Talk:Variable (mathematics)

Changes in the lead

An editor is starting an WP:edit war for changing the lead. Here are some reasons of the revert.

• The change of the order of the paragraphs makes the lead confusing, since in the new version, "This is in contrast to ..." refers grammatically to the sentence on mathematical logic.
• There is a WP:SUBMARINE link to domain of discourse, which is a philosophical concept not used in modern mathematics.
• The source is more 100 years old and the edit does not take into account that mathemaical logic has dramatically evolved since Russel's time.
• Changing "a mathematical object" into "an arbitrary object in some specified domain" is a clear disimprovement: a variable may represent a matrix, which is clearly a mathematical object, but one can define A as a variable representing a matrix without specifying the domain set (called "domain" in the edit) to which the represented matrix belong.

These reasons explain why I'll reverting again this edit. D.Lazard (talk) 14:37, 23 August 2024 (UTC)

I agree with your comments, D.Lazard. There are also other problems with the editing you refer to; for example "Some authors also consider more abstract objects to be constants, for instance and [sic] the identity element of a group" misses the point: since a group has only one identity element, if I use a symbol meaning precisely that one element and no other then that symbol is a constant, just as much as a symbol used to represent precisely one number is a constant, or a symbol to represent precisely one object of any other kind. Any author who doesn't "consider" that to be so would, it seems to me, be using the word "constant" in a very unorthodox sense; or have I misunderstood? JBW (talk) 21:50, 25 August 2024 (UTC)

That's a fair objection. But the meaning of "an identity element of a group" is meant more abstractly, not as any specific construction of a group.

For instance, take the groups (1) addition over the integers, and (2) multiplication over the integer powers of 2. These two groups are isomorphic, so I might define the group that describes them both. Here, the identity element can technically denote either 0 or 1, depending on the construction but some may consider in the abstracted group in isolation that the identity element is itself a constant. In the same way that 0 and 1 themselves do not denote any particular construction of 0 or 1 objects. Farkle Griffen (talk) 22:51, 25 August 2024 (UTC)

The edit has been made in accordance with what you say here. Specifically,

• no paragraphs have been moved
• no link to “domain of discourse” has been added
• modern sources are used
• only a small piece of the definition was changed to clarify the topic

The reason for the edit: One should make clear the difference between variables, constants, and general symbols in mathematics. Referring to constants as a kind of variable is confusing for most readers and not how most people use those terms.

A note on the domain of discourse article: it should be noted that it is listed as a mathematics article; it is introduced as “In the formal sciences” which mathematics is a part of, and links to the articles related in mathematics (such as Universe set); and uses mathematical examples to describe the topic. So it really shouldn’t be written off so quickly as “a philosophical concept not used in modern mathematics.” In regard to your example, you have exactly specified the domain set; that is the set of matrices. Farkle Griffen (talk) 22:37, 25 August 2024 (UTC)

@Farkle Griffen: I agree with your comment that "Referring to constants as a kind of variable is confusing for most readers and not how most people use those terms." I have previously expressed my disliking of referring to them in this way. As for "the domain of discourse", I partially agree with your comments, but the expression is not currently in common use in mathematics (in fact I'm not aware that it is currently in use in mathematics at all), and also the article goes into the topic in ways which are not really relevant to the content of the article on variables; for both of those reasons I don't think that linking to it is likely to be helpful to most readers. JBW (talk) 14:58, 26 August 2024 (UTC)

I agree the link could be better, but Wikipedia doesn't seem to have an article about the domain/range of a variable in mathematics, and that was the closest article I could find. Linking to domain of a function would be even more confusing. It's possible that one could be created, but I don't think that's necessary.

In many of the sources I've recently found (which user: D.Lazard appears to have removed), they often use the term range of a variable, rather than domain, to refer to the set of constants it can represent. I think a sentence could be added to introduce this term. Farkle Griffen (talk) 15:16, 26 August 2024 (UTC)

D.Lazard's revisions

User D.Lazard continues to revert the article to a version that is not supported by any sources, and contradicts all sources that define the term. Enforcing the definition "a symbol that represents a mathematical object", and asserting that constant symbols are variables.

Source 1) "A variable a symbol representing an unspecified element of a given set"

Source 2) "A variable is a symbol that holds a place for constants."

Source 3) "We use the term variable in keeping with the usage of Collis (1975), Küchemann (1978), and a number of other researchers, such as Philipp (1992), who terms it a varying quantity. When a letter is used as a variable, the letter is seen as representing a range of unspecified values, and a systematic relationship is seen to exist between two such sets of values” (Küchemann 1981, p. 104)."

This author distinguishes between variables and placeholders, using them to mean what this article calls free variables and bound variables respectively.

