the formulas in ZFC "mean" anything. If you give the theory an interpretation, i.e., the individuals literally represent sets (in the same way the word "egg" represents a real egg), then function application as defined is the usual function application, as you use it.

That is to say that definition is literally function application, insofar as ZFC is literally about sets. – Farkle Griffen (talk) 21:20, 11 September 2025 (UTC)Reply

If you work in a world like ZFC where everything is sets, then everything is sets: function application is just what we call certain usages of certain sets. There is no need to formalize that something is "literally" function application: that is just our interpretation of certain arrangements of set operations. To say that something is "literally" function application you need to be working in a world where there is a class of mathematical objects that are "literally" functions and their application, and not merely sets that we interpreting as those things. A world such as, say, type theory. But then if you have functions built into your foundation as first class objects, there is no longer a need to specify how to formalize them as something else. As far as I can see, the need to both have literal functions and to have a formalization of those things in sets only comes up in specialized occasions when you are showing how to interpret one formalization in another. —David Eppstein (talk) 21:43, 11 September 2025 (UTC)Reply

I don't think I understand that last sentence, but I don't really see how this objects to my previous reply to you above at all.

What I originally wrote in the article is well-sourced as being the definition of "function application" in ZFC for that particular definition of function. Unless you have a source saying that is not "real" function application, I don't see an issue with taking the authors at their word, beyond philosophical differences. – Farkle Griffen (talk) 22:01, 11 September 2025 (UTC)Reply

https://us.metamath.org/mpegif/df-fv.html includes 1 and links to 4–7 alternative definitions of function application. I think all of them are too technical for anywhere in the top several sections of this article. Maybe you can add something in § In the foundations of mathematics (perhaps with details in a footnote) or at History of the function concept, or perhaps it might be relevant somewhere in Zermelo–Fraenkel set theory. Edit: Function application is probably the most relevant place to delve into this. –jacobolus (t) 03:49, 12 September 2025 (UTC)Reply

Huh, interesting... following the sources, a couple link to Takeuti & Zaring, Monk, and Enderton, which use NBG and class comprehension to do it, and a couple use the foundations in Principia Mathematica, so definitely not ZFC.

The one using the union operator is super contrived but technically would be usable in ZFC. It doesn't link to a source though, unfortunately.

Cool link! I haven't seen that before!

I think all of them are too technical for anywhere in the top several sections of this article.

I agree. My only motivation is to mention it somewhere in the article. I'm happy to work in § In the foundations of mathematics if it is accepted there.

In response to your edit, I agree. I really don't want to delve into anything here, just a pointer to the main article so interested readers know where to find it. – Farkle Griffen (talk) 04:23, 12 September 2025 (UTC)Reply

I oppose adding A more formal definition of function application can be given in terms of first-order logic (cf. Function application § Set theory). for several reasons.

• This section is about the definition of functions, not of function application.
• "More formal" does not apply since function application is not defined previously.
• The definition of functional application given in the linked article involve only set theory, not first-order logic.
• This definition uses the definition of functions and is exactly as formal as the definition of functions that is given here.
• Wikipedia is not a source, especially when the linked article has many issues (in particular here, where ⁠${\displaystyle f(x)}$⁠ is defined when ⁠${\displaystyle f}$⁠ is not a function or ⁠${\displaystyle x}$⁠ is not an element of the domain of ⁠${\displaystyle f}$⁠; applied to the square root function, it defines ⁠${\displaystyle {\sqrt {-1}}=\emptyset }$⁠).

D.Lazard (talk) 20:49, 11 September 2025 (UTC)Reply

@D.Lazard, To respond to the last point, "Wikipedia is not a source...", my original edit contained three sources, which included that fact. The choice of the empty set is by convention, and due to certain technicalities in how function symbols need to work in general. Mendelson and Levy talk about this quite explicitly, and every other source I found uses that convention.

It doesn't really cause issues further up the chain of abstraction, aside from some "technically true but weird" trivialities like that one. – Farkle Griffen (talk) 20:59, 11 September 2025 (UTC)Reply

The definition of the function symbol ${\displaystyle X(Y)}$ does not define ${\displaystyle {\sqrt {-1}}=\emptyset }$. They are different symbols. It is very hard to this article to ever get to be properly written, when the self-appointed owner cannot do formal mathematics. ~2026-12778-52 (talk) 15:02, 26 February 2026 (UTC)Reply

I said it many years ago, the arguments in the talk about exactly the same point of the definition(s) of function will keep happening over and over again. How many years is it now? 5? 10?. There are multiple, non-equivalent, definitions of functions in the literature. That is the reality. The article should present them both, and acknowledge that they are not equivalent. Currently, and it has been like that since Lazard needs entertainment after retirement, there are two paragraphs in the section of Formal Definition, that follow one after another, that pretend to be one a more formal version of the other, but that define non-equivalent concepts. Hilarious. ~2026-12778-52 (talk) 15:02, 26 February 2026 (UTC)Reply

The current definition states

A function f from a set X to a set Y is an assignment of one element of Y to each element of X.

First off, there might be elements of Y that are left unassigned. This would still be a function, just not surjective. Second, in general, we assign more than one element of Y to some value of X. Jollykk (talk) 21:59, 21 August 2026 (UTC)Reply

First, where in the cited sentence do you see a claim that f must be surjective?!

Sencond, what you describe unter "in general", is called a multi-valued function, and is handled under Function_(mathematics)#Multi-valued_functions. You have a point: this section should state more clearly that they are a proper generalization of ordinary function. - Jochen Burghardt (talk) 06:09, 22 August 2026 (UTC)Reply

Because the cited sentence states "is an assignment of one element of Y", which means that all elements of Y must be assigned? One and only one element of Y is assigned? At least one element of Y is assigned? What "A function f from a set X to a set Y is an assignment of one element of Y to each element of X"? And what does it mean "to each element of X"? That an element of Y can be assigned to more than one element of X ("each element of X")? That also contradicts the definition of function because each element of X must be assigned to one and only one element of Y.

The common definition of function is that each element of X is assigned to one and only one element of Y. The Wikipedia definition I cited above turns it the other way around. Jollykk (talk) 06:52, 22 August 2026 (UTC)Reply