Talk:Mathematics/Archive 15

Proposal for new structure (Sep 2020)
Hi everyone, I still plan to come around and work on the article, but in the meantime, I wanted to float a proposal to chew on for the rest of the year. It's a little far-reaching but could lay groundwork to improve the article in several ways (add more content beyond math research, handle the "Mathematics as science" section more cleanly, etc.)

Essentially, I'd consider sections 2-4, as they are now, all different takes on what math is:
 * The Definitions section first notes it's practically undefinable, but then throws in what are really 3 philosophies of math
 * It then swings back to address the common notion that it's also a science (at least in the German sense)
 * On top of that, the brief comment "Mathematics is what mathematicians do" arguably leans towards a 4th philosophy (instrumentalism)
 * The next two sections describe wide-ranging properties of mathematics
 * To me, these also seem to emphasize how math can't be reduced to any one of the previous aspects

I think all of this content could be harmonized though if we're willing to shuffle it a bit. Instead, we can just give each aspect/outlook a section & pull in from the old outline. I'm not suggesting using these exact names, but what about an outline something like this?
 * Mathematics: what is it? Aspects of mathematics
 * Definitions ← Only the introductory paragraphs from the current Definitions... section (emphasizing it's undefinability)
 * A way of reasoning ← The Logicist sub-section and details on proof and abstraction from throughout the article
 * A creative process ← The Intuitionist sub-section, the Inspiration... section, the bits about aesthetics & "math as a liberal art" from other sections, and any new content related to how math / numeracy involves intuition or psychology
 * A formal language ← The Formalist sub-section, the Notation... section, and maybe some notes on how math is unusually good at jumping across time & place
 * A body of knowledge ← Most of the current ...as Science section, especially the notes on empiricism, maybe a brief mention of platonism, plus new content on how mathematical knowledge accumulates and interconnects
 * A skilled practice ← New content on how both mathematicians and other people use math in their lives, notes on math education and mathematical tools, and maybe a brief discussion of instrumentalism

What do you think? --Zar2gar1 (talk) 23:30, 3 September 2020 (UTC)


 * To be honest, I think it's excessive. This is the main article on mathematics, not on how to define mathematics.  --Trovatore (talk) 07:21, 6 September 2020 (UTC)


 * Ah, maybe I could have made my main point clearer, but moving away from narrow definitions to a full description of mathematics is the whole goal. You could even use "Description" as the main header instead of "Mathematics: what is it?".
 * It just happens that the current arrangement partly casts some aspects of math as conflicting definitions. Under my proposal, any "Definitions" section would be reduced to at most what's in the lead of the current "Definition of mathematics" section.
 * At the same time, moving towards describing distinct (but non-exclusive) aspects would allow consolidating at least sections 3 & 4 from the current arrangement. The descriptive schema also makes coverage gaps more recognizable; based on previous discussions, I think something like the "Skilled practice" section would be a particularly nice addition. --Zar2gar1 (talk) 15:19, 6 September 2020 (UTC)


 * In that case, I'm afraid I'm opposed to your goal. That's not what this article is or should be about. --Trovatore (talk) 15:35, 6 September 2020 (UTC) Hold on, maybe I haven't understood your point well enough yet. --Trovatore (talk) 16:23, 6 September 2020 (UTC)
 * I guess what I'm saying is I don't think we should focusing as much as your draft suggests on the question of "what mathematics is". That's more appropriate at the definitions of mathematics article, or to some extent at foundations of mathematics.  I'm not opposed to expanding coverage of foundations in this article (why a "brief" mention of Platonsim, by the way? It's one of the main schools).  But I'm not convinced that "what is mathematics?" is the right organizing principle for the discussion of foundations.  --Trovatore (talk) 16:29, 6 September 2020 (UTC)

Actually, that's a good point; I've seen in the archives how often arguments over definitions or the lede come up, so I don't blame you for keeping a wary eye out for that. I probably should have thought more about the main header because "what is it?" does typically imply a definition, doesn't it?

Since that's ultimately the opposite of what I was hoping to go for, I've struck & replaced the header to emphasize describing facets of the field, not defining. If the reorganization did go forward, we could also add a sentence somewhere in the new section intro stating the subsections are mutually reinforcing, not exclusive.

Oh, definitely not to downplay Platonism. On the contrary, if these changes happen & work somewhat like I'm picturing, I think we could outlink & pare back most of what's already there about the other "-isms". So discussions on all the philosophical schools would be equally brief.

Each subsection would still have a corresponding philosophy or 2 to mention. By moving the emphasis from semantics to description though, there's less need to linger on those ideas that belong more in the other articles you mentioned. --Zar2gar1 (talk) 22:05, 6 September 2020 (UTC)
 * I understand a little better now. It seems worth discussing at least.  I'd have to look through the article and try to figure out how and where it would apply before I could offer a clear opinion.  Maybe I'll try to do that tomorrow. --Trovatore (talk) 06:01, 7 September 2020 (UTC)


 * I appreciate that you're trying to improve the clarity of the article. But I'm not sure that your plan doesn't also encompass Section 5 on the Fields of mathematics. Aren't these the "body of knowledge" actually? And then the only sections that your plan doesn't encompass are the comparatively trivial Etymology and Awards sections.
 * In other words, your plan seems to be pretty close to rewriting the article. And that's not a crime --- surely a better version of this article exists in some Platonic realm --- but the article as it stands is the result of compromise among many editors, and there are many devils in the details. It would help me to see a more detailed draft of your rewrite. Mgnbar (talk) 13:55, 9 September 2020 (UTC)
 * +1. Paul August &#9742; 17:25, 9 September 2020 (UTC)

First off, I really like the draft suggestion; it may not happen until the winter, but if everyone is at least willing to consider the changes, I can definitely whip one up.

Yes I suppose, looking at it very maximally, you could include section 5 under that if you wanted to. Besides spacing out the sections though (pure copy-editing), I think this schema could emphasize how section 5 is distinct: "aspects" vs. "parts", form vs. subject matter, the activity vs. the result, that sort of thing.

A draft will definitely be good to clarify for everyone & provide a sanity-check on the idea in my head. In advance though, I'd reassure you that for immediate changes, I'm hoping for minimal rewriting (just sentence & paragraph transitions). Looking back, my list mixes the more immediate splicing with what I think the new schema could grow into.

If everyone resolves to go forward, my intent is only to reorganize what's already in sections 2-4 at first. After that, I would honestly prefer to give other editors several months to work their magic before I touch those sections to add or subtract more. --Zar2gar1 (talk) 03:26, 10 September 2020 (UTC)

Presumably the mathematicians referred to who define mathematics as; "Mathematics is what mathematicians do", do so with tongue firmly in cheek. Given they of all people would be aware of the circularity of such a definition. What is mathematics? It's what mathematicians do. Who are mathematicians? People whose vocation is the practice of mathematics. Ah, got it ;) 165.73.55.32 (talk) 16:25, 10 October 2021 (UTC)
 * Just a word about "Mathematics is what mathematicians do": it is not a philosophical approach. It is the observation that nobody knows how to define mathematics, but mathematicians well know what is mathematics and what is not mathematics. There are rarely disputes for deciding wheter a scientific article is about mathematics or not. So, although seeming circular, this definition is probably the most accurate of those that have been ever given. Also, mathematics evolves with the time, and new areas are included in mathematics, which at the beginning were not mathematics (for example cryptography, numerical analysis, mathematical economics and theoretical computer science). Another way to explain the sentence is the following: Mathematics Subject Classification is a list of mathematical subject that has been elaborated by consensus among mathematicians. This means that a subject belongs to mathematics if mathematicians have decided to include it in the classification. D.Lazard (talk) 17:44, 10 October 2021 (UTC)

Opening sentence
I wanted to change the opening sentence to say "Mathematics (from Greek: ) is the use of symbols to reason in the abstract.", but I see that this sort of edit has the potential to irritate people who have put a lot of effort into perfecting that portion of the article. I would also edit the following portion to be consistent with this edit, most likely rewording it to highlight that more exact definitions are hotly debated. In almost every Wikipedia article, the opening line says "[article title] is [definition]", but Mathematics, one of the most important articles on the site, is left nebulous and poorly defined, which is both not helpful to readers and massively annoying to mathematicians, who love good definitions. If I could get some people to sign off on this edit, I'll go ahead with it. But if I face resistance, I don't have much interest in fighting over it. BlackEyedGhost (talk) 05:56, 7 September 2021 (UTC)
 * This first sentence is factually wrong: Mathematics predates for centuries reasoning with symbols, and the basic mathematical objects (numbers and geometric shapes) are not symbols, even if their epistemologic nature is hard to characterize.
 * Apparently, you have not any source for your definition. This is thus what Wikipedia calls original research (see WP:OR), and including your first sentence would go against one of the fundamental Wikipedia policies.
 * It is certainly good to begin an article by the definition of its subject, but this requires the existence of a commonly accepted definition, which is not the case here. D.Lazard (talk) 08:41, 7 September 2021 (UTC)
 * Your motivation is noble, but D.Lazard's objections are overwhelming and correct. Any proposed definition must not only be supported by a reliable source, but also must reasonably summarize the "consensus" view from many reliable sources. The article's definition is muddled precisely because the philosophy of math is muddled. Mgnbar (talk) 12:17, 7 September 2021 (UTC)
 * Cool. Have fun with your article about (undefined). BlackEyedGhost (talk) 22:37, 7 September 2021 (UTC)
 * Thanks. And, if you haven't read it yet, you might enjoy Mathematics. Mgnbar (talk) 01:35, 8 September 2021 (UTC)
 * After reading this discussion, I thought it would be better not to link to this article unnecessarily even if the lead sentence of the article starts with "In Mathematics…". For example, in the Holomorphic function seems to have removed the link to this article. So, I checked What links here, but there are too many … --SilverMatsu (talk) 01:03, 14 October 2021 (UTC)
 * You're systematically removing links to the Mathematics article throughout Wikipedia? If so, please don't. Mgnbar (talk) 01:28, 14 October 2021 (UTC)
 * Thank you for your reply. I'm not going to systematically remove the link. I thought removing the link might be one way to solve the discussion in this section, but I haven't thought about any details yet.--SilverMatsu (talk) 06:45, 14 October 2021 (UTC)
 * (Edit conflict) It may be better to try changing the lead text on other pages rather than removing the link. (e.g. In complex analysis …)--SilverMatsu (talk) 23:26, 14 October 2021 (UTC)
 * Can you please how this discussion is related to links to this article? And give examples where you think links are inappropriate? Paul August &#9742; 23:10, 14 October 2021 (UTC)
 * Thank you your reply. I guess the proposer of this section made this proposal by looking at the opening sentences in other articles. By the way, is it better to link to this article again in Holomorphic function?--SilverMatsu (talk) 23:26, 14 October 2021 (UTC)
 * I see no reason not to link to the first use of "mathematics" in Holomorphic function. Paul August &#9742;

This discussion started about the opening sentence of this article, and is now whether the opening phrase "In mathematics" of many mathematical articles must be linked here. Because of the high number of articles that are involved, this must be discussed in WT:WPM, and, before any change in the articles, MOS:MATH must be edited for reflecting the consensus (if any). Until that, linking or unlinking "mathematics" in opening sentence are two admissible options, and thus MOS:VAR applies.

I'll not open myself the discussion on WT:WPM, because this is a minor issue, and none option is harmful. My opinion is that the link is not useful because the phrase is here only for informing readers that the subject of the article belongs to mathematics, and everybody knows enough of mathematics for knowing whether they may be interested in such an article. On the other hand, the blue link makes reading easier, as allowing readers to know that it is mathematics, without reading really the first phrase. D.Lazard (talk) 09:04, 16 October 2021 (UTC)


 * Thank you your reply. I agree that the place to discuss "What links here" is not this page, and WT: WPM is appropriate, and also that it is minor issue. And I also agree that linking to explain "In mathematics" (rather than "mathematics") makes it easier to understand. I think the current opening sentence is appropriate to explain "What is mathematics ?" So I guess that why they think this article is inadequate and ambiguous is that they need to define of mathematics, probably for other articles related to mathematics. Since the current consensus is that there is no need to change the opening sentence, so I thought one of the possible methods would be to remove the link, that is, explain definition in another article that needs its definition. However, if the link explains that the subject of the article belongs to mathematics, there is no need to remove the link.--SilverMatsu (talk) 15:03, 16 October 2021 (UTC)

Lead sentence
User:D.Lazard I think the lead sentence, in its current state, seems like original research, or a synthesis of existing material. Whilst the sources do define math as "the science of space, number, quantity, and arrangement" and of "abstract structures", they don't explicitly relate "quantity" to "number theory" or "structures" to "algebra". --L&#39;âne onyme (talk) 17:38, 27 October 2021 (UTC)


 * If L&#39;âne onyme's summary of the sources is correct, then I agree with him/her/them. It is precisely because the opening definition (or lack thereof) is so contentious that we should hew so conservatively to the sources. Mgnbar (talk) 18:19, 27 October 2021 (UTC)


 * "Mathematics" is a very abstract concept and very difficult to define precisely. The contention of how to define it in this article has been going on for years. (See previous posts on this talk page and in the archives.) I don't think we should make wholesale changes to the lead at this time unless there is agreement that something is seriously incorrect.—Anita5192 (talk) 19:01, 27 October 2021 (UTC)


 * I agree with you in general, but the criticism here is quite specific and justified: The opening sentence's parenthetical remarks are not supported by the cited sources (but appear to be supported, to the casual reader). Mgnbar (talk) 21:13, 27 October 2021 (UTC)


 * Until relatively recently the lead sentence had&mdash;for a very long time&mdash;read:
 * Mathematics ... includes the study of such topics as quantity, structure, space, and change.
 * While I agree that the association of the parentheticals "number theory", "algebra", "geometry", and "analysis" are generally correct and useful (although one might quibble why not arithmetic, instead of, or in addition to, number theory? or calculus, instead of, or in addition to, analysis?), I agree with  that we should hew conservatively to the sources. Paul August &#9742; 21:56, 27 October 2021 (UTC)

First paragraph
I have edited the first paragraph without changing its whole structure. Here are my motivations. I was uncomfortable with it (and with a large part of the article) because I do not recognise in it the mathematics that I know. I have kept the structure of the paragraph. So, I guess that my edit does not go against a previous consensus. D.Lazard (talk) 10:42, 30 October 2021 (UTC)
 * The four areas mentioned in the paragraph were described each by a linked word. This suggest wrongly that this word is used here in a technical meaning, and that the word is commonly used in the corresponding area. For example number theory was presented as the studies of quantities, when treatises of number theory never use the term. Also, the use of "structure" for algebra, refers clearly to algebraic structures, but is clearly ambiguous for most readers. So, I have replaced the description of "what is included" in mathematics, by phrases that better correspond to the reality, while accessible for most readers.
 * Sentence about the "definition": In this sentence, "definition" is linked. This suggest that a domain of knowledge should have a formal definition. This is wrong, at least because mathematics, as every domain of knowledge, evolves with the time.


 * I like the first paragraph the way you have it now. Thank you!


 * I wonder, should we also mention "arrangement," which encompasses, for example, permutations and structure? See the wording of citation (1) from the Oxford English Dictionary.—Anita5192 (talk) 14:45, 30 October 2021 (UTC)
 * "Structure" is already mentioned ("formulas and relates structures") I think that the first paragraph must be limited to the four basic areas that occur in every basic curriculum. Otherwise, there is no way to limit the list (arrangements, but also probabilities, discrete mathematics, ...). Mathematics Subject Classification has more than 60 first level areas! D.Lazard (talk) 15:52, 30 October 2021 (UTC)

| Lead Sentence - revert of my edit
"D.Lazard (talk) Undid revision 1052725126 by Emdosis (talk) It has been asked explicitly to not link these terms. Adding them change completely the meaning of the paragraph." Does anybody who actually bothered to look at my edits agree with that statement and can those people confirm which version they think is better? Thanks.