They define placeholders as "We use the word placeholder to mean a letter standing for a number that will be provided in a particular problem or context. A placeholder is often called a given or a constant; in specific instances it is a parameter or a coefficient. Like a variable, a placeholder is indeterminate, but whereas a variable can stand for an entire set of values, here the point is that the equation or expression really stands for an entire set of equations or expressions. For instance, in the equation “ax² + bx + c = 0,” a, b, and c are placeholders (in particular, coefficients). This equation in fact stands for an entire set of quadratic equations (if a≠0), and it is understood that in a specific context these letters will be replaced with specific numbers"

To D.Lazard's edit summary: "The removed paragraph does not says that constant are variables, but the letters that denote them are variables."

None of the sources say that a variable is simply a symbol. All of them are very clear a variable has the property of being, at the moment, unspecified. Constant symbols are not usually considered "unspecified". While, as he has noted, the symbols for pi and e have been used as variables, this does not mean that they are always used as variables, it is just that some contexts use the symbol as a variable, while others use it to denote a particular constant. Constants and constant symbols are not variables, and this should be made very clear. Farkle Griffen (talk) 12:27, 28 August 2024 (UTC)

@D.Lazard, please do not edit war. I am trying to engage with you outside of the article space, but you have not responded. Farkle Griffen (talk) 18:30, 28 August 2024 (UTC)

@D.Lazard, I have added four more sources, totaling 7 sources which very clearly state that a variable represents an unspecified constant. None of which support "variable" denoting a particular constant. Do you have any sources supporting this definition? Farkle Griffen (talk) 06:05, 1 September 2024 (UTC)

@D.Lazard, Have you considered that the term may be used diffrently in English versus French? Farkle Griffen (talk) 16:57, 1 September 2024 (UTC)

@D.Lazard, I have found a source, ISO 80000-2: https://www.iso.org/standard/64973.html, which explicity notes e, pi, and i as not variables. It is for this reason I will editing the article to be more inline with the sources. Farkle Griffen (talk) 04:33, 5 September 2024 (UTC)

ISO is not a reliable source for mathematics. In particular it promotes roman fonts for the mathematical constants e and i, while in Mathematics and Wikipidia, the common convention is italics (see MOS:MATH#Roman versus italic). Also, for saying that e and i are or are not variables, you must be clear whether you are talking of the symbol or the number. D.Lazard (talk) 08:23, 5 September 2024 (UTC)

Thank you for responding.

While you may disagree about the validity of the source, you have not provided a single source. While other conventions exist, the purpose of that source is not about the italicization, but rather it is a source which explicitly notes that the symbols e, pi, and i, when denoting constants, are not considered variables.

I have asked you to provide a source, I have provided 7 sources (now 8), I have left two talkback requests on your talk page, and all of my edits have been in accordance with your previous discussion. If you revert my edit again without supplying even a single source, or responding to each of my sources showing how, (as you continue to say in your edit summaries) "I am misinterpreting them", I will be reporting this as WP:Edit warring behavior.

If you disagree with what I have written, you are free to edit it. However, you should not imply that the symbols denoting constants or constants themselves are variables without doing one of the options mentioned above.

Thank you. Farkle Griffen (talk) 13:43, 5 September 2024 (UTC)

On the description of "unknown"

The article currently seems to describe the unknown of an equation as a kind of bound variable, but this interpretation is not always true. For instance, one may regard the unknowns in an equation to be free variables, we call the "solutions" the values of the variables that make the equation true, and "Solving an equation" refers to finding an equivielnt formula that makes the solutions more obvious. For instance, the formula $x^{2}-3x+2=0\iff (x=1{\text{ or }}x=2)$ is a tautology.

I'm not too upset if the current description doesn't change though, it is just something I'd like to note. Farkle Griffen (talk) 17:26, 28 August 2024 (UTC)

The unknowns of an equations are variables such that the value is unknown before solving. Solving an equation is not to find an equivalent formula; it is to find the value(s) of the variable that satisfy the equation. Also the above equivalence is not a tautology; it is a theorem whose proof is a very simple computation in one direction, but is less obvious in the other direction (for proving that there are no more solutions). D.Lazard (talk) 17:38, 28 August 2024 (UTC)

A tautology is a formula which is always true. If you plug in any real number for x, that formula will be true. Needing a proof does not mean it's not a tautology.

"Solving an equation is not to find an equivalent formula; it is to find the value(s) of the variable that satisfy the equation."

These two statemnets are not necessarily disctinct. One just needs specify the form the formula must be in to consider the equation "solved". Farkle Griffen (talk) 17:54, 28 August 2024 (UTC)

We are talking of elementary mathematics and not of formal logic. "Tautology" is a term that is not used in mathematics, except in mathematical logic. So I understood the term as in tautology (language). Even if one use the sense of tautology (logic), your example is not a tautology, since the formula becomes false if x interpreted as a square matrix. D.Lazard (talk) 09:29, 29 August 2024 (UTC)

I am talking in terms of mathematical logic. "Free and bound variables" is not common terminology outside of mathematical logic.

And you're kinda missing the point here. This conversation isn't meant to be about the definition of "tautology". My point is that it doesn't make much sense to define unknowns as a kind of bound variable.

Nevertheless, I have found a source that specifies exactly what I'm saying here... [R. Wolf] A Tour Through Mathematical Logic, p.17, specifies that in basic algebra, the unknowns in an equation are free. Farkle Griffen (talk) 20:22, 8 September 2024 (UTC)

The term "bound variable" is not presented in the article as being related to the unknown of an equation, and nobody but you use this term in the context of equation solving. D.Lazard (talk) 19:56, 9 September 2024 (UTC)

Edit revert

@D.Lazard: re [1]. "Variable (mathematics)" is not ambiguous. "Mathematics" is not "computer science". No other use at disambiguation page Variable refers to mathematics. The hatnote is not helpful, and is not required per WP:NAMB. Please further explain. Shhhnotsoloud (talk) 15:32, 11 November 2024 (UTC)

Maybe the distinction between these two concepts is clear to you, but certainly not for most readers. This is why a hatnote is useful. However the template {{distinguish}} is more adequate in this case, and I have changed the hatnote for using it. D.Lazard (talk) 16:05, 11 November 2024 (UTC)

Article priority for Statistics project

Why is this of High‑importance to the Statistics project? Relevant mentions here are cursory and there is a separate Random variable article. I'm wondering if the latter article grew from this one, so the rating was once more relevant? NeilOnWiki (talk) 16:24, 5 January 2025 (UTC)

What's the best way to improve this article?

The history shows that there was a surge in editing in 2024; yet somehow this article seems to have left 2024 in a worse state than it entered it. Much of the reason for this stems from edits in August/September, which sparked a potentially bruising and energy-sapping sequence of reverts plus their accompanying discussion threads. I'd like to find a way to fix this and wading-in repeating the same reverts isn't the answer I want to choose, because reverts and unreverts are so binary. There are reasons why the editors involved did what they did and wholesale reverts wouldn't really capture those reasons.

If this article were a physical document, then we might highlight specific areas with a marker pen and tackle each one by one. Assuming Wikipedia has no nifty built-in tool for emulating this, perhaps one way forward might be to re-revert the edits which are in dispute (such as [2]) and for the editor(s) to take a more incremental consensus-seeking approach by tackling the issues that they've identified one-by-one, treating each as a separate edit.

In no particular order, and so far as I understand them, the main issues include:

• domain/range of a variable;

• variable vs. constant vs. symbol;

• free variables and bound variables (there may be a misunderstanding involved);

• probably others I've missed;

• plus questions suggested implicitly by the relevant discussions; eg. whether a variable is a mathematical object, and if not (or not always), why would it be wrong to describe it in terms from other disciplines such as philosophy (or perhaps linguistics)?

NeilOnWiki (talk) 09:08, 6 January 2025 (UTC)

@Farkle Griffen: would you be willing to revert your edits in their current form and adopt a slower more stepped approach, perhaps taking WP:CAUTIOUS into account? Another strategy might be to allow time for a smaller edit to bed in before moving to a new issue. I'm suggesting all this as a person who hasn't really witnessed this kind of situation that much, so the solution I'm suggesting may be totally outrageous. If so, there may well be other more tried and trusted ways of improving it (which hopefully aren't too brutal!).

I'd agree that the article in its newly reverted form will still require improvement, and have my own thoughts on the content of its incarnations past, present and future; but I felt that concentrating on method rather than content might be more productive in these opening comments.

NeilOnWiki (talk) 09:14, 6 January 2025 (UTC)

I agree. I stopped the edit war because I was tired to try to convince the other editor. Reading the article again after a while, it appears the the edit war resulted in a better (and longer) lead than previously. Nevertheless it has the following issues:

• A use of "constant" in the first paragraph that seems contradicts the paragraph on constants.  Fixed
• A circular definition in the last paragraph of the lead ("a variable is a variable"). Fixed
• The paragraph on constant is controversial and difficult to understand for most readers. I suggest to replace it with It depends on the authors and the context whether the symbol representing a constant is a variable or not. A discussion could begin here in view of a consensus for expanding this sentence. Nevertheless I prefer the the treatment of constant in the last version reverted by Farkle Griffen.
• Too many sources for the first sentence.

In any case, I would agree with the revert you suggest. D.Lazard (talk) 12:24, 6 January 2025 (UTC)

@D.Lazard, I think your most recent edits were improvments, however, on the topic of your third bullet point, I would oppose adding that sentence to the lead. Based on the current sources, it would be more accurate to say "It depends on the authors and the context whether a symbol represents a constant or a variable" so that we don't accidentally imply that constants (or the symbols denoting them) may be considered variables.

I can remove some of the sources, as you suggest. – Farkle Griffen (talk) 19:53, 6 January 2025 (UTC)

@NeilOnWiki, I'm happy to help, but what exactly are you asking me to do? Though there were lots of edits, mosts of them were reverted/reverts. My total addition to the article only really accumulates to about a paragraph's worth of change, and that was only really to remove insinuations that pi and e are variables even when denoting constants. I don't think undoing that change would be an improvement.