Emdosis (talk) 20:31, 30 October 2021 (UTC) "D.Lazard (talk) ...change completely the meaning of the paragraph." I don't mean to be rude but is English your first language? I'm asking this because none of the links change the meaning of the paragraph in any way. Emdosis (talk) 20:39, 30 October 2021 (UTC)
 * I can't believe we're talking about the opening sentence yet again, but I kind of agree with Emdosis. Why don't people want these terms linked? Mgnbar (talk) 21:06, 30 October 2021 (UTC)
 * This article was written by consensus; please be prepared to defend your word choices, link by link, on this talk page, before changes to the article. That means you will have to explain the rationale for each of your changes, which might eventually appear in the article; be patient. --Ancheta Wis   (talk  &#124; contribs) 23:45, 30 October 2021 (UTC)
 * A large part of my rationale for not linking these terms is explained in WP:OVERLINK; in particular
 * here, the important links are those to mathematics areas; the links added by tend to hide this by adding numerous minor links.
 * Providing links for "shape", "quantity", etc. does not help for understanding what is mathematics.
 * . Here the intended meanings is the everyday meaning, even when it is not the same as the mathematical meaning. This is the reason why linking changes the meaning of the sentence.
 * adding these links confuses the meaning of the sentence by emphasizing minor terms.
 * D.Lazard (talk) 10:18, 31 October 2021 (UTC)

I see. Now at least I can understand. The first two guidelines are valid. The third isn't..in my opinion (in this case). I wouldn't have added those links in the first place if instead of "Please, do not link words that are used in their non-technical meaning," you didn't use the words ' non technical' (for they're all used in relevance to mathematics in this case). Emdosis (talk) 12:20, 31 October 2021 (UTC)

Lead rewritten
I have rewritten the lead for a better structure and a focus on the most important things. I have removed minor facts such as the mention of Peano and the mentions of behaviour and psychology of mathematicians. I have also added some examples for clarifying important facts that can be obscure to the layman (the difference between validity of a proof and truth. Also the previous structure was poor and rather confusing. The new lead is structured in 5 paragraphs: I have tried to keep the relevant facts of the previous version, and to present them in a more structured and less controversial way. I hope having succeeded. D.Lazard (talk) 15:22, 31 October 2021 (UTC)
 * Introductory paragraph
 * Proof vs. experimentation (IMO, this is the main specificity of mathematics)
 * Importance of mathematical rigor for experimental sciences
 * Ubiquity of applications for applied mathematics as well as for pure mathematics
 * Very sketched history


 * Quick thoughts:
 * I like it in general, although it could use (IMO) more on pure math(s).
 * The parenthesis at the end of the second paragraph seems out of place - maybe that information later?
 * Minor: relies should be rely in the last sentence of the second paragraph.
 * To my (US) ears, modeling phenomena rings better than modelization of phenomena.
 * --John (User:Jwy/talk) 15:40, 31 October 2021 (UTC)
 * I have fixed the two last points.
 * The distinction between pure and applied mathematics is controversial for many mathematicians. This is why I have tried a more neutral formulation. Moreover, more emphasize (in the lead) on pure mathematics could be misleading. In fact pure mathematics are generally thought as mathematics that are developped without being drived by non-mathematical applications. A large part of mathematics that is qualified as pure is drived by applications to some parts of mathematics. A typical and explicit example is scheme theory. So, more emphasis on pure mathematics would need explanations that do not belong here.
 * About the parenthesis at the end of the second paragraph: I agree that the worlking methods of mathematicians do not belong to the lead, but in this case, this may be useful because of the common misconception that experimentation does not belong to mathematics. D.Lazard (talk) 17:03, 31 October 2021 (UTC)
 * Thanks. The balance will be difficult.  When I think math(s), I think what you have in most of the second paragraph - which describes the 'pure' part.  The rest qualifies it - correctly.  Not sure I have further suggestions! --John (User:Jwy/talk) 00:56, 1 November 2021 (UTC)

Section
This section contains a lot of controversial WP:OR. An example is the first sentence: "Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change". Other examples are the names of the subsections.

Rewriting the section will take some time. I could prepare a new version in my sandbox space, and move it only when it will be ready. This will make difficult to compare the versions and to discuss the new version, as this will require to discuss many points simultaneously. So I uses work in progress for distinguishing the two versions. Parts of the old version will be suppressed when their relevant content will have been included in the new version.

For the moment, I have written a new introduction to the section. Comments are welcome. D.Lazard (talk) 14:14, 2 November 2021 (UTC)

Areas of mathematics
I have started to rewrite this section. For the moment the job is done for and.

Reasons: The former section was controversial, since it makes difficult for a mathematician to recognize mathematics in the description. In particular, the main classification is based on a non-mathematical classification (shape, structure, change, ...)

Guidelines:
 * This article is aimed for a general audience, and thus must be understandable for people who know only mathematics taught in school.
 * This section must provides a description of mathematics that replaces a lacking definition
 * As mathematics evolves, the section must present the main trends in the past evolution. For example, it must explain why the modern scope of geometry is much wider than that of Euclid's Elements
 * WP:NPOV must apply; this was not the case of the previous version, which privileged a philosophical point of view

This is the principles that I try to follow. As I do not know any source that provides a description of the areas of mathematics that follows these guidelines, the new vesion is certainly an original synthesis (this why it is so difficult to rewrite). This was also the case of the previous version. As this section is absolutely needed in this fundamental article, and, IMO, original synthesis is unavoidable, this is a case of WP:IAR.

I hope that this new version will not be challenged. However, it can certainly be improved, and improvements are welcome. D.Lazard (talk) 11:54, 8 November 2021 (UTC)

Etimology of word "MATEMATIKA"
The word mathematics consists of a verb of the Albanian language repeated twice and the possessive pronoun "have" the third person "Mãt e mãt i ka" (Geg dialect) "Matë e matë i ka" ( lit. Albanian) In english mean: "He has measured them twice" 92.53.57.90 (talk) 21:29, 29 November 2021 (UTC)


 * Please provide a Reliable source for your claims, and propose a specific alteration to this Wikipedia article that uses that source. Mgnbar (talk) 00:58, 30 November 2021 (UTC)
 * Not a credible etymology: Albanina is derived from Latin and did not exist when ancien Greeks  introduced the word. Probably, the Albanian word "matë" is derived from "mathematics". D.Lazard (talk) 08:41, 30 November 2021 (UTC)
 * Please review WP:NOTAFORUM -  FlightTime  ( open channel ) 18:06, 30 November 2021 (UTC)

Galleries
In, there are galleries at the end of subsections. Each image has a caption that consists generally as a link to an area of mathematics. There are several problems with these gallery: Before editing the other subsections, I have not yet an opinion on how illustrating them. D.Lazard (talk) 10:22, 9 November 2021 (UTC)
 * In, the images illustrated and were linked to number systems that are not mentioned in the section text. Moreover, some of these number systems (real and complex numbers) do not belong to number theory. So, I have removed the gallery, and I left open the question of more convenient images.
 * In, I have removed the image illustrating measure theory, as this area of mathematics does not belong to geometry. All the other images are clearly related to geometry, but the captions linked to areas of geometry in an obscure way (if one would ask a mathematician what the image represents, he would not answer the previous caption). So, in a first step, I have removed the caption. In a second step, I have titled the gallery "Examples of shapes encountered in geometry", and provided descriptive captions with links to the article where the shapes are explained.


 * For number theory, pictures fro the main article number theory, or/and from (subcategories of) commons:category:number theory (to be handled with care; commons math cats are often a mess) might be used. - Jochen Burghardt (talk) 10:55, 2 December 2021 (UTC)

Semi-protected edit request on 14 December 2021
On the Algebra section there is the line "So, the scope of algebra evolved for becoming essentially the study of algebraic sructures." Change "sructures" to "structures". Sonpahien (talk) 04:52, 14 December 2021 (UTC)
 * Done. Thanks for telling us about this. You can fix such errors yourself in future. HiLo48 (talk) 05:20, 14 December 2021 (UTC)

Semi-protected edit request on 22 December 2021
The bibliography entry "Riehm, Carl" says that it's from pages 778–72, which seems unlikely (should be 778–82). Michael Aurel (talk) 09:16, 22 December 2021 (UTC)
 * ✅ ScottishFinnishRadish (talk) 11:57, 22 December 2021 (UTC)

Semi-protected edit request on 24 December 2021
The link on the "p. 56" for the book "Curry, Haskell (1951)" should be changed to: https://books.google.com/books?id=tZHrBQgp1bkC&pg=PA56 so that it sends you to the specific page cited.

The link for Wigner's Unreasonable Effectiveness should be changed from:
 * http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

to
 * https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

so that it works correctly.

Behind "Raymond L. Wilder, Evolution of Mathematical Concepts..." can be added: ISBN 978-0-486-49061-8

Also, a link can be added: Raymond L. Wilder

Thanks, Michael Aurel (talk) 05:27, 24 December 2021 (UTC)


 * ✅: . - DVdm (talk) 10:55, 24 December 2021 (UTC)

Notation, language, and rigor
I don't dare to add a to this article, so I describe here my issues in section Mathematics.

The sentence Mathematical symbols are also more encrypted than words, in the sense that a single symbol can encode multiple operations or ideas. doesn't convince me:
 * 1) A word in natural language often has several different meanings, too.
 * 2) An example should list several meanings of (e.g.) the multiplication sign, while symbolizing repeated addition is irrelevant.

The last paragraph (Traditionally, axioms were ... formulas within set theory.) has several flaws, imo: Possibly, the whole paragraph can simply be deleted, since all its contents is discussed elsewhere in the article; I didn't check that. - Jochen Burghardt (talk) 21:39, 27 January 2022 (UTC)
 * 1) "Traditionally" could be made more precise; I guess "starting with Euclid, ending with Hilbert" would be OK; the relevance of the Russell quotation ("theft vs. honest toil") is not obvious.
 * 2) Being "a string of symbols" is not a contradiction with the traditional use. Instead, since Hilbert a shift of responsibility for axiom validation has occurred: it is now no longer of the theory-developping mathematicians, but the duty of those who want to use a mathematical theory for some particular real-world purpose.
 * 3) Despite Godel's result, axiomatization is still possible, and is actually done in all(?) subfields of mathematics; however, completeness cannot be achieved. The latter property obviously cannot be expected in any formal system that can express (a counterpart of) a proposition like "this proposition does not have a proof"; Godel's achievement was to construct such a counterpart by means of arithmetic.
 * 4) The last sentence seems to suggest that restricting oneself to set theory can achieve completeness/axiomatizability; this is wrong.


 * Your objections are reasonable. I don't follow Oakley's argument, but the reason could be that I'm reading a snippet of it out of context. I think that the last sentence about set theory is intended to mean something like, "Most mathematicians work within a version of set theory, on concepts that can be formalized as sets with additional structure." I don't follow how completeness enters into it. The average mathematician doesn't often worry about whether a conjecture is formally provable, because, even if it is, it might not be practically provable. --user talk:Mgnbar 23:04, 27 January 2022‎
 * I am in favor to remove immediately the part of the first paragraph of the section that starts with "Barbara Oakley". There are several reasons for that: WP:NPOV, since the point of view of this single author is far from reflecting a consensus, and this is out of scope, as this is intended to be a philosophical analysis of languages; also WP:BIASED, since the use of "encrypted" in place of "encoded" suggest a non-existent secret.
 * The remainder of the section would require to be completely rewrittten. IMO, this applies to most of the article, from section on. Rather recently I started to rewrite this article which gave a highly biased view of mathematics. In particular it started with non-mathematical views of the subject before describing the subject. This is for this reason that I moved this section (and other similar ones) toward bottom, for starting the article with a description of what is mathematics. I stopped the job before  with the intention of continuing later. I still have the project to resume the job. D.Lazard (talk) 10:45, 28 January 2022 (UTC)

Contrast with "physical laws"
I noticed a recent back-and-forth over this passage:
 * Contrary to physical laws, the validity of a theorem (its truth) does not rely on any experimentation but on the correctness of its reasoning (though experimentation is often useful for discovering new theorems of interest).

Users and  disagreed on whether it should say "scientific laws" or "physical laws", and while Jochen's preference for "physical" was well-motivated, in my opinion the entire sentence is problematic, for two reasons.

First, it's true that the truth of a theorem does not rely on experimentation, but neither does the truth of physical laws. Experimentation is how we find out that they're true; it isn't what makes them true. This is a fundamental confusion between truth and knowledge.

Relatedly, the validity of (the proof of) a theorem must not be confused with its truth. A valid proof from false assumptions can give a false conclusion, and a mathematical statement can be true even if no proof exists.

For now I've just commented the claim out. I think there is some point along these lines that might reasonably be made, but it would need to be carefully worded to avoid the problems pointed out above. --Trovatore (talk) 07:08, 19 January 2022 (UTC)
 * I'd have written: Contrary to physical laws, proving a theorem does not rely correctly predicting natural phenomena, but on the soundness of its reasoning. Pabsoluterince (talk) 07:19, 19 January 2022 (UTC)
 * That needs polishing but it's getting there. I think what we want to contrast here is the method, not the content.  There is a reasonable distinction to be made between the scientific method and the method of mathematical proof; the leap from that to supposing that it distinguishes mathematical from natural-science semantics is much shakier. --Trovatore (talk) 07:26, 19 January 2022 (UTC)
 * I agree that truth and knowledge should be distinguished. Following (my understanding of) Karl Popper, I'd suggest something like "While the truth of a mathematical law can be established once and for all by a proof, a physical [or: natural science] law can only be confirmed [or another word expression probability, not certainty] by withstanding continued falsification challenges (i.e. experiments)".
 * While in English "scientific laws" is mostly read as "natural science laws" (God knows why), some international users of en.wikipedia (in particular German language people like me) subsume "mathmatical" under "scientific". That's why I prefer "physical laws", or "natural science laws". - Jochen Burghardt (talk) 07:28, 19 January 2022 (UTC)
 * Hmm, I don't think "mathematical law" is a very common expression. Some mathematical truths can be proved in a particular axiom system, but then again some can't, and for statements that can be proved, what happens if one of the axioms is wrong?  The theorem doesn't stop being a theorem in that formal theory, but it might not mean the statement is actually true.  That's why I think we should be careful not to phrase this passage so that it sounds like it's about content or semantics.  It is (or should be) about method. --Trovatore (talk) 07:38, 19 January 2022 (UTC)


 * The final wording should not indicate that mathematicians avoid experiments. Mathematicians do tons of experiments. It is through experiments that a mathematician formulates a conjecture and builds confidence in it.
 * What's fundamentally different about math vs. the natural sciences is that the mathematician can sometimes do the extra step of actually proving the conjecture. Mgnbar (talk) 21:52, 19 January 2022 (UTC)
 * I agree we don't want to downplay experimental mathematics.
 * I'm actually a bit skeptical that there is anything "fundamentally" different, "different in kind", about math from the natural sciences. Sure, math has proofs, but they're only as good as their assumptions, and you can do that in natural science too.
 * However there's definitely a difference in degree. I think we can agree that mathematics puts more emphasis on formal proof, and less on experimentation, than physics, and that this is worth saying. --Trovatore (talk) 22:39, 19 January 2022 (UTC)
 * It seems to me a key difference is the assumptions of mathematics are applied to abstract concepts that can be absolutely deemed to meet the assumptions whereas the assumptions of natural science are never as absolutely evident. --John (User:Jwy/talk) 04:47, 20 January 2022 (UTC)
 * It might seem like that for some of the very simple axioms, but justifications start getting more involved as you work your way up the ladder. The axiom of choice, for example, seems pretty bloody obvious to me &mdash; if you don't accept it then I start to wonder if you've actually understood the motivating picture of set theory.  And yet it was somewhat controversial for most of the 20th century.  I gather that it isn't so much anymore, but I don't think that's because it's become more "absolutely evident".
 * Some workers have raised doubts about even much more concrete-seeming things (induction, say) in ways that you can't necessarily dismiss out of hand.
 * The only real fundamental "difference in kind" that I see is that natural science deals with physical objects that have mass-energy and spatiotemporal extent, whereas mathematics studies abstract objects that don't. --Trovatore (talk) 06:13, 20 January 2022 (UTC)
 * Imo, the fundamental difference between natural sciences and (modern) mathematics is that the former (attempt to) make propositions about the real world, while the latter does not. In particular, mathematics doesn't care about "validity" of axioms; a mathematical theory has the form "If the following axioms hold [list them], then the following theorems also hold [list them]". After e.g. Cohen showed that the axiom of choice (AC) can neither be proved nor disproved from the other ZF set theory axioms, reasoning about the actual validity of AC ("in the real world") may still be a philosophical task, but it is not mathematical task.
 * I agree that experiments are important in mathematics to come up with conjectures. However, only formal proof is accepted as method to establish mathematical truth. - Jochen Burghardt (talk) 09:04, 28 January 2022 (UTC)
 * Jochen, I completely disagree with your statement that mathematics doesn't care about "validity" of axioms. That's simply not true.  That's a version of formalism (so-called "if-thenism") that I think is just wrong.  This is not really the place to defend why I think it's wrong, but I do want to point out that it's not by any means a consensus view. --Trovatore (talk) 18:00, 28 January 2022 (UTC)
 * Both positions may be correct, depending on the meaning given to "validity of the axioms". If the validity of the axioms means that they represent well some aspect of the reality, I totally agree with Jochen, and one can refer to non-Euclidean geometry for justifying the assertion (the fact that non-Euclidean geometries are used in physics postpones the introduction of these geometries). But "validity" may have other meanings. An axiom system may be considered as valid, if it is consistent; it is clear, since Gödel at least, that mathematics take care of consistency. There is another meaning of validity, which is more subtle. Mathematicians say commonly that a axiom system is "good" when it allows to solve preexisting problems or when it leads to a "rich" theory (in practice, this is not really different). This goodness is a sort of validity. The experience (in particular, in abstract algebra) shows that, with randomly chosen axioms, all theorems that can be proven are trivial, and such an axiom system is definitively uninteresting. So "goodness" is a sort of validity.
 * If one consider only the first meaning of "validity", it is possible that there is no consensus among philosophers, but there is a clear consensus among mathematicians that discussing the validity of the axioms does not belong to mathematics. D.Lazard (talk) 18:52, 28 January 2022 (UTC)
 * Well, you agree with Jochen and I disagree, but it's not really about our opinions. There are quite a lot of realist mathematicians who are not willing to take the view that axioms are arbitrary, or that the only differences are in the richness of the consequences.
 * I do agree that even these usually consider arguments about the truth (let's say what we mean; no need to weaken it by calling it validity) of axioms to be somewhat distinct from the usual practice of mathematics.
 * But what's relevant to this page is that not everyone is willing to characterize mathematics as simply proposing axioms and working out their formal consequences. Mathematics is (at least sometimes) the attempt to elucidate the properties of a more-or-less-intuitively-well-specified structure.  The axiomatic method is the tail, not the dog. --Trovatore (talk) 19:16, 28 January 2022 (UTC)
 * Yes, the natural sciences are about the real world, while math is about the mathematical world, and this is what allows mathematicians to prove things sometimes. But the case of the (generalized) Riemann hypothesis is instructive. It is an unproven conjecture, but tons of work (e.g. in algorithmic number theory) is predicated on its being true. While keeping in mind the Platonic ideal of what math should be, we should also acknowledge the messy reality of what math is. Mgnbar (talk) 12:46, 28 January 2022 (UTC)

Abstract objects and axioms
On a related note, I have some concerns about this sentence:
 * These objects are either abstractions from nature (such as natural numbers or lines), or (in modern mathematics) abstract entities that are defined by their basic properties, called axioms.