I agree that, in general, this article needs work, but could you be more specific about what exactly you are unhappy with?

To reply to your last bullet point, it wouldn't be totally correct to say "variables are mathematical objects" in the sense that mathematics doesn't study variables, it only uses variables as part of its language. The difference there is analogous to the difference between eggs and the word "eggs". This is the distinction between an object-language (a language that talks about objects) and a meta-language (a language that talks about a language).

The object-language of mathematics, usually ZFC, does not contain any notion of "variables", only sets; so in a strict sense, no, variables are not "mathematical objects". However, a meta-language of mathematics, such as mathematical logic, does talk about variables. Thus it would be more accurate to say that variables are part of the language of mathematics or that variables are a meta-mathematical object. But if one considers mathematical logic as a bona-fide branch of mathematics, then in that sense, one could say variables are a mathematical object.

I don't think any of this is relevant for the lead, unfortunately. – Farkle Griffen (talk) 19:39, 6 January 2025 (UTC)

Unfortunately, I tend to be slow in replying and it's often not as fully as I'd like. When you ask "what exactly are you asking me to do?", I'm extremely grateful for exactly what you've done so far by replying with your willingness to help. We all want to improve the article.

You've also described something very close to my own view with your very clear analysis of whether a variable is a mathematical object. If "variables are part of the language of mathematics", rather than mathematical objects in their own right, then it affects both how we should consider the lead and the constant/variable question. There's a tension between definition and usage, not helped by the suggestiveness of the historical term "variable": the language of mathematics is much messier than we might wish.

Thanks for your edit introducing the range of a variable. Your making it a small localised edit made it easier to tweak and I hope what I've written takes into account both your own concerns and those of D.Lazard (talk · contribs) : both had validity. I returned object to singular, since I think this reads better - and your helpful extracts from the sources refer to an individual. It's expanded for a wider readership, which is a personal obsession. More importantly, I think there's a virtual symbiosis between doing this and the collisions of opinion encountered when trying to deal with the evident messiness of linguistic usage of the word 'variable' in mathematical discourse. NeilOnWiki (talk) 10:02, 7 January 2025 (UTC)

This page needs better sources and organization for the body so that the lead can summarize unambiguous and well-sourced material. Currently, there could not be clearly-grounded consensus on content because too much of the article is unsourced (and is probably not WP:DUE).

It might be effective to first establish a talk page consensus of some list of good, widely-available sources for this article that would cover the most important contemporary pedagogical schools of thought on what a "variable" is. RowanElder (talk) 02:38, 7 January 2025 (UTC)

I agree to list sources and discuss them and discuss them, However, this is not an article on pedagogy; it is an article on mathematics. "Pedagogical schools of thougnt" would be interesting here only if this article would have a section on variables in education. What is needed here, is how variables are defined in major textbooks on algebra and calculus. D.Lazard (talk) 08:19, 7 January 2025 (UTC)

I am suggesting exactly to find "major textbooks on algebra and calculus" but then thinking one step past that to "how many major textbooks and why." I did not say this was an article on pedagogy or suggest it. I suggested covering the most important pedagogical schools because In the history of mathematics, the span of pedagogical schools of thought is typically a fair proxy for the span of major conflicting points of view within mathematics (especially on a subject as basic as this one). NPOV would not be satisfied by using only Bourbaki texts, for instance, and ignoring controversies around the Bourbaki group's highly opinionated choices. That would at best create a brief illusion of consensus for non-experts. Neutral experts would recognize it as a sign of memetic capture by a minority interest and they would then either (a) revive the conflict, making new edit wars or (b) avoid or disdain Wikipedia, which is the current default.

And yes, once the list of textbooks were assembled via consensus on the talk page, I would not recommend the pedagogy itself should go into this article. Unless, as you say, it were for a subsection on variables in education. RowanElder (talk) 13:36, 7 January 2025 (UTC)

Thanks for suggesting another possible strategy for unblocking these discussions. As someone who has a big problem with sourcing, I can see value in it.

I'd also observe that, so far, the thread is working better (is more cooperative) than the previous ones, so I'm personally naively optimistic that we'll find a positive way through, whatever that ends up being. NeilOnWiki (talk) 10:21, 7 January 2025 (UTC)

Yes, in my editing history here around 500–700 edits I was in a similar naively optimistic phase. You can find that at Series (mathematics), Geometric series, and Rate of convergence, for instance. I hope yours goes better than mine did! Optimism is great when it works out. RowanElder (talk) 13:45, 7 January 2025 (UTC)

@RowanElder: My last comment was meant for you. (I also agree there's scope for reorganisation — not just within the article, but considering where it sits within the ecology of articles involving the more general concept of variable.) NeilOnWiki (talk) 12:32, 7 January 2025 (UTC)

Thanks for the clarification and yes, the reorganization of the ecology seems promising to me as well. RowanElder (talk) 13:52, 7 January 2025 (UTC)

I've a possibly ill-conceived urge to create a new section in the main article, calling it Symbolic constants (current working title!).

My intention is to embrace the statement that "It depends on the authors and the context whether the symbol representing a constant is a variable or not."

—Although I probably wouldn't choose this particular wording, I'm assuming the sentiments behind the statement are relatively uncontroversial; or, if deemed necessary, that there are reliable sources to back it up.

As I see it, the new section would aim to:

• Focus on describing usage in practice, rather than what the answer to this dilemma should be.

• Avoid reproducing the kind of complicated technical details engaged in in this Talk page (or at least my own contributions would);

• ...but be mindful of the considerations raised by both sides in the various discussion threads.

• Proceed incrementally, so the content can be modified easily; or reverted, if necessary, eg. if it appears to be proceeding in the wrong direction and require discussion.

• Be phrased with a wider readership in mind (eg. non-mathematicians). (But I'd consider eventually moving the current Moduli spaces there as a sub-section, if this tactic ever gets that far.)

• Point out, in natural language (and without necessarily claiming they're equivalent), the resemblance of a symbolic constant (ie. a constant represented by a symbol) to a variable whose value-set is a singleton. This would (probably) appeal to the definition of a variable settled at in the lead; contrasting usage and intuitions with the inferences that logically follow.

• Acknowledge, in some probably superficial way, the interchangeability of the term variable for constant that might be encountered in different discussions/texts/contexts.

I might also wish to request that Farkle Griffen (talk · contribs) reverses his deletions of all the occurrences in the original text where constants were described as variables. I'd give my reasons for the request when or if I made it.

Does anyone object to this suggestion? My thinking is that it might be easier and more effective to try some edits in practice to see if this approach might work, rather than engage in a theoretical discussion why it would or wouldn't; or set out more precisely what it might entail. I don't mind disappointment if it fails part-way: discovering something is impossible has had a good track-record in maths.

I would, by the way, be characteristically slow; not least because of other commitments.

NeilOnWiki (talk) 17:46, 7 January 2025 (UTC)

"Focus on describing usage in practice, rather than what the answer to this dilemma should be."

"I might also wish to request that Farkle Griffen (talk · contribs) reverses his deletions of all the occurrences in the original text where constants were described as variables."

I would not be willing to do so on the grounds that I have not found any source claiming "the symbol representing a constant is a variable". In fact, this goes against all sources currently in this article. I don't know what you mean by "this dilemma", what dilemma, exactly?

If you're looking for sources that distinguish variables and constants, I would suggest any book on mathematical logic. In every book I've found, they state very explicitly that every First-Order Language has two distinct sets of symbols: one for individual variables, and one for individual constants. – Farkle Griffen (talk) 18:46, 7 January 2025 (UTC)

This is why I would like to suggest a definite list of textbooks with a well-chosen span. Within logic, particularly first order languages, I don't doubt that this is what you have found, but in engineering mathematics and numerical mathematics, my experience has been that it is common to call "symbolic constants" also "variables." (Which is not a sourced statement, so I mean this only to contribute to the talk page discussion about why there is or is not yet consensus. I would also need to provide a source before I added anything I believe about this to the page itself.) RowanElder (talk) 19:20, 7 January 2025 (UTC)

The exact title can be changed beforehand or in situ if deemed controversial. I didn't mean it as a compound noun, though I'm aware that there's a name-clash with its use as a term in programming.

Your experience echoes mine. NeilOnWiki (talk) 10:47, 8 January 2025 (UTC)

Sould I recall the title of the article? It means that the article must focus on the common use of the term in mathematics. Certainly, the use in education, engineering, physics, logic, computing must be mentioned when it differs from the use in mainstream mathematics, but mainstream mathematics must be the base. As the use in mathematics deserves to be the subject of an article, every variant that deserves an article must have its own article. We have already Variable (computer science).

About variable vs. constant, there are several things to say:

• In mathematics, "variable" has lost its meaning of "changeable", which it had at Newton, Leibniz and Euler time.
• Historically, a variable represented a number, but, presently, this is not always the case. For example, S is a variable in the sentence "Consider a set S.
• A variable can be instanciated. This is commonly done with sentences such as "let ⁠$x=3$⁠".
• Being a constant or not is context dependent: the identity element of a group is often denoted e. I is a constant in the group, but it is no more a constant if the group is allowed to change.
• The possible values of a variable are often restricted to a set, often called the "range" of the variable. So, a constant may be viewed as a variable that has a singleton set as it range.
• The set of natural numbers ⁠$\mathbb {N}$⁠ is a constant for the definition given in the linked article and for all definitions of "constant" that have ever heard of. However, I have never heard of the use of "constant" for refering to this set.
• There is no real difference of nature between the sentences "Let ⁠$\mathbb {Z}$⁠ be the ring of the integers, with the number 0 as additive identity" and "Let ⁠$R$⁠ be a ring, with the 0 as additive identity". The only difference is that, the first sentence is obtained from the second one by instantiating ⁠$R$⁠ to ⁠$\mathbb {Z}$⁠.