Now, it's sufficiently broad that I can't say it's actually wrong. I'm worried that it's misleading. For example, where would sets fall into this rubric? They might well be taken to be "abstractions from nature", but many readers are going to assume that they fall into the second disjunct, "defined by their basic properties, called axioms".

And even that is not quite wrong, taken broadly enough (for example, allowing axioms to be interpreted in second-order logic, and then mumbling something about the ordinals to keep the hierarchy from closing off too soon).

But there's going to be a temptation to read the language as suggesting that sets are "defined" by the first-order axioms of ZFC. And that is wrong. To the extent that they have a definition, it's given by the iterative hierarchy. --Trovatore (talk) 23:36, 28 January 2022 (UTC)

Rewrite of Sections 3-6: Be Bold?
Hi everyone, I talked about possibly reorganizing / rewriting the later part of the article a while back. Other things in life came up, but I've recently found time to work on it, and my draft is practically finished. It is a major article though, so I wanted to check first: Anyone mind if I'm bold and just do the edits here to kick off any discussion? Zar2gar1 (talk) 02:16, 10 February 2022 (UTC)
 * I agree that sections 3–6 need to be rewritten. This also true for sections 1.7, 1.8 and 1.9. In particular applied mathematics is not really an area of mathematics, but results on the epistemological (and historically dated) distinction between pure and applied mathematics. So, I suggests to merge section 1.7 into section 4.
 * In any case, it would be useful to have an access to your draft, for allowing discussion before implementation. D.Lazard (talk) 11:44, 10 February 2022 (UTC)
 * Aside: I wouldn't call the distinction between pure and applied math "epistemological" so much as "teleological". I do tend to agree that applied math is not an "area of mathematics" per se.  That said, I think it is frequently listed as one of the "areas of mathematics", and we might have to take that into account. --Trovatore (talk) 21:46, 10 February 2022 (UTC)

OK, I got my last changes together and put them up as a user draft. In advance though, I'll point out a few things: If you have any comments, I can respond to them here or at the draft. I'll let it percolate for at least a week, but after that, if nobody has any complaints or open change suggestions, I'll go ahead & update the article. Zar2gar1 (talk) 00:53, 12 February 2022 (UTC)
 * This is just a rewrite of what's currently in sections 3 through 6
 * I agree other parts could be refactored with these sections, but I wanted to focus on this as a 1st phase
 * Philosophy of math is now more implicit and spread across the sections by theme
 * I know the "proposed definitions" have bounced around under different names
 * I think a distinct section is best for now if everyone agrees with the reorg
 * I actually don't picture it staying a separate section, but again, I wanted to limit the 1st round of changes
 * I tried to focus on the section & paragraph level more than rewriting at the sentence level
 * In order to improve flow and clarity though, I wound up rewriting or adding a decent amount
 * I tried to be especially conservative with sourced sentences
 * The few sourced details I've removed (like quotations) have actually already been merged into other articles
 * So far it looks very good! But I haven't been through it in detail.  Please wait the full week at least. --Trovatore (talk) 04:26, 12 February 2022 (UTC)
 * I strongly support the change: as a mathematician, I recognize mathematics that I know, which was not the case previously. Even if there could be disagreements in details, they become easy to fix with local edits. D.Lazard (talk) 08:31, 12 February 2022 (UTC)


 * I'm glad you both like it so far; feel free to edit it in place or wait until I move it here. I'm also in no rush so I can leave it as a draft longer, say until mid-March. Zar2gar1 (talk) 17:20, 12 February 2022 (UTC)

I've gone back and re-synced my draft with changes over the past month. I also reworked the blurbs about "unreasonable effectiveness" and Platonism so they're more to the point and veer into philosophical details less. Does anyone have any other thoughts, should I give the draft more time for review, or does everyone think it's good to go? Zar2gar1 (talk) 23:27, 18 March 2022 (UTC)

Platonism and "unreasonable effectiveness"
Although the reference to Platonism is much better in the draft than in the current version of the article, it seems still incorrect to me, or, at least, somehow outdated (it was correct before 20th century). However, for the moment I am unable to propose something better. So, I will only explain my concerns.

Almost all modern mathematics are developed inside Zermelo–Fraenkel set theory. This means that all theorems and proofs are logical consequences of the axioms of this theory. So the fact that theorems and proofs are discovered or proven (not invented) has nothing to do with Platonism, but with the fact that many consequences are far to be immediate.

For a specific mathematical theory, the axioms are, they are chosen. The fact that a choice is "good" and an axiom system is useful means that it allows to unify existing mathematical theories and proving new theorems. A typical example is scheme theory, invented by Grothendieck, who discovered that it allows the unification of algebraic geometry and number theory in view of the proof of several conjectures (including, eventually, Fermat's Last Theorem. Again this has nothing to do with Platonism.

On the other hand, "unreasonable effectiveness of mathematics" and "Platonism" are two aspect of the unsolved philosophical question of the relationship between reality and logical constructions of thought. This is not mathematics, and I do not know how giving it its due place in this article. Suggesting that modern mathematicians are implicitly Platonists, means that they have a collective opinion on this question. This is blatantly wrong. — D.Lazard 11:29,12 February 2022‎

I think maybe it's important here that I say what I think this should imply for the actual article. I am not asking that the article state that most mathematicians are Platonists (even "without knowing it").
 * I won't argue with any of that. The best option is probably to trim down the philosophical bits further (moving fragments to Philosophy of mathematics if necessary). Previously there was no direct mention of Platonism whatsoever though, so I just wanted to put something in that flowed and actually motivated the view fairly. A fresh pair of eyes could probably do a better job of compressing what's in the draft now. Zar2gar1 (talk) 17:20, 12 February 2022 (UTC)
 * I suspect that D.Lazard is correct that mathematicians in general do not think they're Platonists. To put it polemically, I would say they are Platonists and just don't know it, but that's probably not a thing that needs to be said here.
 * I think it's quite wrong though to suppose that mathematicians in general work in a framework of "chosen" axioms, and in particular ZFC. I don't think most mathematicians could even tell you what those axioms (and axiom schemata) even are without looking them up.  And when was the last time any of you wrote down an instance of the axiom of replacement?
 * I think it's worth recalling here that when Wiles proved Fermat's Last Theorem, he technically used Grothendieck universes, which can't be formalized in ZFC. Now, almost no one really thinks the full power of Grothendieck universes are necessary to the proof.  But no one to my knowledge has officially managed to eliminate them.
 * Does anyone care? Mostly it's only logicians who care very much.  Because it's not really about whether FLT is a formal theorem of ZFC (or PA; it's probably a formal theorem even of PA, but again, not many people are really interested).
 * What they care about is that FLT is true in the natural numbers, not that it's a formal theorem of this or that set of strings and rules for manipulating them.
 * This is the sense in which most mathematicians behave as Platonists, whether they call themselves that or not. They're interested in the behavior of the structure, not in sequences of formal strings that follow certain rules. --Trovatore (talk) 02:42, 13 February 2022 (UTC)
 * If I understand well the above reasoning, mathematicians are Platonists without knowing it, but do not use ZFC because they do not know it. Should I recall that all textbooks on linear algebra prove that every vector space has a basis, and that the proof requires both the axiom of infinity and axiom of choice. Also, as far as I know, Platonism has to do with the nature of abstract objects such as numbers and infinite sets. Mathematicians work with these abstract objects without considering their philosophical nature, and this nature has no influence on their work. IMO, discussing on Platonism here is similar to discussing whether God created the Universe, in an article about physics: both discussions are about the nature of the objects of study, and are unrelated with the study itself and the methods of study. D.Lazard (talk) 09:20, 13 February 2022 (UTC)
 * The point I want to focus on here is that mathematicians do work with the abstract objects themselves, and not merely with the axioms that partially describe their behavior. (It's important to note that the axioms only partially describe the behavior, but that the objects themselves are taken as well-specified, generally with a minimum of fuss.)  To behave as a realist (maybe that's a better word here than Platonist) it's not necessary that you make specific commitments to the nature of the abstract objects.  Treating them as existing is enough "nature" to make you a realist. --Trovatore (talk) 17:54, 13 February 2022 (UTC)
 * Rather, there are a couple of things that I think it's important that the article avoid.
 * * First, we all agree that the axiomatic method is fundamental to mathematics. The article should avoid claiming that it's foundational to mathematics.  That might seem like a subtle distinction but it's important.  Specifically, the article should not imply that the axiomatic method demarcates mathematics from non-mathematics, nor that axioms in any way "define" their objects of discourse.  (It's not that those positions can't be represented, but they should be attributed to particular schools rather than stated as fact.)
 * * Second, while the article should discuss how mathematics was increasingly described in terms of set theory during the 20th century, it should avoid over-emphasizing a specific axiomatization such as ZFC. Mathematicians carried on doing what they had done before; they just now had a new language that facilitated communication between subfields. In most cases it didn't fundamentally change what they were doing.  Moreover, the set-theoryization of math predates ZFC by quite a lot &mdash; it really starts at least by the mid-19th century, not just before Zermelo but even before Cantor, with Dedekind and Weierstrass and Cauchy and Bolzano.  The set theory used in the bulk of math is indistinguishable for most purposes from so-called "naive" set theory.  Note in passing that, as I understand it, Zermelo's conception of set theory was to be understood in second-order logic, whereas ZFC is a first-order theory.
 * --Trovatore (talk) 22:05, 13 February 2022 (UTC)
 * For me, Trovatore has put things perfectly in perspective. I agree with all of the above. Paul August &#9742; 23:06, 13 February 2022 (UTC)

Rewrite of later sections done
Hi everyone, since it's been over a month & a half without any comments on the draft and a couple weeks since my last RfC above, I went ahead & applied my rewrite of the later sections in the article. Just in case there's a consensus the changes need to be reverted & kicked back to a draft though, I'm leaving the original draft page up for a bit at User:Zar2gar1/Math rewrite draft.

I hope everyone likes the changes though, and I think the different structure will make further improvements easier. I actually have other ideas from here, and if nobody minds, I'll probably restore the To-Do list here on the talk page. Otherwise though, I'll be taking a long wiki vacation before trying more rewrites. Ramadan mubarak, y'all! Zar2gar1 (talk) 17:36, 2 April 2022 (UTC)
 * After a quick look at the new version, it seems much better than the previous one. Thanks. D.Lazard (talk) 21:03, 2 April 2022 (UTC)

Semi-protected edit request on 25 April 2022
In the first sentence, could you replace "knowledge, which" with "knowledge that"? Right now it's a non-restrictive clause (see English relative clauses), and a restrictive clause would be better. It sounds as if it's saying "maths is a branch of knowledge, which by the way happens to include..." instead of the intended meaning "maths is the branch of knowledge comprising..." 49.198.51.54 (talk) 20:27, 25 April 2022 (UTC)
 * You're right. Thanks. Done. Mgnbar (talk) 01:33, 26 April 2022 (UTC)

Opening sentence
Anyone wishing to change the opening sentence might want to discuss proposed edits here, after reading archived discussions. The fact that mathematics has no agreed-upon definition makes it difficult to come up with concise, accurate opening text &mdash; desirable as that may be. Mgnbar (talk) 12:16, 20 May 2022 (UTC)
 * In particular, any claim that math is a science is controversial and not an accurate summary of the reliable sources. I wish that editors of the lede would discuss proposed changes here, before disrupting consensus (however imperfect) constructed over years of debate. Mgnbar (talk) 18:54, 20 May 2022 (UTC)
 * My concern relates primarily to concision and readability, not to the substance of the definition in the lede. No definition is perfect, and concision always sacrifices a degree of accuracy. Yet, an average reader might easily agree that 12 words are superfluous when one word will do. It's immaterial that D.Lazard and I agree mathematics is a science; Oxford English Dictionary says mathematics is a science and Merriam-Webster says mathematics is a science. So, the lede merely attests that common usage, and the link to science adequately explains what "science" means rather than torturing this article with the "an area of knowledge that includes the study of such topics as ...(blah, blah, and blah)" periphrastic, circumlocutory, namby-pamby monstrosity. If someone edits the current lede by substituting and citing an alternatively concise definition, I'm all for it. Kent Dominic·(talk) 19:07, 20 May 2022 (UTC)
 * P.S. Mathematics: "The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols," per American Heritage Dictionary. Quoting and citing that definition has its merits, but limiting the definition to "study" (i.e., in the same manner as the prior lede definition) excludes the practice and operations associated with mathematics. --Kent Dominic·(talk) 19:19, 20 May 2022 (UTC)
 * Accurately summarizing the reliable sources is necessary; it is essentially the second pillar of Wikipedia. Concision is desirable but not strictly necessary. So we sacrifice concision when we must. Mgnbar (talk) 21:05, 20 May 2022 (UTC)
 * I reverted the good faith edit, but 3RR is now in effect for K.D. for 31 hours. --Ancheta Wis   (talk  &#124; contribs) 21:14, 20 May 2022 (UTC)
 * I guess that you intended to revert Kent Dominic's edit of the opening sentence, but it is another one of their edits that you reverted. Nevertheless, I agree with this revert also. Thus, I have restored the stable version of the opening sentence. D.Lazard (talk) 17:44, 22 May 2022 (UTC)
 * See my reply below --Ancheta Wis   (talk  &#124; contribs) 21:33, 22 May 2022 (UTC)

I have restored the stable version of the opening sentence. A consensus is required here for changing it again. D.Lazard (talk) 17:44, 22 May 2022 (UTC)


 * You undid an edit by Mgnbar. I had already thanked Mgnbar for the sake of maintaining concision, so there's a 2:1 consensus for the edit among those who have expressed an opinion.
 * Rather than revert your edit on the whole to what Mgnbar and I agreed, I'm going to momentarily tweak the current lede to remedy what was problematic from square one: wordiness and syntax errors.
 * Specifically, "an area of knowledge that includes the study of (something)" doesn't make sense regardless of the topic. Meaning, the reversion restores verbiage where the "that" clause corresponds solely to knowledge, creating the equivocal "knowledge ... includes study" phrase.
 * Instead, a sensible interpretation is, "an area of knowledge and study that includes (blah blah)." As always, the coming edit is intended as an improvement, not an item of perfection. I'm going to ID the edit as minor since there'll be no change to anything except function words.
 * Everyone, rather than mere reversions to what had been there, please ensure any subsequent changes semantically comport with common sense. Kent Dominic·(talk) 19:43, 22 May 2022 (UTC)
 * In my edit, I was compromising between the long-standing text and the text that Kent Dominic was proposing. I replaced "science" with "study", just to prevent the article from violating neutrality. I have no special love for Kent Dominic's version or my tweak of it.
 * Moreover, although consensus does not have to be unanimous, we do not have consensus here for Kent Dominic's version or my tweak of it. Anyone, who has not been following this article for a long time, may not understand how contentious this lede is, and how long it takes to build consensus. Mgnbar (talk) 20:00, 22 May 2022 (UTC)


 * I found in your first edit summary of 20 May 2022 your argument "Although I personally agree with the word "science", there is no consensus that mathematics is a science. Moreover, "science of ..." is much too restrictive". I have several questions / remarks:
 * (1) I wonder who challenges mathematics being a science.
 * (2) Should we give the contemporary understanding of (contemporary) mathematics, or summarize all understandings the word had throughout history (i.e. including when numerology was considered part of mathematics)?
 * (3) Defining mathematics by enumerating its "typical" topics is unsatisfactory to me; I'd prefer to give an abstract characterization of what makes a topic belong to (contemporary) mathematics. Imo, it is the use of rigorous (or even: formal) proof as the main (or even: only) method of obtaining knowledge. However, I see that none of the above sources mention the concept of proof. (This seems to answer my question under (2) as: "summarize all history, including the pre-scientific one".)
 * (4) I agree that "area of knowledge and study that includes such topics as" is rather clumsy. 's most recent edit ("study of") seems to convey the same information; "knowledge" might go without saying in the lead, and "area" and "topics" are redundant. We might insert an "e.g." as a replacement for "such as". I'd prefer, however, if my suggestion under (3) was acceptable, and we could avoid enumeration completely. - Jochen Burghardt (talk) 20:20, 22 May 2022 (UTC)
 * Jochen, search the archives for the "science" issue; it's very contentious. As for your point 3, I strongly disagree.  Any abstract characterization of mathematics is inevitably going to be POV, and it isn't at all necessary in the lead section.  We can and should defer the (multiple) abstract characterizations in play till a deeper dive somewhere in the body of the article. --Trovatore (talk) 20:26, 22 May 2022 (UTC)
 * Jochen, thank you for summarizing; my personal motivation for the 3RR was to restore the place of the talk page in WP:BRD. It appears that the talk page is now being used for its intended function.


 * D.Lazard, I have been watching this article's development (e.g. for Combinatorics, which is still in process, as most of us are aware; my personal interest is in your summary of Ramsey's work for the article section). An up/down vote of individual positions, while momentarily satisfying, is not my personal interest. If it were possible to reconcile your views with Jochen's (3) above, as well as Trovatore's characterization of (3)'s limits, perhaps we might get further follow-up development of the article beyond this roadblock. --Ancheta Wis   (talk  &#124; contribs) 21:10, 22 May 2022 (UTC)
 * , let me state my view plainly. I am utterly opposed to anything resembling (3) in the lead section.  An abstract characterization should not appear in the lead section at all. --Trovatore (talk) 05:16, 23 May 2022 (UTC)
 * I'm not thrilled with Mgnbar's version, but I support it. I'm not thrilled with my "science" version either because of the linked language in the respective science article and mathematical sciences article, but I indeed think mathematics is a science. My "an area of knowledge and study that includes such topics as ..." edit is purely formulaic re where the "that" clause should go. Yet, one particular editor keeps reverting, accusing me of edit warring, insisting on discussion here, but has been absent from the discussion. What gives? Kent Dominic·(talk) 20:28, 22 May 2022 (UTC)
 * Kent Dominic, thank you for using the talk page. --Ancheta Wis   (talk  &#124; contribs) 21:10, 22 May 2022 (UTC)
 * Jochen, thank you for summarizing; my personal motivation for the 3RR was to restore the place of the talk page in WP:BRD. It appears that the talk page is now being used for its intended function.
 * D.Lazard, I have been watching this article's development (e.g. for Combinatorics, which is still in process, as most of us are aware; my personal interest is in your summary of Ramsey's work for the article section). --Ancheta Wis   (talk  &#124; contribs) 21:12, 22 May 2022 (UTC)
 * If a some object is not science, then is it a not-mathematics ?--SilverMatsu (talk) 22:29, 22 May 2022 (UTC)
 * It is possible for a subject of study to not be science, and yet remain mathematical; for example the work of Pappus of Alexandria, see Pappus's hexagon theorem. (But here you see my bias, what about the other formal sciences? and what about the role of conjecture?, or mathematical imagination? As Feynman puts it: where is this place?) To the other editors: I can reply to SilverMatsu privately, from now on. I suspect I should. --Ancheta Wis   (talk  &#124; contribs) 00:29, 23 May 2022 (UTC)
 * As for my above (3), I'm afraid I have to agree with you, all the more as all sources are against my personal taste (sigh!). - Jochen Burghardt (talk) 08:00, 23 May 2022 (UTC)

Here are some comments that can be view as an answer to Jochen, but are more general. This is the reason for not indenting them. D.Lazard (talk) 10:00, 23 May 2022 (UTC)
 * "Science", "area of knowledge" or "study": Numerous archived discussions show that it is controversial to qualify mathematics as a science. "Mathematics is the study ..." is not acceptable, as suggesting that the following list is complete. "Mathematics includes the study ..." is acceptable. However, with the overuse or the term "study" for qualifying new academic disciplines (gender studies, ...), the term may be misleading as hiding that learning mathematics is needed for talking about mathematics. This is why I prefer the term "area of knowledge" that was introduced by another editor in the short description.
 * Definition of mathematics:In previous discussion it appeared clearly that there is no non-controversial definition of mathematics. This is why I have restructured the lead for explaining the main specificities of mathematics: Proofs and rigor, proofs vs. experimental evidence, ubiquity of applications, etc. Here, I agree with Jochen on the importance of proofs, but it seems very difficult to include this in the first sentence
 * List of areas in the first sentence:It is clear that the list is incomplete. It is here for the link with dictionary definitions and what is mathematics for the layman. Maybe, one could changes "includes" into "includes (but is far to be limited to)". But this would make the first sentence more clumsy.
 * Numerology and non-rigorous aspects:The last paragraph began by saying that before Euclid, there were no proofs in mathematics. This has been removed by Kent Dominic. IMO, this must be restored, but this is another discussion.
 * Two comments: (1) I deleted "Mathematics has been a human activity from as far back as written records exist" as WP:UNDUE, as part in the wrong section, and as awkwardly worded. The assertion that Euclid is among the first re mathematical proofs is still intact. (2) The word "includes" needs no clarification since the plain meaning connotes items beyond those that are mentioned. It's unnecessary to say, e.g., "includes, but is not limited to." By analogy, "my bike includes new brakes" implies that my bike comprises things beyond mere brakes. --Kent Dominic·(talk) 10:33, 23 May 2022 (UTC)

Opening sentence, part II
I propose two changes to the current lede, from this:
 * Mathematics is an area of knowledge that includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry),and quantities and their changes (calculus and analysis).

to this:
 * Mathematics is an area of knowledge, study, and learning that includes such topics as number operations (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

My immediate concern relates solely to concision and syntactic cogency. Any takers? Pot-shotters? Hand grenaders? --Kent Dominic·(talk) 20:50, 22 May 2022 (UTC)
 * I'm fine with that. Paul August &#9742; 21:29, 22 May 2022 (UTC).
 * I've struck my comment above because the proposed change was modified after my comment about it. I no longer support the new sentence. Paul August &#9742; 13:27, 23 May 2022 (UTC)
 * Thanks Paul, but I just now noticed an incongruity: the máthēma definition includes "learning" but it's omitted in the mathematics definition. Also, I think the mention of numbers needs clarification to specify their operational sense (i.e., excluding their ordinal sense). Please weigh in again after I tweak the definition accordingly. Kent Dominic·(talk) 09:03, 23 May 2022 (UTC)
 * Please don't modify your comments after they've been replied to! Paul August &#9742; 13:27, 23 May 2022 (UTC)
 * Alerting re modification is standard procedure. Thanks for noting you've withdrawn your support. Anyhow, the pertinent proposal had already been mooted by the version that appears further down in this thread. Kent Dominic·(talk) 13:36, 23 May 2022 (UTC)
 * Fine, but please don't do that again. What you did made it appear that I had supported your proposed changes when in fact I hadn't. For the applicable quideline please see WP:TALK, which says that "if anyone has already replied to or quoted your original comment, changing your comment may deprive any replies of their original context, and this should be avoided." Paul August &#9742; 14:11, 23 May 2022 (UTC)
 * @Paul August: Forgot to mark it as (five tildes+ edited). Oops. My bad. No harm, no foul? Kent Dominic·(talk) 14:23, 23 May 2022 (UTC)
 * @User:Kent Dominic It's more prudent to alter a proxy for the lede, here in this section, rather than to alter the article lede itself. Please wait for consensus on the article's lede before touching it there. --Ancheta Wis   (talk  &#124; contribs) 10:31, 23 May 2022 (UTC)
 * I strongly oppose to add "study and learning" to "area of knowledge": this is implied by the phrase "area of knowledge" and it is misleading as it may be understood as excluding theorems and proofs.
 * I would agree with removing "the study of", giving "is an area of knowledge that includes such topics as ..."
 * I strongly oppose with replacing "numbers" with "number operations", which excludes unduly number definitions and number properties.
 * I agree with adding "the" in "shapes and the spaces in which they are contained. So, my proposal is
 * D.Lazard (talk) 11:04, 23 May 2022 (UTC)
 * I support that definition. I'm not insistent on "number operations." It's just that numbers, broadly construed, includes words like first, second, etc., or Chapter One, or Part IV numbers that typically aren't construed as mathematical. --Kent Dominic·(talk) 11:14, 23 May 2022 (UTC)
 * Note for KD: There is a kind of number (the ordinal number) which is accepted in the corpus of mathematics. There are even computer systems that understand mathematical objects such as the ordinals, including infinite ordinals. Some mathematicians use these computer systems as proof assistants before publication. There are mathematical objects (such as operators that are well-behaved enough for laymen to rely on) that can be unattended. (But I have never encountered a mathematical object that can construe things.) --Ancheta Wis    (talk  &#124; contribs) 12:24, 23 May 2022 (UTC)
 * You make an interesting argument, but I'm 50/50 whether to agree. I deem my use of ordinals via set theory algorithms to be logical analysis per linguistics, not mathematical analysis per "science," however one construes that word. I don't know if Kurt Goedel would say ordinal numbers are mathematically intrinsic or analytically  extrinsic to his incompleteness theorem. Computers' understanding or ordinals doesn't convince me that ordinals are mathematical, however. Computers understand my voice commands, but that doesn't make my voice mathematical unless you want to echo the person who first said that "everything is measurable and mathematical." Kent Dominic·(talk) 14:04, 23 May 2022 (UTC)
 * Some punctuation changes to delete the nested parentheses for readability's sake:
 * --Kent Dominic·(talk) 11:22, 23 May 2022 (UTC)
 * This version is fine with me. Paul August &#9742; 13:42, 23 May 2022 (UTC)
 * I would suggest simplifying to "...is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas (algebra), shapes (geometry), and how quantities change (calculus and analysis). I don't think the words I took out add much understanding. Perhaps also add "and reasoning itself (mathematical logic)." The second paragraph deserves attention as well, it's use of "abstract" four times is sloppy.--agr (talk) 12:27, 23 May 2022 (UTC)
 * This shortened version is fine for me. For the second paragraph, I suggest to replace "abstract entities of which certain properties, called axioms, are stipulated" with "entities that are stipulated with (by? through?) certain properties, called axioms". This makes clearer that axioms "define" the entities. D.Lazard (talk) 12:57, 23 May 2022 (UTC)
 * I am skeptical about the addition of "and reasoning itself (mathematical logic)": proofs and rigor are already very present in the lead. Adding this would put to much emphasis on an area that is indeed important, but not more than other parts of mathematics such as Discrete mathematics. D.Lazard (talk) 13:12, 23 May 2022 (UTC)
 * @agr and @D.Lazard Any iteration absent "knowledge that includes the study" works for me, and I'm indifferent about including mathematical logic, but I wouldn't equate mathematical logic to "reasoning itself". Kent Dominic·(talk) 14:27, 23 May 2022 (UTC)
 * I am skeptical about the addition of "and reasoning itself (mathematical logic)": proofs and rigor are already very present in the lead. Adding this would put to much emphasis on an area that is indeed important, but not more than other parts of mathematics such as Discrete mathematics. D.Lazard (talk) 13:12, 23 May 2022 (UTC)
 * @agr and @D.Lazard Any iteration absent "knowledge that includes the study" works for me, and I'm indifferent about including mathematical logic, but I wouldn't equate mathematical logic to "reasoning itself". Kent Dominic·(talk) 14:27, 23 May 2022 (UTC)

The new lede: further considerations
The current lede reads as follows:
 * Mathematics is an area of knowledge that includes such topics as..."

That phrasing addresses my concern how the previous "knowledge that includes the study of" is grammatically and semantically combobulated, distended, and just plain weird. I don't plan to edit the lede further since I'm no longer interested in linking the article to my own publication, which instead defines mathematics as "a science that..." IMHO, however, this article's current lede is too narrow. Consider how: Consider how we use "mathematics" in everyday parlance: Accordingly, I don't see any semantic issue with changing the lede to this:
 * 1) Mathematics implies knowledge (i.e., "the fact or condition of knowing something with familiarity gained through experience or association," per per Merriam-Webster).
 * 2) Mathematics implies study (i.e., "application of the mental faculties to the acquisition of knowledge," per per Merriam-Webster).
 * 3) Mathematics implies operations (i.e., "performance of a practical work or of something involving the practical application of principles or processes", per Merriam-Webster).
 * A: Do you know much about physics? B: Not really, I only know mathematics.
 * A: What was your undergrad major? B: I majored in chemistry and studied mathematics as a minor.
 * A: Is A ⊆ B a mathematics operation? B: In one sense, yes. In another sense it's a symbolic logic statement about sets.
 * Mathematics is an area of knowledge, study, and operations that include such topics as..."

The issue of consensus I leave to whomever else cares to follow up. --Kent Dominic·(talk) 13:18, 24 May 2022 (UTC)
 * Wikipedia is not a dictionary. This implies that the article is not a bout the common meaning of the word, but about the area of knowledge (or science, if you prefer). As every area of knowledge can be known (at least partially; I doubt that anybody can know all mathematics), studied and used (through operations or other means), your edit sugggestion is confusing, as implying a meaning of "area of knowledge" that is not the common one.
 * Said otherwise, there is a fundamental semantic issue in your suggestion: If you replace "area of knowledge" with "science", the semantic of the current sentence is not really changed, while your sentence becomes nonsensical. D.Lazard (talk) 13:51, 24 May 2022 (UTC)

Geometry
The edit with a --commented note-- "For example, it is not sufficient to verify by measurement that, say, two lengths are equal; such a property must be proven via abstract reasoning from previously accepted (theorems) and basic properties that are considered as self-evident because they are too basic for being the subject of a proof (postulates). -- This sentence requires further revision, starting from "and": (1) "properties" is not the correct word; (2) the run-on sentence with four separate clauses needs parsing into smaller segments without repetition of "basic"  in order to become comprehensible; (3) "they" doesn't seem to correspond to whatever is intended by "properties", which is the nearest syntactical referent. --" might be rewritten thus:
 * For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems), and from statements which are self-evident. The self-evident statements  (postulates) need not be subject to proof.

--Ancheta Wis   (talk  &#124; contribs) 00:17, 25 May 2022 (UTC)


 * Good. Kent Dominic·(talk) 01:45, 25 May 2022 (UTC)
 * I suggest instead:
 * For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms).
 * Reasons: the unproved statements may be self-evident, but they may also be counter-intuitive in the case of axioms; emphasis that there is a small number of unproved statements. D.Lazard (talk) 07:55, 25 May 2022 (UTC)

Observations on current lead
I haven't had attention to spare for the details of the current discussions, but I thought I'd take a look at the current lead paragraph and see what I thought. I don't actually like it much, but the changes I object to don't seem to be a result of the Kent Dominic intervention. Some time ago, it read as follows:
 * Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity, structure, space, and change.  It has no generally accepted definition.

Now, this was certainly not beyond improvement, but I don't really think the current version is an improvement:
 * Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).   There is no general consensus about its exact scope or epistemological status.