All these non-controversial remarks leads to consider that a symbol refering to a constant is a variable whose range is a singleton set. For example e is a variable that refers to the Euler's number, a mathematical constant, in some contexts and to something else in other contexts. In other words, in the case of Euler's number, one must distinguish between the number, a constant, and its conventional name, a variable (the word is not the thing).

I recommend to rewrite in this way the part of the article related to constants D.Lazard (talk) 15:46, 8 January 2025 (UTC)

I do not find it plausible that all of these remarks are non-controversial. I will agree that describes a coherent and admirable practice in mathematics and that it would be agreed on by many academics, especially those associated with Bourbaki, and I plan to teach my own children the conventions that you're naming here, but I do not believe this would be representative of the entire discipline. Natural language mathematics has not fully given way to this sort of strict set-theory-based structuralism in English language practice.

Would you know what I was talking about if I began a conversation about the political motivations and impacts of Saussure, Serre, Althusser, and Levi-Strauss in French intellectual history? Or the relations to Lang's politics or Lawvere's? These have been understood as highly controversial ideological programs in the English-language scholarship I'm familiar with. RowanElder (talk) 16:30, 8 January 2025 (UTC)

Sorry, I should have written "non-controversial in mainstream mathematics". Other views on variables must be mentioned if they can be sourced, but not at the same level of emphasis, per WP:FALSEBALANCE.

By the way, I do not see what this discussion has to do with the "political motivations and impacts" of French philosophers (I believe that you are talking of Michel Serres rather than Jean-Pierre Serre). D.Lazard (talk) 17:35, 8 January 2025 (UTC)

I was intending to refer to Serre, in fact, and thank you for answering the question in the negative: you did not know what I was talking about as I began a conversation concerning the French intellectuals I mentioned.

The relevance is that in Anglophone mathematics, set-theory-based structuralist-like approaches did not become identical to "mainstream mathematics." Anglophone mainstream mathematics is strongly influenced by the pragmaticism and pragmatism of Peirce, Davidson, and Sellars (among others) as well as the ordinary language philosophy and process philosophy that followed the collapse of Bertrand Russell's strict logical program. I would suggest that this issue one of the reasons why consensus on these pages has been particularly difficult to form effectively: these assumptions about "mainstream mathematics" you're making are not, in my experience, typical of Anglophone mathematics. Here, more informal ordinary language and pragmatic approaches to mathematics are also mainstream. However, I am not insisting that you put this into the page without sources. I am just informing you of a likely reason why other editors here do not find you convincing. RowanElder (talk) 18:05, 8 January 2025 (UTC)

Everything that you are talking about in your last post is philosophy of mathematics, or, more precisely, opinions of some philosophers on mathematics. Please, do not confuse mathematics and philosophy of mathematics. If you disagree with the existence of mainstream mathematics founded on Zermelo–Fraenkel set theory, you must explain why almost all mathematical Wikipedia articles use set theory as a basis. Personally, I see Zermelo–Fraenkel theory as a great accomplishment of Russel's logical program, certainly not a collapse.

Also, I do not know any text of Serre (Jean-Pierre) that is related to intellectuel debates D.Lazard (talk) 19:04, 8 January 2025 (UTC)

No, what I wrote was intellectual history of mathematics, and it was more about opinions of mathematicians regarding philosophy than vice versa. I am not confusing these. The "structuralism" I am writing about here has three supporting sources cited in the lead section of Bourbaki group.

Set theory is excellent, mainstream mathematics and it makes a convenient formal foundation for many other branches of mathematics. I have no question why almost all Wikipedia articles that need formal foundations use it. However, in Anglophone mathematics, the formal foundations do not play the role you seem to take for granted in this conversation. And I would argue that it is a likely reason why the pages most relevant to formal foundations are consistently difficult to find strong consensus on, and why they drive away the majority of editors who (a) do recognize the pages need extensive work but (b) can't agree with the accidental and poor-quality consensuses that emerged before strict sourcing of material became normal on Wikipedia.

Russell's logical program had lasting successes that are still worth celebrating, certainly, but it also collapsed. RowanElder (talk) 19:29, 8 January 2025 (UTC)

D.Lazard: I've previously considered whether the article title was creating differing expectations. Even if so, I think we should allow the article to evolve and not make the decision until it has had a chance to be reorganised. It isn't the primary problem here and we've arrived quite productively at a single issue that we need to resolve.

After restructuring Sign function, I believe that different levels and contexts can exist side-by-side. A good structure can make clear to the reader what to expect, so they can make their own choices. It would also help editors to judge what criterion is the most sensible one to specialise on if the need arose.