Some specific concerns: Moving on, I continue to detect a bit of a formalist POV in the second paragraph, which glosses axioms as "entities that are stipulated". First of all, "entities"? They're not ghosts, they're assertions. "Stipulated" seems to exclude the possibility that they are self-evident or discovered empirically, and even suggests an extreme "glib formalist" conception according to which they're actually arbitrary. --Trovatore (talk) 05:54, 25 May 2022 (UTC)
 * Glossing "numbers" with "arithmetic and number theory" tends to suggest something specific to the natural numbers or integers.
 * "[F]ormulas and related structures" is a very odd phrasing. Formulas are syntactic whereas structures are semantic.  To be sure, lots of readers aren't thinking of "formulas" in the sense of "formulas of the lower predicate calculus", and if they were, then I suppose you could point out that abstract algebra does also treat models that satisfy the formulas.  Still, it strikes me as wording that's trying to sound precise without actually being precise.  I have no objection to some vagueness early on, but it should be clear that it's vague, and ideally in what way.


 * I essentially agree with 's views here. In particular, I prefer the older lede sentence (a much wrangled over and well-established consensus). Paul August &#9742; 13:24, 25 May 2022 (UTC)

Introductory sentence: synopsis of other sources
For those who wish inspiration by existing introductioins to mathematics, here is an overview. Extend it by adding your favorite introduction if it is still missing here. - Jochen Burghardt (talk) 08:37, 27 May 2022 (UTC)


 * Mathematik (translation by www.deepl.com)
 * Mathematics (Federal High German: [matemaˈtiːk], [matemaˈtik]; Austrian High German: [mateˈmaːtik];[1] Ancient Greek μαθηματικὴ τέχνη mathēmatikē téchnē 'the art of learning') is a formal science that arose from the study of geometrical figures and arithmetic with numbers. There is no universally accepted definition for mathematics; today it is usually described as a science that uses logic to examine self-created abstract structures for their properties and patterns.


 * Mathématiques (translation by www.deepl.com)
 * Mathematics (or mathematics) is a body of abstract knowledge resulting from logical reasoning applied to various objects such as mathematical sets, numbers, shapes, structures, transformations, etc.; as well as to the mathematical relations and operations that exist between these objects. They are also the field of research that develops this knowledge, as well as the discipline that teaches it.


 * American Heritage Dictionary
 * The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.


 * Merriam-Webster
 * the science of numbers and their operations (see operation sense 5), interrelations, combinations, generalizations, and abstractions and of space (see space entry 1 sense 7) configurations and their structure, measurement, transformations, and generalizations


 * Oxford English Dictionary
 * The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics)

Overreaching re "no general consensus"
The lede paragraph says about mathematics, "There is no general consensus about its exact scope or epistemological status." The Proposed definition section likewise says, "There is no general consensus about the exact definition or epistemological status of mathematics." In both instances, the no general consensus phrase warrants an Attribution needed flag. The corresponding cites given for the statements aren't similarly overreaching: Roberta Mura limits her statement to a lack of consensus among university teachers of mathematics; Robies & Neunzert limit their statement to illustrious scholars having reached no consensus. The most that can be said, based on those two cites, is that university mathematics teachers and illustrious scholars disagree about the exact definition, scope, and epistemological status of mathematics. Or, alternatively, there is no consensus among university mathematics teachers and illustrious scholars re blah blah. --Kent Dominic·(talk) 03:03, 25 May 2022 (UTC)
 * I suggest to remove this sentence. It is not encyclopedic to list what a subject is not. The sentence was useful when the first sentence tried to define mathematics. Presently, the aim of the beginning of the article is to describe mathematics instead of defining it. So, it makes no sense to discuss opinions on mathematics in the first sentence: considering that mathematics is or is not a science is an opinion; every definition of mathematics is an opinion of its author, as the general opinion of mathematicians is that no true definition is possible. D.Lazard (talk) 08:20, 25 May 2022 (UTC)
 * Agree on removal. I would suggest a replacement: "Mathematicians have found important connections between its different topics."--agr (talk) 12:45, 25 May 2022 (UTC)
 * Or, "Mathematicians have found important connections between its different topics but widely agree about its exact definition, scope, and epistemological status." Kent Dominic·(talk) 14:03, 25 May 2022 (UTC)
 * but widely [dis]agree? Ancheta Wis   (talk &#124; contribs)
 * Yeah, that's what I meant: widely disagree. If not, wildly disagree. =/ --Kent Dominic·(talk) 16:01, 25 May 2022 (UTC)
 * @agr: Perhaps the Domain of discourse could be used to limit the statement "There is no general consensus about its exact scope or epistemological status". Thus:
 * There is no general consensus about its exact scope or epistemological status, without also stating (or constructing) its domain of discourse.
 * --Ancheta Wis   (talk  &#124; contribs) 14:14, 25 May 2022 (UTC)
 * I suggest keeping an amended version of the sentence, but I'm not averse to removing it. The lede sentence still defines mathematics albeit in a manner quite shy of lexicographic rigor. So, I think it's fair to somehow explain the lede's definition-qua-non-definition by saying university mathematics teachers and illustrious scholars disagree about the exact definition, scope, and epistemological status of mathematics, as the relevant cites point out. Subtextual apology: We, as editors, tried but failed to provide a definition on which mathematics instructors and scholars widely agree. Kent Dominic·(talk) 14:01, 25 May 2022 (UTC)

Can someone name a field of study with a "general consensus about its exact scope or epistemological status"?--agr (talk) 00:51, 26 May 2022 (UTC)


 * ☐ ☒ Kent Dominic·(talk) 10:49, 26 May 2022 (UTC)


 * I think that the historical reason for the "math is not clearly defined" text in the lede is that both readers and editors tend to be surprised when they see that the lede does not define mathematics. So then they ask about, or even boldly insert, their favorite definition &mdash; which, even if it's supported by reliable sources, conflicts with other reliable sources. And then that leads to hundreds of person-hours of discussion (often resulting in no net improvement of Wikipedia). Mgnbar (talk) 18:21, 26 May 2022 (UTC)


 * To respond to ArnoldReinhold's question just above: I think that more concrete subjects are more easily and accurately definable. For example, biology is the study of life. This definition is not perfect, because there are gray areas about what's living and not, etc., but it's still highly accurate and understandable.
 * On a related note, an editor once complained to me that Manifold (mathematics) was too abstract. They pointed to Cell (biology) as an example of a technical scientific article that was nevertheless understandable. My response what that you can ask many concrete questions about cells &mdash; how big they are, what they're made of, how many there are, etc. &mdash; that you can't ask about manifolds, because manifolds do not exist (outside the mind). I think that disagreement, about what math is, arises partially from its abstractness. Mgnbar (talk) 18:35, 26 May 2022 (UTC)
 * Is there a universally agreed definition of what life is? Do viruses count? Must it contain RNA? Does it have to be carbon-based? Is psychiatry part of biology? Anthropology? Political science? SETI? The Biology article's lede links to life which in turn says "There is currently no consensus regarding the definition of life." Our article has a section on Definition which links to a whole article on the different definitions of mathematics. I don't think a disclaimer needs to be our second sentence and especially not in such pretentious language.--agr (talk) 20:04, 26 May 2022 (UTC)


 * My post, to which you were responding, already acknowledged the grayness of the definition of life, but okay. One problem with leaving the definition to the Definition section is that many readers read only the lede, and hence many readers complain before ever getting to the Definition. One problem with having a defective (meaning, in violation of Wikipedia's second pillar) definition in the lede is that people tend to cite Wikipedia article ledes a lot. (Admittedly this is anecdote not data.) Mgnbar (talk) 20:37, 26 May 2022 (UTC)
 * We do not need a definition of mathematics in the lead. We need a description of what is mathematics. This is in this spirit that the whole lead and the first section have been written. So, I agree with the removal of the disclaimer. D.Lazard (talk) 20:45, 26 May 2022 (UTC)
 * I'm okay with having no definition in the lede, as long as the wording makes it clear that no definition is being given. In my experience, it is difficult to find such wording. Our consensus (as of two weeks ago or so) accomplished the job through its clumsy "includes"/"including"/"such as" language. Mgnbar (talk) 20:53, 26 May 2022 (UTC)
 * (And I see that our current versin also includes that language. So I'm okay with deleting the offensive sentence.) Mgnbar (talk) 20:57, 26 May 2022 (UTC)
 * I've started a synopsis of other introductions to mathematics below; feel free to extend it. In particular, I like the German Wikipedia approach ("arose from", "no universally accepted definition", "science that uses logic"[disagreed by ]). - Jochen Burghardt (talk) 08:42, 27 May 2022 (UTC)
 * I'm not sure why I was called out here, but perhaps to avoid misunderstandings I should respond briefly. I don't personally object to "science".  I said it was contentious, which it is. The usual objection is that science is empirical whereas mathematics is purely deductive.  Given that I don't think mathematics is purely deductive but consider it to be partly empirical (in particular the axioms of infinity, powerset, replacement, and large cardinals fit nicely with a falsificationist empiricist epistemology), I am happy to call it a science, but others are not. If I did object personally to something in the de.wiki approach, it would probably be "self-created structures", as I'm not sure what that's getting at, but it might tend to suggest that mathematics is invented rather than discovered. --Trovatore (talk) 16:31, 27 May 2022 (UTC)

Incorrect history of mathematics
In the introduction it is claimed that "Mathematics developed at a relatively slow pace until the Renaissance, when algebra and infinitesimal calculus were added to arithmetic and geometry as main areas of mathematics." Mathematics did develop at a relatively slow pace until the Renaissance. Where does one draw the line with slow. I'd argue mathematical infancy ends in the 19th century. There's way more mathematics after than before that. Algebra pre-dates the Renaissance regardless of whether it starts with Diophantus or Al-Khwarizmi and calculus comes slightly later. — Preceding unsigned comment added by 101.112.251.146 (talk) 07:29, 29 May 2022 (UTC)

Request to move draft to page article
This page will be placed in the following categories if it is moved to the article https://en.wikipedia.org/wiki/Draft:Soufia_Taloni 196.80.38.18 (talk) 11:16, 30 May 2022 (UTC)


 * This talk page exists to discuss the article Mathematics. If you wish to discuss creating articles related to mathematics, then try Wikipedia talk:WikiProject Mathematics. However, I cannot detect that Soufia Taloni is mathematical at all. Mgnbar (talk) 11:47, 30 May 2022 (UTC)

A proposed tweak to the lead
I think there's an important element missing from the lead. But having no intention to be a bull in the china shop that is this article, I first come here to seek consensus (or perhaps a world-weary chorus of "Don't touch a thing!"). What would people say to my modifying the final sentence in the first paragraph to read


 * There is no general consensus about its exact scope or epistemological status,[6][7] though among contemporary mathematicians it is common to conceive of it as the science of patterns[8][9].

where [8] and [9] would link to Hardy's A Mathematician's Apology and Steen's 1988 article in Science?—PaulTanenbaum (talk) 20:24, 8 June 2022 (UTC)
 * "Science of patterns" is highly controversial, since "pattern" is a term that is rarely used in mathematics, except maybe in specialized areas like combinatorics. Moreover, "among contemporary mathematicians it is common to conceive ..." is not only WP:OR, it is pure invention: I know many mathematicians, and do not know any who would agree with this sentence.
 * By the way, this sentence has been discussed above . There is a consensus there to remove this sentence, and I'll do that. D.Lazard (talk) 08:25, 9 June 2022 (UTC)
 * PaulTanenbaum's proposed text and sources could be valuable in the "Definitions of mathematics" section. Mgnbar (talk) 13:58, 9 June 2022 (UTC)
 * There is no section "Definitions of mathematics". The section is perfectly neutral, which is very difficult for such a controversial subject (I am not the author of this section). Please, do not break this neutrality, as PaulTanenbaum's suggestion would do. D.Lazard (talk) 14:32, 9 June 2022 (UTC)
 * Yes, I was talking about the "Proposed definitions" section, which links to the "Definitions of mathematics" article, and I mixed up the wording. A Hardy quote is already in that article. I don't know the Steen quote, but it could be added to that article. And then there's always the question of how much of that article to summarize in the "Proposed definitions" section.
 * In short, PaulTanenbaum is trying to add content to Wikipedia, and he is doing it in just the right way (building consensus on the talk page), and I'm trying to help him find the right wording and place for his content, rather than dismissing his contribution out of hand. Mgnbar (talk) 15:37, 9 June 2022 (UTC)

Discrete Mathematics Subsection
I'm not sure that there should be a discrete mathematics subsection (which is currently present, but unfilled) in the areas of mathematics subsection, but I wanted to determine the consensus before removal. The reason is that I don't believe this term is used in most current classification schemes. For example, under MSC2020, it is a second-level topic (68R) considered as a subarea of computer science, and within Wikipedia's Math topics TOC it is not listed. I have written a basic combinatorics subsection, which could be expanded to include its relations to discrete math, and there is good reason to include the term somewhere on the page, but I don't think its independent enough from the other areas of mathematics to be included in the Areas of mathematics section. Juto20 (talk) 20:17, 28 April 2022 (UTC)


 * The section got added on 13 November, 2021, and has remained empty since then. Looking through previous versions, the mention of discrete mathematics was indeed just a passing mention of the term (E.g. this revision {Search for "discrete"}). Id agree with simply removing the section and having it be a passing mention again. Aidan9382 (talk) 20:22, 28 April 2022 (UTC)

I strongly disagree with the removal of the section. On the opposite, the section, must be included in it, after being rewritten in a style that is compatible with that of the preceding sections. Here are the reasons.

This article is for readers who know very little of mathematics. Its large first section is here for explaining what is mathematics. That is, it plays the role of a "Definition section". Due to the large number of areas of mathematics, and for having an acceptable size, the subsections must be as wide as possible, and, when possible, cover several first-level sections of AMS2020 classification.

Section was partly rewritten in this spirit in November 2021, but, for several reasons, this was not finished for sections, , , and.

Discrete mathematics is an area of mathematics that emerged recently. It includes combinatorics, graph theory, and many other areas concerned with discrete behaviours that are not well solved by continuous methods of mathematical analysis. One may say that it consist of all areas in which NP-completeness is commonly encountered (this is my personal opinion). The fact that discrete mathematics is an established area of mathematics, is attested by the fact that there are two mathematical journals whose titles start with "Discrete Mathematics", which have an article in Wikipedia.

So, "Discrete mathematics" is the good classification granularity for having most mathematics covered withs a small numbers of subsections in.

I'll try, in next days to fill the section, and to merge in it the recently created section. D.Lazard (talk) 15:02, 30 April 2022 (UTC)


 * After composing and deleting a summary of how messy these categorization issues are, let me just say: Juto20 raises a very important point. It would be great if D.Lazard would fill in that section --- at least to draft quality --- so that we could see how much material, overlapping with how many other mathematical topics, might end up in such a section. Mgnbar (talk) 15:34, 30 April 2022 (UTC)


 * I agree with Mgnbar that it would be nicer to work from a draft. I will say am still hesitant about its inclusion, mainly as this area overlaps so heavily with others. That said, a well-written section could fix another major problem with this section. Namely that a large number of more applied areas (even in the TOC) are omitted from the general discussion. Though the inclusion of everything individually would admittedly be a lot of work and possibly too long for the page. In either case, I will not edit anything then until something hase been added somwhere. (PS: This is mostly irrelevant, but responding to the personal opinion of D.Lazard. A think a place where the NP-completeness is commonly encountered that is not discrete math is continuum optimization, while something like integrable combinatorics would be the opposite. These claims are of course my own personal opinion) Juto20 (talk) 18:08, 30 April 2022 (UTC)

I have written a first version of section, and moved as a subsection of it. Clearly my work requires many improvements, and must be removed or merged, but this has to be discussed here. Also, I have included, as a blind comment, the scopes of the two major journals on the subject. This can and must be used as sources, but, for the moment, I do not know how format such citations. D.Lazard (talk) 11:23, 1 May 2022 (UTC)
 * Since more than a month, nobody has objected to my suggestion of removing the subsection from section . Presently, this subsection gives a WP:UNDUE weight to combinatorics, with respect to other subareas of discrete mathematics, especially graph theory. On the other hand, removing the section will be harmless, since combinatorics will stil be shortly described in, and the removed information will be still available in the linked article. For these reasons, I'll remove . D.Lazard (talk) 14:16, 10 June 2022 (UTC)

Semi-protected edit request on 12 June 2022
to edit math definition Dmentornuh (talk) 12:57, 12 June 2022 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. ScottishFinnishRadish (talk) 13:06, 12 June 2022 (UTC)

Incorrect history in the lede
"In the history of mathematics, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed at a relatively slow pace until the Renaissance, when algebra and infinitesimal calculus were added to arithmetic and geometry as main areas of mathematics."

This does not summarise the history of mathematics very well. Complex use of mathematics was widespread in urbanised societies millennia before Euclid (that's thousands of years being skipped here); the earliest mathematical texts are from the 20th century BC, 1,600 years before Euclid. While the first sentence is inaccurate by omission, the second sentence here is just a pure mischaracterisation. Algebra was codified in medieval Persia, whereas calculus was developed firmly in the Enlightenment, not the Renaissance. Tigernose (talk) 04:09, 15 July 2022 (UTC)


 * I agree with you about the second sentence. It has always bothered me. Notably it lacks citation.
 * The first sentence is not about how complicated math is. It is not about when texts appeared. It is about proof. So I do not see how your comment contradicts it. Mgnbar (talk) 12:09, 15 July 2022 (UTC)
 * I agree with.
 * As far as I know, this was in Baghdad, and Baghdad was never in Persia. In any case, "codified" seems an inappropriate term, and the truth of this sentence depends from the meaning given to "algebra".
 * I have edited the article for
 * removing the mention of the so called "relatively slow pace";
 * specifying in a footnote that "algebra" is meant as the art of manipulating formulas.
 * As formulas with symbolic variables were not invented before the Renaissance, this clarifies the mention of algebra. Care must be taken when talking of algebra, since the meaning of the term has changed since the 19th century: before, "algebra" referred the theory of equations, which can by dated back to Diophantus; todays, there are as many theories of equations as type of equations.
 * As it is unclear whether the infinitesimal calculus must be dated for the Renaissance or to the Age of Enlightenment, I'll replace "Renaissance" with "16th and 17th century". D.Lazard (talk) 17:45, 15 July 2022 (UTC)
 * "17th and 18th century", surely? --Trovatore (talk) 18:29, 15 July 2022 (UTC)
 * Viète was from the 16th century. Descartes, Newton and Leibniz published during the 17th century. So, this is during the 16th and 17th centuries that algebraic computation with symbols (variables) and differential calculus were introduced. D.Lazard (talk) 20:14, 15 July 2022 (UTC)
 * Ah, I missed that you were including algebra. --Trovatore (talk) 21:31, 15 July 2022 (UTC)

Mathematics
What is the meaning of mathematics.And stand for of math. 103.179.240.193 (talk) 01:11, 26 July 2022 (UTC)
 * Please start your study of mathematics here. Are you keeping a notebook of your questions? Pay attention to the little words (a, the, all, some, none, ...) also! Can you ask your questions in your own words? --Ancheta Wis   (talk  &#124; contribs) 02:32, 31 July 2022 (UTC)

Semi-protected edit request on 21 July 2022
I am looking to add to the "Society" part of the article, which currently needs extra content, and more precisely, on the subject of education. Here is what I would add under "and so on".

In education, mathematics are a core part of the curriculum. While the content of courses varies, many countries in the world teach mathematics to students for significant amounts of time. IEA’s Trends in International Mathematics and Science Study TIMSS 2019 p450-p451

Since it is my first time editing, I am sure there could be a lot to improve here (despite it being only a few words...). Furthermore, more content on education could be added; for instance, how advanced chemistry and physics classes require solid mathematic knowledge. LuneDeCuivre (talk) 18:29, 21 July 2022 (UTC)
 * Red information icon with gradient background.svg Not done for now: Could you please specify the page of the source? Aaron Liu (talk) 14:53, 30 July 2022 (UTC)
 * @Aaron Liu Pages 448 to 451 are revelevant, but pages 450 and 451 hold the crucial data. LuneDeCuivre (talk) 09:43, 2 August 2022 (UTC)
 * Correct me if I'm wrong, but these pages are about science excluding math? Aaron Liu (talk) 12:39, 2 August 2022 (UTC)
 * @Aaron Liu I see the issue. The page number I was referring to is the one on the bottom right of the pdf. Since the PDF has more pages, you should find it on pages 466-469 if you go by the computer's count.
 * The title on the correct page is "instructional time in mathematics", so you can also use ctrl+f as a last resort if I was not clear enough. I apologize, it is my mistake. LuneDeCuivre (talk) 13:06, 2 August 2022 (UTC)
 * ✅ Ah, sorry, my fault. Aaron Liu (talk) 19:16, 2 August 2022 (UTC)

Semi-protected edit request on 11 August 2022
In the first sentence, please change "arithmetic, number theory" to "arithmetic and number theory". It flows better, and the last example in the sentence is "calculus and analysis" rather than "calculus, analysis". 49.198.51.54 (talk) 19:54, 11 August 2022 (UTC)
 * ✅ &#8209;&#8209; El Hef  ( Meep? ) 21:20, 11 August 2022 (UTC)

Applied mathematics
Currently, is a subsection of. It is tagged with expand section with the comment "the connections between mathematics proper and the other sciences (enough for an entire first-level section)". I agree that these connections deserves a first-level section; however, his section exists already under the name. On the other hand, applied mathematics is no more an identified area of mathematics.

Thus, I suggest to rename as, and to merge there. Because of the controversial content of this section (identified by several tags), and of the much richer content of the target of the merge, this merge could consists in adding simply a hatnote.

Some thoughts? D.Lazard (talk) 14:45, 1 September 2022 (UTC)


 * @D.Lazard No, according to
 * "A distinction is often made between «applied» mathematics and mathematics oriented entirely towards «abstract» questions and concepts, known as «pure» mathematics."


 * "Abstract" and "Pure" are synonym and "Abstract" is antonym of "Applied", so the rename is not correct. Hooman Mallahzadeh (talk) 16:21, 1 September 2022 (UTC)


 * You're right, but so is D.Lazard. The point is that the section Abstract Knowledge, as it is currently written, is not actually about "pure" math. It's about the relationships among pure math, applied math, and the sciences. And I definitely agree that applied math deserves some kind of top-level section. So I agree with D.Lazard's plan, with the caveat that it will require much editing of the Abstract Knowledge section. Mgnbar (talk) 16:54, 1 September 2022 (UTC)
 * After adding a Gauss citation for "durch planmäßige Tattonieren" (NOT Tattowieren. 'Tat' can mean action; but German terms are elastic, just as in English, and could connote 'grope' as well), I wanted to follow up with some likely historiography: The phrase is likely to exist in correspondence between Gauss and Schumacher in 1850 (see Campbell (1977) p.399), regarding Gauss' approach to the 8 queens puzzle, which pops up in computer science programming problems nowadays. Gauss' role was strictly confidential, being recreational; Schumacher agreed to this. Gauss conjectured "a careful enumeration of solutions via trial and error, taking only an hour or two", he estimated (see Paul J Campbell (1977), "Gauss and the eight queens problem" Historia Mathematica 4 p.399). Gauss' estimate of 76 solutions was too large by 4, in 1850, the accepted value of 92 solutions for an 8x8 chessboard was not yet known. Gauss transformed positions on a chessboard to integers; the size of the chessboard now variable n X n, and known as OEIS A002562: "Number of ways of placing n nonattacking queens on n X n board (symmetric solutions count only once)". Experimental mathematics, applied mathematics, recreational mathematics, and science lie in a lattice, it seems. --Ancheta Wis    (talk  &#124; contribs) 04:03, 5 September 2022 (UTC)

I have rewritten with a subsection. By the way, I have removed some misplaced considerations and controversial philosophical opinions, and created a subsection. This can certainly be improved, but the new version will certainly improvements easier than the previous version. D.Lazard (talk) 10:17, 19 September 2022 (UTC)

Semi-protected edit request on 5 October 2022
Image result for math The longest-standing unresolved problem in the world was Fermat's Last Theorem, which remained unproven for 365 years. The “conjecture” (or proposal) was established by Pierre de Fermat in 1937, who famously wrote in the margin of his book that he had proof, but just didn't have the space to put in the detail. 2405:201:E047:E057:1FCD:C759:E1D:6431 (talk) 12:39, 5 October 2022 (UTC)
 * This is wrong: duplication of the cube, angle trisection, and quadrature of the circle remainded unresolved during more than 2,000 years. D.Lazard (talk) 13:05, 5 October 2022 (UTC)
 * Red information icon with gradient background.svg Not done: Per D.Lazard. (not that I understand this but I'd assume they're right) ― Blaze WolfTalkBlaze Wolf#6545 14:42, 11 October 2022 (UTC)

Areas or branches of mathematics
An editor has changed "areas of mathematics" into "branches of mathematics" in the whole article including in a section heading. I have reverted this because it is a long-standing terminology, and such a change requires a discussion.

Moreover, I disagree with the change, because "branch" suggests a tree structure. This is misleading, because of many overlaps between areas.

By the way, I have added an anchor for fixing the two redirects and  D.Lazard (talk) 11:27, 14 October 2022 (UTC)


 * @D.Lazard The term "branch" is used for most of the English Wikipedia articles' definition for mathematics branches (areas), like Number theory, Geometry, Mathematical analysis, etc., you can inspect this claim, and some minor, like definition of Algebra, uses the term "area", instead.
 * Yes, there is a tree structure between mathematics and geometry and other branches. Because mathematics includes geometry and also geometry is distinct from number theory because different axioms hold in each, so semantically the term "branch" is better than "area". The term "area" conveys no meanings of inclusion and no meaning of distinctive of each branch from each other.
 * In the definition of formal sciences, we have:
 * "Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, game theory, systems theory, decision theory, and theoretical linguistics."

Hooman Mallahzadeh (talk) 12:25, 14 October 2022 (UTC)
 * Here it uses the term "branch" to distinguish "formal science" from "science".
 * Another point is that for example, statistics or is a "formal science" and not a branch of mathematics. So the definition of mathematics (the first line of article) should be more precise. I suggest something like
 * "Mathematics is a branch of formal science studying disciplines concerned with abstract structures described by formal systems, including numbers, shapes, etc."
 * WP:Wikipedia is not a reliable source. So, all these considerations are WP:original research and cannot be used for supporting your suggestion of changing "area" into "branch". D.Lazard (talk) 10:38, 15 October 2022 (UTC)

Dear Please discuss and write your opinions about this and the two following sections. Thanks, Hooman Mallahzadeh (talk) 10:44, 15 October 2022 (UTC)
 * We have consensus for "areas". Saying "branches" seems no better and probably worse. Let's direct our energy at more important improvements, such as adding citations and content. Mgnbar (talk) 13:44, 15 October 2022 (UTC)

Science or Knowledge
Aside from the above discussion about branch and area, mathematics is a «science» not a knowledge. Please read definitions of knowledge and science carefully. Mathematics «builds» knowledge. So please retrieve my edit. Thanks, Hooman Mallahzadeh (talk) 08:59, 15 October 2022 (UTC)
 * Firstly, nobody has said that mathematics is a "knowledge". The phrase "area of knowledge" must not be split.
 * "Mathematics «builds» knowledge". What is built when one learns mathematics?
 * As mathematics can be taught and learnt, it is undoubtly knowledge (science is also knowledge).
 * It has been discussed here many time whether mathematics is or is not a science. The result of these discussions is summarized in the first paragraphs of and . As this is philosophy and not mathematics, this does not belong to the introduction.
 * In summary, the term "area of knowledge" is a compromise between divergent opinions on the philosophical status of mathematics. It must not be changed without a deep discusion among the concerned editors. D.Lazard (talk) 09:56, 15 October 2022 (UTC)
 * @D.Lazard The definition of science is:
 * "Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe."

Hooman Mallahzadeh (talk) 10:27, 15 October 2022 (UTC)
 * Assume we can conclude from the rule
 * a * b = a + ... + a (b times)
 * that
 * 2 * 2 = 4
 * because
 * 2*2 = 2+2 = 4.
 * Here «2*2 =4» is a conclusion of this rule so it is a «knowledge». But it is the «science of mathematics» that prove definitely that 2*2=4 and also concludes the knowledge 2*3=6. Mathematics is not "2*2=4" but also it explains it and predict it and similar productions.
 * You are right, an axiom in mathematics is some knowledge, but no, the act of prediction of «new knowledge» is science.
 * No, this sentence is definitely wrong. Please read the definition of them carefully.
 * I really persist that the phrase "area of knowledge" is definitely wrong about the definition of mathematics. Because «area of knowledge» means "part of knowledge". Right?
 * We cannot define mathematics is correct. But we can at least categorize this science in its definition, i.e., "Mathematics is a formal science".
 * Anyway, if you really think that mathematics is not a formal science, please change the first line of the article Formal science too:
 * "Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, game theory, systems theory, decision theory, and theoretical linguistics."
 * Anyway, if you really think that mathematics is not a formal science, please change the first line of the article Formal science too:
 * "Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, game theory, systems theory, decision theory, and theoretical linguistics."
 * Please, stop using Wikipedia as a source. This is unreliable. In particular, the article Formal science lacks clearly of recent sources discussing the subject in general. So there is no recent definition of the subject that is commonly accepted. Also, as far as I know, there is no recent source that discusses whether mathematics is commonly considered as a formal science. D.Lazard (talk) 13:02, 15 October 2022 (UTC)
 * Hooman Mallahzadeh, some of your arguments are insightful. However, what matters on Wikipedia is reliable sources.
 * If you actually want to make improvements to this article &mdash; as opposed to arguing for the sake of arguing &mdash; then there is a clear strategy that you should follow. Gather reliable sources, summarize them here, convince us that the current article is not accurate summarizing reliable sources, and thereby achieve consensus for changing this (long contentious) article. Regards Mgnbar (talk) 13:53, 15 October 2022 (UTC)

Structure of the sidebox
@D.Lazard I suggest to use the terms "First-order branch" and "Second-order branch" and "merged branch" instead of the term "Areas" and "Relation", as a tree that has first and second order branches from a main branch. This yields in  :

Do you agree? Hooman Mallahzadeh (talk) 07:26, 15 October 2022 (UTC)
 * Another option is using tree structure for this template, like:  :


 * — Preceding unsigned comment added by Hooman Mallahzadeh (talk • contribs) 10:25, 15 October 2022 (UTC)
 * Both suggestions are not acceptable as "merged with" does not correspond a any merger, and you have no reliable source for the term "second-order branch", for the repartition of areas/branches in two levels, and for such a tree structure. Also, many of your choices would be disputed by every specialist of the concern "branch". Please stop trying to introduce your personal opinions in the article. This is WP:OR. D.Lazard (talk) 10:26, 15 October 2022 (UTC)
 * @D.Lazard
 * See in the article Set theory is written:
 * "Set theory is the «branch» of mathematical logic that ..."
 * "Set theory is the «branch» of mathematical logic that ..."


 * So this tree is not an artifact of my mind. But also, this tree implements reliable definitions that exists in Wikipedia. Hooman Mallahzadeh (talk) 10:32, 15 October 2022 (UTC)
 * Again, WP:Wikipedia is not a reliable source. Also, you will never find a reliable source asserting that number theory is a subarea of discrete mathematics, and that probability theory includes statistics. Even the inclusion of set theory into logic can be disputed, as the theory of ordinals is set theory and not mathematical logic. So, a flat presentation of branches/areas is the only reasonable choice for the template. The only place for a useful discussion is the choice of the areas that are listed: too many woud make the template non-useful, and too few could be confusing. D.Lazard (talk) 10:58, 15 October 2022 (UTC)
 * I agree with that the subfields of mathematics do not form a tree hierarchy. For example, while Number Theory indeed intersects with Discrete Mathematics, it also intersects with Algebra; the only possibility to force this into a tree structure would be to claim that both Discrete Mathematics and Algebra are subbranches (children in the tree) of Number Theory, which is certainly nonsense.
 * I reverted 's recent changes at Template:Math topics TOC; we should settle the discussion first. - Jochen Burghardt  (talk) 13:01, 15 October 2022 (UTC)
 * @Jochen Burghardt@D.Lazard I used tree structure because:
 * "Number «is a» Discrete Object"


 * The subject of study in number theory is "integers" which is itself a discrete object (studied in discrete mathematics). So I really persist that there exists an "is a" relation between them. See article Hyponymy and hypernymy. I think discrete mathematics is hypernym and number theory is a hyponym. Am I wrong?
 * I really think that we should create an ontology for the concepts of mathematics. i.e., making a hypernym-hyponym for all of its concept.
 * Hooman Mallahzadeh (talk) 13:15, 15 October 2022 (UTC)
 * Editors have been arguing over categorization in math for ages. The ideal categorization would be supported by reliable sources. Unfortunately, even the sources disagree. The basic reason is that human endeavors are messy, and any attempt to impose order on them is imperfect. So, as I recommended in the preceding talk page section, let's direct our energy at other goals. Mgnbar (talk) 13:47, 15 October 2022 (UTC)
 * I still feel that I've shown above that no tree structure is possible. Since you apparently disagree, I wonder how your tree would look like, in particular near the nodes I mentioned (Number Theory, Discrete Mathematics, Algebra). Apart from that, I agree that reliable sources have to be provided eventually. - Jochen Burghardt (talk) 14:23, 15 October 2022 (UTC)

First sentences
Wikipedia has to have one of the worst defining sentences about mathematics. For comparison, here are some other examples: Surely we can do better than: Per MOS:FIRST: Thanks. Praemonitus (talk) 04:07, 31 October 2022 (UTC)
 * The abstract science of number, quantity, and space. -- Oxford
 * The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. -- Encyclopedia Brittanica
 * An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them. -- Wiktionary
 * The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. -- Merriam-Webster
 * The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. -- dictonary.com
 * The science that deals with the logic of shape, quantity and arrangement. -- Live science
 * The study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them. -- Cambridge
 * Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).
 * The first sentence should tell the nonspecialist reader what or who the subject is, and often when or where. It should be in plain English.