A Symbolic constant section could begin in a fairly elementary fashion. If an editor wishes to go more deeply, then they could; perhaps in a subsection (or elsewhere if there's extensive notable material). NeilOnWiki (talk) 21:28, 8 January 2025 (UTC)

This also sounds good and feasible to me. RowanElder (talk) 23:12, 8 January 2025 (UTC)

@D.Lazard, ignoring the first paragraph of your reply for a moment, I don't have to reply to your assertion about what a variable ought to be, given the fact that you still have not provided a source to discuss. It doesn't matter what you or even a hundred editors believe "variable" ought to be, if it can't be sourced, it shouldn't be in the article. Please provide a reliable source saying "a constant is a variable whose range is a singleton." Nevertheless, this would also imply that numerals like 1, 2, 3, are also variables, which is certainly controversial.

To the first paragraph, this isn't necessarily the case. Variable (computer science) is a separate article because (1) there are enough sources stating how it and conventional variables are distinct, and (2) there is enough information justifying its own article. As far as I'm aware, engineers, physicists, and teachers don't generally consider their understanding of "variable" to be different to that used in "mainstream math", as you call it. – Farkle Griffen (talk) 22:30, 8 January 2025 (UTC)

Seems right to me. RowanElder (talk) 23:11, 8 January 2025 (UTC)

It doesn't to me. Eg. not...

Please provide a reliable source saying "a constant is a variable whose range is a singleton."

We should be working to improve the article, not silencing anyone on a Talk page for presenting an unsourced argument. I think these discussions have missed opportunities to get things posted on the main article; and I think we've unnecessarily attacked an opposing view, rather than looking for positive ways for improving the article from our differences.

Eg. I think the emphasis by Farkle Griffen (talk · contribs) on domain and range has been very helpful to improving the article, but his wording needed modifying. It would be absurd to demand a source for this, even though it may seem Pollyannaish. The lead paragraph now seems better for it and has survived a surprisingly long number of hours. RowanElder (talk · contribs) has pointed out the wider context that variables are used or encountered, which I consider an important observation towards improvement. Et cetera, et cetera.

As authors and/or mathematicians we have other strategies than sources to tease out the way that a particular difference can be used to improve an article. We can examine the logical argument if a respected Wiki author such as D.Lazard (talk · contribs) comes to a conclusion that appears so controversial from assumptions he believes to be valid. I don't necesarily agree with Imre Lakatos, but he eloquently expresses why proofs and refutations can lead to better outcomes when embraced positively.

And as a second example, I disagree that we've enough information to judge whether the following appeal is valid ...

Nevertheless, this would also imply that numerals like 1, 2, 3, are also variables, which is certainly controversial.

We don't know whether or not D.Lazard agrees that the consequent is true. I would be surprised if he didn't recognise that however logical his thinking is, the personal conclusion he has come to is controversial. (It would also have been odd otherwise for him to have suggested: "It depends on the authors and the context whether the symbol representing a constant is a variable or not.")

Therefore this argument loses its power. NeilOnWiki (talk) 13:24, 9 January 2025 (UTC)

I agree insofar as I would not have said "seems right to me" if I had thought @Farkle Griffen was "silencing someone on a talk page for presenting an unsourced argument." However, I didn't think that's what was happening there; I thought it was more an expression of "please ante up if you would like to continue playing this game, otherwise I'm not going to spend more time." However respected they are, @D.Lazard often tests people's patience [not an aspersion, I assume good faith behind their doggedness]: I quit editing math largely because of frustrating interactions with them.

I do think even the "domain and range" issues would need sourcing to establish a lasting, high quality, and maintainable consensus, but yes, marginal improvement can also occur before that quality of consensus as well: no argument (or discouragement!) there.

Take an example. Above, in earlier talk page discussion here, you can see "ISO [International Standards Organization] is not a reliable source for mathematics" from @D.Lazard. I'm not highlighting that in order to contest it, though I do think it's wrong (I think it misrepresents what "reliable source" means, which is not "it matches Wikipedia consensus in each detail"); I highlight it because it should indicate how opinionated and fractured "mainstream math" is, in reality, outside classrooms where individual teachers can present "their way is the way." My opinion is that Wikipedia's approach to math has not been less opinionated or fragmented than mainstream mathematicians usually are themselves, but then also that this is more of an issue in a general encyclopedia with an NPOV policy than it is for the ISO or the Bourbaki group or whatever other math standards-setters and structure-definers are busy building, improving, and advocating their own particular PsOV. RowanElder (talk) 14:33, 9 January 2025 (UTC)

Please replace "but his wording needed modifying" by "despite my believing his wording needed modifying". Still not quite right: by "this" I meant my optimism it was helpful. NeilOnWiki (talk) 13:33, 9 January 2025 (UTC)

Replying to: "I think the emphasis by Farkle Griffen (talk · contribs) on domain and range has been very helpful to improving the article, but his wording needed modifying. It would be absurd to demand a source for this [...]"