 * This first sentence has been discussed many times here, and is the result of a consensus. If you have an idea for improve it, please provide it for discussing it. None of your quotations is convenient for Wikipedia because (1) there is no agreement in reliable sources whether mathematics is a science or not, and (2) most of your quotations are misleading, as excluding implicitly from mathematics most areas of modern mathematics. D.Lazard (talk) 08:58, 31 October 2022 (UTC)


 * I was expecting your summary of the reference works to make Wikipedia look worse. It's impressive that our opening sentence contains substantially the same content as the others'. The biggest difference I see is that we parenthetically mention some of the big fields of mathematics. Mgnbar (talk) 11:53, 31 October 2022 (UTC)
 * To my eyes the parenthetical comments do disrupt the flow. It is awkward to read out loud, in contrast to many of the examples above. Even the Wiktionary definition is cleaner. Perhaps it should paraphrase the Wiktionary definition and place the fields of study in a separate sentence? Praemonitus (talk) 12:46, 31 October 2022 (UTC)
 * I am not opposed to moving the parentheticals to another sentence. I suspect that the core of the Wiktionary definition is a non-starter for editors here. "Abstract representational system" is vague, intimidating, and probably controversial (as it seems to emphasize symbolic notation).
 * Praemonitus, it seems that you are an experienced editor, and I don't want to talk down to you, but you might not realize how fought-over this sentence is. This sentence is a classic Wikipedia hornets' nest. Do not expect rapid, easy change. :) Mgnbar (talk) 14:32, 31 October 2022 (UTC)
 * Not to worry Mgnbar, I'm quite familiar with how contentious certain topics can be on Wikipedia. I was just taking a look at some of the top Vital Articles and the lead sentence for this article just stood out for me as not really providing a concise definition compared to the other articles. If there is a heavy resistance to change, then I'll go away. (I do have a graduate-level background in mathematics, so I'm not completely ignorant on the topic.) Thanks. Praemonitus (talk) 16:29, 31 October 2022 (UTC)
 * Most readers of this page have already heard of mathematics, number theory, calculus, geometry, etc. but they are not supposed to be able to relate the names of the historically main areas of mathematics and the given informal descriptions. This is the reason of the parenthetical explanations. However, these informal descriptions are likely to be controvesial. So, I will not be opposed to change the sentence into . By the way, the second sentence should be the beginning of a new paragraph, and, as it is, is partially wrong (the discovery of property is rarely the result of "pure reason"). I'll fix it soon. D.Lazard (talk) 15:14, 31 October 2022 (UTC)
 * First, I agree about splitting the paragraph at the start of the second sentence. Then the first paragraph is a single sentence, which is not a crime, but which may be an opportunity. What about something like...
 * "Mathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics as subdisciplines such as number theory, algebra, geometry, and analysis."
 * Second, I don't agree about your proposed typical reader. For example, I suspect that most adults in the USA have never heard of number theory or (mathematical) analysis. Mgnbar (talk) 16:22, 31 October 2022 (UTC)
 * Perhaps then the Wiktionary definition could be used in a third sentence, so it avoids the apparent contention here? Praemonitus (talk) 16:57, 31 October 2022 (UTC)

Lead
The (now) third paragraph of the lead devotes a full paragraph to the mathematics of scientific modelling, whereas the body barely mentions it. Per WP:LEDE:
 * "The lead serves as an introduction to the article and a summary of its most important contents."

I think that paragraph should be moved down to the "Relationship with Science" section. That would keep the lead within the heuristic limit of four paragraphs. Praemonitus (talk) 15:07, 1 November 2022 (UTC)


 * IMO, the section must contain an expanded version of this paragraph, but the paragraph must be kept, possibly in a shortened version. Indeed, the independence of mathematical truth from any experimentation is fundamental for the usefulness of mathematics in so many scientific areas. As such, this deserves to be mentioned in the lead and expanded in a section. D.Lazard (talk) 16:45, 1 November 2022 (UTC)


 * I, too, have problems with the contents of the 3rd paragraph:
 * (1) What is an experimental law? A law that was generalized from experimental results? A preliminary law that is going to be checked if it helps to explain the observed results? (In the latter case, I suggest to avoid "experimental".)
 * (2) The statement ... implies that the accuracy of such predictions depends only on the adequacy of the model is misleading, imo. The adequacy of any model (mathemaical or not) is defined as the accuracy of its predictions. For example, a wooden model of the Eiffel tower may be used to predict length ratios that will be found when measuring the original, but not to predict weight per volume ratios (specific weight). For this reason, we say that the model is adequate for the former, but inadequate for the latter purpose. There is nothing specific to mathematics in this. Nevertheless, I feel that the mercury example has a point, but I'm unable to say, what it exactly is.
 * After the paragraph has been fixed, it might be sufficiently short to merge it into the (now) 4th paragraph.


 * Beyond that, although I know that the current status of lead discussion is "better do not touch anything here", I'd like to comment on a few more sentences; maybe some of my issues can be settled without re-opening the debates:
 * In the 2nd paragraph, I'd like to see at least a citation that supports abstractions from nature that don't have the form of axioms, and show a proof that is based (at least in part) on non-axiomatic basic properties abstracted from nature.
 * In the 4th paragraph, I suggest to change "sciences" to "natural sciences", to emphasize the distinction to "social sciences".
 * In the 5th paragraph, I doubt that the interaction between mathematical innovations and scientific discoveries was the cause for the rapid increase in the development of mathematics since the 16th/17th century. Both events (if "interaction" can be seen as an event) undoubtedly occurred at the same time, but the causal relation is at least not obvious. Again, I'd be content with a citation supporting the sentence. In the last sentence, I suggest to change "An example of this is" to "This can be seen, for example, in the contemporary", since the increase of number of areas cannot have an example. - Jochen Burghardt (talk) 16:57, 1 November 2022 (UTC)


 * I agree that the independence of mathematical truth from any experimentation is an important point and should be mentioned in the lead. However, the 3rd paragraph in its current form fails to make that point (or I'm unable to understand it). - Jochen Burghardt (talk) 17:00, 1 November 2022 (UTC)


 * I disagree. The paragraph isn't even sourced, and the point is receiving undue weight by spending an entire paragraph in the lead. At best it deserves a sentence with more coverage below. Praemonitus (talk) 00:37, 2 November 2022 (UTC)


 * Because the lead is a summary of the body, I was WP:BOLD and copied said paragraph to the body for citation and further expansion. I inserted a candidate replacement sentence that makes the core point of the paragraph. I propose that the paragraph is now redundant. Praemonitus (talk) 15:44, 2 November 2022 (UTC)

Images

 * Completely new to Wikipedia editing, but I should point out that the image at the top-right of the article could be seen as outdated by some, considered the trend of logo simplification by organizations. Though the image is vectorial, I can't shake off my head the presence of 2010s internet with the shade and lighting selection by the author. Please don't bully me. SpinozaDios (talk) 13:18, 2 October 2022 (UTC)
 * I agree. But to find a single image that says "mathematics" borders on impossible. Suggestions, anyone?
 * dino (talk) 00:40, 3 October 2022 (UTC)
 * My view is that we ought not strain to have an image just to fit some general layout expectation. The idea of taking a very broad and very abstract field of study and turning it into a representative image is just not really plausible.  Some effort might be put into finding images for each of the "areas" sections, or at least as many as possible (maybe Cantor's diagonal for math logic/set theory?), but even there I'd say, if it's too hard to make it fit, then it just might not be the right thing to do. --Trovatore (talk) 16:50, 3 October 2022 (UTC)
 * Maybe I misunderstand, but I don't think that SpinozaDios is complaining about the content of the image. They are complaining specifically about the aesthetics. The specular highlights, the translucency, the drop shadows, etc. are all fashionable circa 2001. These effects might put off some readers in 2022.
 * Personally, I tend to agree, but updating the image aesthetics is of low priority. Maybe SpinozaDios would like to take on the job? Mgnbar (talk) 18:17, 3 October 2022 (UTC)
 * Sure, that was the original complaint, but it's also true that the content is not representative of mathematics as a whole. It looks like an image that was put there just to have an image.  I'm suggesting that, between having an image that's there just to have an image, and not having an image at all, I prefer not having an image at all. --Trovatore (talk) 18:24, 3 October 2022 (UTC)
 * I'm not doing any artwork wtf SpinozaDios (talk) 11:48, 18 October 2022 (UTC)
 * Okay, but it wasn't a crazy idea. All of Wikipedia is the result of people noticing that something missing or wrong and fixing it. This place doesn't appear by magic. We make it. :) Mgnbar (talk) 12:59, 18 October 2022 (UTC)

I have removed the image from the infobox, and I have restructured the infobox for compliance with its main article (Mathematics). D.Lazard (talk) 20:55, 4 October 2022 (UTC)


 * The image has been restored in the navbox. I'll not dispute this revert. D.Lazard (talk) 14:28, 5 October 2022 (UTC)
 * I think we should remover the image of Euclid as well, since general image has been removed. Now, if you hover over Mathematics in an article, that is the picture that shows up, which I think is even less representative of mathematics today than the removed image. Rogalendingen (talk) 14:38, 11 October 2022 (UTC)
 * Leonhard Euler identitatea.png
 * Good point. Instead of removing Euclid's image, I suggest to add the image of Euler's identity. Even if it represents a small part of mathematics, it is clear to everybody that it is mathematics. D.Lazard (talk) 16:29, 11 October 2022 (UTC)
 * @D.Lazard I suggest the image of four elementary arithmetic operations for this purpose: i.e.: [[file:Arithmetic symbols.svg|thumbnail|120px]] Hooman Mallahzadeh (talk) 16:36, 11 October 2022 (UTC)
 * This may be misleading, as elementary arithmetic is often considered as not really being mathematics.
 * By the way, I have added the image to the article, with the caption Euler's identity, sometimes called the most beautiful formula of mathematics. I remain open to better suggestions, if any. D.Lazard (talk) 17:01, 11 October 2022 (UTC)
 * I like the picture of Euler's identity and think that is a good suggestion to illustrate the article.
 * However, I still think we should remove (or rather replace) the picture of Euclid. One of the arguments above for removing the general picture was that it looked like an image that was put there just to have an image, which is also true for the picture of Euclid. He is also on the Mathematician page, so it is unnecessary to have the same picture of him in an article about the subject, not the people working with it. Rogalendingen (talk) 16:43, 12 October 2022 (UTC)
 * I have moved Euclid's image (with a shorter caption) to the history section, for replacing the image about the method of exhaustion, since Euclid is historically much more important than this method. I have also moved Euler's identity to the top. Hoping that reaches a consensus. D.Lazard (talk) 17:22, 12 October 2022 (UTC)
 * As far as I remember, the previous caption of the Euler identity picture was a result of some discussion about the question "who said this?" and "can you supply a source for this?"; and it answered these questions. Therefore, I'd prefer the previous caption. - Jochen Burghardt (talk) 18:52, 12 October 2022 (UTC)
 * The article Euler's identity contains several citations that say essentially that Euler's identity is the most beautiful mathematics formula. So, there is no need to refer specifically to Feynman, and "something called the most beautiful" is a convenient formulation. Nevertheless, I have added a citation, and changed "formula" to "theorem" for agreeing with the quotation in Euler's identity. D.Lazard (talk) 11:21, 13 October 2022 (UTC)
 * I changed the equation image with something that include a bit more geometry and relationship, and perhaps a bit more clear on casual readers. I changed the image to Euler's formula unit circle instead. CactiStaccingCrane (talk) 10:54, 16 November 2022 (UTC)
 * @CactiStaccingCrane According to this reference, the most beautiful formula is Euler's formula in the form of e^iπ = -1, not the version of Euler's formula which you mentioned. So please revert your edit about image and its caption. Thanks, Hooman Mallahzadeh (talk) 11:27, 16 November 2022 (UTC)
 * It's a general formula of the identity. When x = pi, then you get e^iπ = -1. CactiStaccingCrane (talk) 11:29, 16 November 2022 (UTC)
 * @CactiStaccingCrane The same concept can be expressed in many forms, but only one of these forms is the most beautiful. Hooman Mallahzadeh (talk) 11:32, 16 November 2022 (UTC)
 * Well... I agree. But I kinda like more geometry... CactiStaccingCrane (talk) 11:33, 16 November 2022 (UTC)
 * I suggest to add the image of commutative diagram next to the image of Euler's identity. But, I can't choose just one … --SilverMatsu (talk) 12:29, 11 November 2022 (UTC)

Euler's identity
The illustration of Euler's identity, which has been called the "most beautiful equation in mathematics", was recently changed to Euler's formula with the caption "Euler's formula, which is sometimes called the most beautiful theorem of mathematics". The citation remains unchanged, even though it was intended to cite the original illustration. The point of the first was that it connects "the five most fundamental constants of mathematics via two of its most fundamental operational symbols." I'm dubious about the second, and believe it needs a better cite. Praemonitus (talk) 16:22, 16 November 2022 (UTC)


 * IMO, the previous image must be restored per WP:LEAST and WP:TECHNICAL: Understanding the meaning of the new image requires technical competences that are far beyond those of the casual reader. The only thing that this image can illustrate in a non-technical article is the strong relationship between different areas of mathematics (here, geometry, trigonometry, and calculus). This could be the subject of a section (although I do not know how to write it, and where to place it). So, this image is not convenient for the present state of the article. D.Lazard (talk) 17:29, 16 November 2022 (UTC)
 * Honestly, you could say the same about the previous image (ei&pi;+1=0). Non-technical readers might get a feeling that they know what it means, because they know what some of the symbols mean in other contexts, but this is a different sort of exponentiation than the sort they're likely familiar with (repeated multiplication), and you need mathematical analysis to know what it means and how it's connected with the more familiar sort.
 * I remain unconvinced we need an image here. We should explore the zero option. --Trovatore (talk) 19:04, 16 November 2022 (UTC)
 * It did seem like the caption for either illustration wasn't very informative for the visiting lay reader. We shouldn't be assuming the reader has a strong math background. My suggestion would be to move the Math Topics template to the top, then place any illustrations below with better captions. Praemonitus (talk) 05:21, 17 November 2022 (UTC)
 * That seems like a good idea. Paul August &#9742; 10:38, 17 November 2022 (UTC)
 * I removed the picture, though do note that in mobile the sidebar won't appear. CactiStaccingCrane (talk) 14:08, 18 November 2022 (UTC)

Relationship or combination
@Praemonitus Hi, according to "Mathematics is used in empirical science for modeling phenomena, which then allows predictions to be made from experimental laws."