I'm not making this up; it is fairly common terminology in meta-mathematical contexts. Source 3 currently in the article states:

"Definition 4. The range of a variable is the set of elements whose names are allowable replacements for the given variable."

The Advanced Calculus source mentioned in the discussion below states:

"A variable used in mathematics is not allowed to take all objects as values; it can only take as values the members of a certain set, called the domain of the variable." – Farkle Griffen (talk) 14:32, 9 January 2025 (UTC)

@Farkle Griffen: Thanks for your reply. I'm not asking you to at the moment—"because reverts and unreverts are so binary" (see my opening comment of this thread). I just wanted to be open about my current thinking, rather than spring any unexpected surprises on you or anyone else involved in this discussion. I'd give my own set of reasons if I did ask, bearing in mind my understanding of your position.

I agree my use of dilemma was ambiguous. I meant "whether the symbol representing a constant is a variable or not"; which is one of several questions which have been fuelling these discussions.

I might also have described the question of how best to proceed as a dilemma, a slightly different one; but that's not the meaning here (and I think I would have chosen a different word that didn't imply there were just two choices). NeilOnWiki (talk) 11:14, 8 January 2025 (UTC)

I've realised my "Does anyone object to this suggestion?" might be ambiguous. The "suggestion" I'm referring to is for me to create the new putative section on the main article page, with the listed intention and characteristics. I wasn't referring to the previous paragraph. (It's interesting how quickly natural language can get us into trouble without a well-defined symbolic reference system.) NeilOnWiki (talk) 11:47, 8 January 2025 (UTC)

Sources

@RowanElder, I'm moving this into a new discussion post since the last one is fairly bloated and hard to navigate. In response to your suggestion about widening the span of sources:

I do think this is a positive way forward, and specifically, "engineering mathematics" is one that I haven't considered.

I haven't gone too deep into this (since most engineering textbooks seem to assume an understanding of variables), but I have found a source Introduction to Engineering Mathematics and Analysis (B. Wood), about modeling, that seems to associate variables strongly with the concept of degrees of freedom and "abstaction" of a formula; and contains the quote: "Abstraction is often done in a universal sense (e.g., using variables instead of numbers for constants) so that the abstracted representation can be generalized beyond the specific case of interest."

I also haven't looked through any sources on any kind of scientific analysis or data analysis, but the articles Observable, Latent and observable variables, and Random variable may be useful for sources there.

On the subject of "major textbooks on algebra and calculus", the article already contains two textbook sources in algebra, one elementary algebra, and one abstract algebra. For calculus, I've found Advanced Calculus (Loomis and Sternberg), which seems to fit the description. Specifically, Chapter 0, section 5 on "Restricted variables" (page 8) and the few paragraphs above describing mathematical objects an their symbols seems useful.

More sources are probably needed, but I hope this works as a start. – Farkle Griffen (talk) 04:45, 9 January 2025 (UTC)

See Help:Talk_pages#Replying_to_an_existing_thread. I hadn't noticed the reply tool before and I've found it wonderful compared to what I was doing manually before. The indenting is logical, but different from a more usual linear thread. NeilOnWiki (talk) 09:53, 9 January 2025 (UTC)

Would this be the right thread to discuss the tension that many mathematicians must feel between the need for strict logical consistency within an article and a statement such as the following:

It doesn't matter what you or even a hundred editors believe "variable" ought to be, if it can't be sourced, it shouldn't be in the article.

And whether "can't" implies it must be sourced by the person making it? NeilOnWiki (talk) 10:02, 9 January 2025 (UTC)

I'm not sure what exactly you mean, but you might want to have a look at WP:Verifiability. Beyond the exact wording used, all information should be attributable to a reliable source. There are some exceptions, like Routine calculations, and Common knowledge, but the general policy is that all information should be verifiable. I am not "silencing" D.Lazard beyond simply asking for a source. If a reliable source is given, the discussion can continue. But as of now, that definition of variable is being WP:CHALLENGED.

Mathematics has the benefit of being a relatively well-documented subject. Unless you're working on an extremely niche article, there's usually at least two authors who have already answered any question one may ask.

– Farkle Griffen (talk) 13:58, 9 January 2025 (UTC)

Thanks. I think you've answered my query regarding this thread. NeilOnWiki (talk) 14:42, 9 January 2025 (UTC)

This does seem useful, yes, and "most engineering textbooks seem to assume an understanding of variables" is exactly why I don't have good sources immediately at hand myself. Engineers and physicists often don't use formal foundations for their mathematics at all, but rather informal "ordinary language" foundations, and it's often hard to find clean sources for that informal ordinary language... or to deliberately formalize consensus on how to describe the informal, since part of the strength of informality is exactly its dynamic potential for ad hoc adaptation and refinement. But it can be done. RowanElder (talk) 14:50, 9 January 2025 (UTC)