The term «is used in» means «is applied» or «is a subset of» or «combined with». Two concepts A and B are «related» if there is some relationship between them. But if one concepts is superset and the other is subset, then the term «combined» is more appropriate. Here «Physics» is the superset science that «combines» some concepts of mathematics in it as the subset science. So in my opinion the term «Combine» is more appropriate here. Thanks, Hooman Mallahzadeh (talk) 15:26, 5 December 2022 (UTC)
 * Hello. Well, generally speaking, empirical results cannot be used to demonstrate a mathematical proof. Mathematics is clearly not empiricism, so "empirical science" is a distinct set. I think calling mathematics a "science" per se is a loaded opinion that can lead to disputes. Is it an art or a science? The cross-over could perhaps be topics such as theoretical physics, although that too still relies on empiricism. Praemonitus (talk) 15:37, 5 December 2022 (UTC)
 * @Praemonitus I mean for a «relationship» that is a type of «inclusion», then the term «Combine» is more appropriate.
 * Here, for example physics «includes» some concepts of mathematics and not uses these concepts separately. For example in physics, the «force concept» that has the formula «combines» some concepts of algebra, and not uses algebra as a separate concept. The physical concepts of mass, acceleration and force are related through an algebraic formula. Without the meaning of F,m and a, this formula as a separate concept is not true in physics. So the force formula combines some algebra, but algebra in isolation is not correct here. Here the superset is the force formula and the subset is the «multiply formula concept» in algebra.
 * So, I still think that the term "combination" is more appropriate here. Tnx. Hooman Mallahzadeh (talk) 16:08, 5 December 2022 (UTC)
 * "Relationship" indicates a close connection; "combination" does not. Praemonitus (talk) 16:17, 5 December 2022 (UTC)
 * @Praemonitus Yes! You are right, and I convinced. They are separate, and therefore the term «relationship» is correct. Hooman Mallahzadeh (talk) 16:25, 5 December 2022 (UTC)
 * Thank you for discussing the issue. Praemonitus (talk) 16:34, 5 December 2022 (UTC)
 * Empirical science redirects to the philosophical concept of Empiricism, which is unrelated to the content of the section. So, if "empirical science" is used, it must be linked elsewhere or not linked. IMO, the phrase is improper, as it is controversial to qualify physics and computer science as empirical sciences (General relativity is far from an empirical theory, and, in computer science, theories do not result generally from experiments, since experiments are used only a posteriori for testing theories). So, I'll the mentions of "empirical sciences", and restore the heading as "Relationship with sciences" (avoiding "other", because of the dispute whether mathematics is a science, and using plural that seems more convenient here). D.Lazard (talk) 16:29, 5 December 2022 (UTC)
 * Fair enough. I suppose it could be titled "Relationship with empiricism". Praemonitus (talk) 16:31, 5 December 2022 (UTC)

Rating of this article
This article is presently rated WP:C-class. This corresponds well to the vesion of the article of October 2021. Since then, the article has been deeply revised, and the rating deserves to be updated. As a major contributor of the present version, I am misplaced to judge which ratting should be given. So, please, give your suggestions or even change the rating yourself. D.Lazard (talk) 14:44, 22 September 2022 (UTC)


 * The only real problem to the article is the lack of citations, which Praemonitus has made significant progress to address this issue. That's the reason why the article is now B-Class. CactiStaccingCrane (talk) 10:53, 16 November 2022 (UTC)


 * I rate it somewhere between a 'B' and an 'A' article; hence the 'GAN'. It seems mostly complete and decently written, but there's always room for improvement. Praemonitus (talk) 15:27, 19 December 2022 (UTC)

Euler formula
It says Euler's identity, sometimes called the most beautiful theorem of mathematics but that sounds unnecessary can't it just say Euler's Identity McDonalds1940 (talk) 02:28, 21 October 2022 (UTC)


 * No it is not unnecessary. It is very commonly called the most beautiful identity/theorem JGHFunRun (talk) 21:09, 19 December 2022 (UTC)

What to expand?
The article is currently 44k characters in prose size. It is quite substantial, but I think we can stuff a lot more content in after bad content is trimmed. What section or topic should we expand first? CactiStaccingCrane (talk) 10:51, 16 November 2022 (UTC)
 * The Psychology section should address the common topic of Mathematical anxiety. The Terminology section should discuss the meaning of the frequently used terms such as Theorem, Corollary, and Lemma. The whole article needs to be properly cited. Praemonitus (talk) 18:40, 16 November 2022 (UTC)
 * Agreed. CactiStaccingCrane (talk) 23:13, 16 November 2022 (UTC)
 * Section requires expansion, and is tagged as such. Not sure whether Mathematical anxiety belongs to  or  (or both). D.Lazard (talk) 16:39, 17 November 2022 (UTC)
 * Possibly the two sections should be under one heading about "Training and practice", since they're essentially both about the human mind? (Done.) Praemonitus (talk) 20:14, 17 November 2022 (UTC)
 * Agree. CactiStaccingCrane (talk) 14:07, 18 November 2022 (UTC)
 * Section must be either removed or completely rewritten: It is written as an essay, and contains many comments that do not belong to it (about terminology, conciseness, etc). In particular, I am uncomfortable with the sentence "The validity of a mathematical proof is fundamentally a matter of rigour", as, in modern mathematics, the validity of a proof is only a matter of correct application of deducing rules. D.Lazard (talk) 16:39, 17 November 2022 (UTC)
 * Nuked, except for the last paragraph about computer assisted proof, which has citations. CactiStaccingCrane (talk) 14:07, 18 November 2022 (UTC)
 * Restored. Can we hold off for a week to see if there is any dissent? There is a main article link now that could be included WP:Summary Style. Praemonitus (talk) 15:43, 18 November 2022 (UTC)
 * I have written a new version of the section on rigor. As usual, I am unable to source it (generally, I cannot remember where I have learnt things that I know), but I strongly believe that it is not too difficult to find sources. D.Lazard (talk) 15:38, 19 November 2022 (UTC)
 * I see it contains yet another repetition of the discussion of the 'foundational crisis of mathematics' and the 'axiomatic method'. This may need to be edited. (Done.) Praemonitus (talk) 16:04, 19 November 2022 (UTC)


 * The content about the 'foundational crisis of mathematics' leading to the development of the 'axiomatic method' is repeated no less than three times throughout the article. I'd say it only needs to be told once, with brief summaries elsewhere for context. There's also a lot of WP:OVERLINKing that I plan to trim. (Fixed.) Praemonitus (talk) 00:28, 19 November 2022 (UTC)
 * The article is lacking a section on proof, which would seem to be crucial to the subject. There is no mention of regression analysis as an important element of statistics. Praemonitus (talk) 14:19, 22 November 2022 (UTC)


 * I'm just swinging by & won't be editing for a while, but if you pop-up open the bottom banner shell here on the talk page, I updated the to-do list a while back. It's pretty much still my own brainstorming ideas since nobody has edited it since, but I think a couple might be interesting. In particular, I think we could go into the interplay between math & technology more, both inventions used for math like the abacus and math modeled on inventions, like Archimedes' method or Carnot engines. And even though it's just a teaser for the main History of Math article, the history section could probably be more global. Zar2gar1 (talk) 05:43, 29 November 2022 (UTC)
 * Re:interplay between math & technology, the entire computer industry and cell phones relies on error correcting codes (mass storage devices would be far more expensive with out them), and cryptography on number theory and elliptic curves and in the future on lattice theory.--agr (talk) 15:38, 29 November 2022 (UTC)
 * I'd argue that these are covered at a high level by "Coding theory, including error correcting codes and a part of cryptography", plus the "Computational mathematics" section. This article is written WP:Summary Style and probably should delve too deeply into any one mathematical topic. Praemonitus (talk) 15:25, 19 December 2022 (UTC)
 * This talk section is about what might be expanded and Zar2gar1 suggested discussing "the interplay between math & technology." We already have a section on the "Unreasonable effectiveness" of Math, which mainly deals with physics, but the impact of mathematics on modern technology is much more direct. Merely listing "Coding theory, including error correcting codes and a part of cryptography" isn't a summary, it's an index with links. No one reading that, who did not already know, would suspect the cell phones in their pockets depend on these results. The audience for this article should be informed about the modern impact of math. I agree wit Zar2gar1 that the article should have a paragraph or two on this topic.--agr (talk) 21:41, 20 December 2022 (UTC)
 * I'm just trying to keep things in perspective. The modern impact of mathematics is basically most of science and engineering. I'd argue that error correcting codes is a small fraction of said. WP:UNDUE still applies. Praemonitus (talk) 22:55, 20 December 2022 (UTC)
 * We have different perspectives. I would argue that the impact of mathematics on most of science and engineering is basically calculus and its children. That brought us railroads, airplanes, electricity, telephones, atomic bombs and the like. The 21st century is more about the explosion in computing and communication (AI, crypto currency, social networks, ecommerce, surveillance state, etc.) and that is driven in large part by more recent math. This impact of modern math is less well known and deserves a paragraph.--agr (talk) 16:35, 21 December 2022 (UTC)
 * For the most part this is a WP:Summary Style article that links to other articles that expand on the topic. What high level article are you referring to in this context? Perhaps Computational science? We've already got a Computational mathematics section, where Computational science is linked. Otherwise, I'd suggest creating and expanding an article that we can incorporate. Praemonitus (talk) 16:44, 21 December 2022 (UTC)
 * In fact, we lack of an article that could be called Mathematics of computer science. Only a small part of this wide subject is covered by the articles cited by . Among the subjects not covered by these articles, I can mention mathematics of data compression (including image compression), image reconstruction, image analysis, semantics and logic of compilers, program certification and proofs, mathematics (mostly graph theory and probability theory) for analyzing and managing large networks (web), geometry of circuits in microprocessors. In summary, it wouls be a good thing to have a subsection of devoted specifically to the relations with computer science. But it is a very difficult task, and I think that we must keep it for future improvements. D.Lazard (talk) 17:15, 22 December 2022 (UTC)
 * The article you are asking for might be Theoretical computer science. - Jochen Burghardt (talk) 18:02, 22 December 2022 (UTC)
 * Alas, that article is itself written summary-style, and much of the content is not math specific per se. I added Mathematics of computer science to WP:RA/M, although there isn't a suitable category. Praemonitus (talk) 18:54, 22 December 2022 (UTC)


 * A glance at fr:Mathématiques (written by our amazing colleagues on frwiki) reveals several topics we could write some more about:
 * The cultural impact of mathematics, as shown by its appearance in popular media (music, art, magazines, books, literature, film, TV shows, etc.). (I think we could even split this into a separate article like Mathematics in popular culture or Mathematics and society. The former may not be viable - plenty of "[arbitrary thing] in popular culture" articles have been deleted here on enwiki. But the latter might - there's plenty of stuff about it on Google Scholar, indicating a likely pass of WP:GNG, and it'd be a natural place to lump stuff like Mathematics education and Mathematics awards.)
 * How mathematics relates to each individual science (physics, computing, chemistry, biology, social sciences, ecology, etc.); this would naturally fit into our own section Mathematics. If we do end up writing about this in detail, we could even hatnote to Relationship between mathematics and physics.
 * Apparently, mathematics was historically associated with astrology and esotericism for some reason. This needs documenting too. Perhaps this could get hatnoted to Pseudomathematics?
 * As you can see, we have a lot to do! Duckmather (talk) 01:04, 23 December 2022 (UTC)
 * Update: I have just boldly expanded our article by translating the relevant sections of Mathématiques (which represents the largest changes to the article structure in a long time). Note, however, that much of this is highly unsourced and unencyclopedic, so it'll need a lot of editing to bring it up to par. Duckmather (talk) 02:40, 28 December 2022 (UTC)

Semi-protected edit request on 28 April 2023
Could we add in the box left... Relationship to Visual Art Add link to Linear Perspective CarvingJoints (talk) 14:56, 28 April 2023 (UTC)


 * I have not found the box that you mention. For an edit request you must be accurate on what must changed, and specify the exact text that you want to insert.
 * Nevertheless, I agree that the section must be completely rewritten and include mention of perspective. But, this cannot be the object of an edit request, unless you provide a draft of the new version. D.Lazard (talk) 15:19, 28 April 2023 (UTC)

"MathematicsAndStatistics" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=MathematicsAndStatistics&redirect=no MathematicsAndStatistics] 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  until a consensus is reached. Duckmather (talk) 19:18, 10 May 2023 (UTC)

"Mathematics and Statistics" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Mathematics_and_Statistics&redirect=no Mathematics and Statistics] 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  until a consensus is reached. Duckmather (talk) 19:18, 10 May 2023 (UTC)

Racist & Eurocentric History Section
The information that mathematics was influenced by the Nigerians is an important fact that should be included in a Wikipedia article about mathematics. The current version of the article is not only incomplete, but it also perpetuates a racist and Eurocentric view of history.

For too long, the contributions of non-European cultures to the development of mathematics have been ignored or downplayed in Western history. This neglect is a form of racism that seeks to erase the achievements of people of colour and perpetuate the false idea that only the West is capable of producing great ideas. By including the fact that the Nigerians were influencers of mathematics, the Wikipedia article can begin to correct this historical injustice. It is essential to acknowledge the contributions of all cultures to the development of mathematics, not just those that are traditionally seen as “the winners” in the Western narrative.

Additionally, by including this information, the article can become more comprehensive and accurate. It is not enough to only focus on the work of European mathematicians; to truly understand the history of mathematics, we must also look at the contributions of other cultures.

In conclusion, the fact that mathematics was influenced by the Nigerians is an important piece of information that should be included in a Wikipedia article about mathematics. By doing so, we can begin to correct the racism inherent in the current version of the article and create a more comprehensive and accurate understanding of the history of mathematics.

---

The ancient Africans also made important contributions to the field of calculus. The Yoruba people of Nigeria developed a system of infinite series in the 14th century, which they used to approximate the value of pi. The series was later discovered independently by European mathematicians in the 17th century. 102.218.103.6 (talk) 05:37, 3 May 2023 (UTC)


 * Please propose specifically how you would work this into the article. 331dot (talk) 08:09, 3 May 2023 (UTC)
 * Four paragraphs of nothing. Either propose specific changes or don't bother at all Kulloth (talk) 12:38, 11 May 2023 (UTC)

Using Venn diagram for PageImage of this article
Hi, following the discussion on https://en.wikipedia.org/wiki/Talk:Mathematics/Archive_15#Images:

I think mathematics is a set that includes "Number theory", "Geometry", "Calculus" as subsets, and some fields like "Mathematical physics" or "Computational mathematics" are outside sets of it but use some parts of it as elements in common. So I propose to use a Venn diagram for PageImage of this article, that show mathematics as a circle, that contains circles for "Geometry", "Number", etc. and some circles like "Mathematical physics" are outside but have some elements in common. Thanks, Hooman Mallahzadeh (talk) 17:07, 1 May 2023 (UTC)
 * There seems to be a common opinion that every article needs an image. I disagree very sharply.  I do not think there is any image that is sufficiently representative of mathematics as a whole to be worth the distraction, and I strongly prefer that this article continue to have no lead image whatsoever. --Trovatore (talk) 17:21, 1 May 2023 (UTC)
 * @Trovatore Textual description is much harder to understand than a visual illustration. So I think existence of image makes this article more user-friendly and more illustrative. Hooman Mallahzadeh (talk) 17:42, 1 May 2023 (UTC)
 * Again, I disagree in the sharpest terms. There is no possible representative image for mathematics; it's too diverse and too abstract.  The article looks much better with no lead image at all. --Trovatore (talk) 18:08, 1 May 2023 (UTC)
 * Although I am not in complete agreement with Trovatore's opinion on lead images, I strongly disagree with the suggestion: the suggested image would imply a classification of mathematics that is WP:OR, and this is definitively forbidden by Wikipedia policies. D.Lazard (talk) 19:59, 1 May 2023 (UTC)
 * @D.Lazard@Trovatore I think these diagrams do not violate WP:OR:
 * https://www.pinterest.com/pin/branches-of-mathematics-poster-free--652529433496268166/
 * https://juliawolffenotes.home.blog/2019/08/18/u-wret313s-map-of-math/
 * https://blancosilva.wordpress.com/2013/10/22/areas-of-mathematics/
 * But I am not sure that they are appropriate for PageImage of this article. What is your opinion? Hooman Mallahzadeh (talk) 14:31, 11 May 2023 (UTC)
 * All these diagram are original research: there are based on the own opinion of their authors about the classification of mathematics, and do not reflect any consensus among mathematicians. As examples: one places topology at the opposite of geometry, while another places it inside geometry; none mention discrete mathematics; one presents pure and applied mathematics as unrelated fields D.Lazard (talk) 15:40, 11 May 2023 (UTC)
 * Hooman, I really wish you'd just let this go. An image is not required.  If there were a good one, we could talk.  So far no good one has been offered, and I see very limited prospects that any ever will be. --Trovatore (talk) 18:21, 11 May 2023 (UTC)

Add logarithms to History
I realize that the history section is a very brief overview and that that are many possible adds, but the impact of logarithms on the usability of mathematical results was massive, leading to Kepler which led to Newton. The text I propose adding is:

"The introduction of logarithms by John Napier in 1614 greatly simplified numerical calculations." agr (talk) 15:55, 18 May 2023 (UTC)


 * I suggest to expand this sentence into "The introduction of logarithms by John Napier in 1614 greatly simplified numerical calculations, especially for astronomy and marine navigation". This (with or without the added phrase) would make sense in the paragraph beginning with "During the early modern period". However, as logarithms are one of the "innovations that revolutionized mathematics", the whole paragraph should be edited, for coherency. D.Lazard (talk) 16:28, 18 May 2023 (UTC)
 * I added the sentence with your addition. Could you be a bit more specific about what the paragraph lacks?--agr (talk) 12:14, 19 May 2023 (UTC)
 * Your edit is fine. However, summarizing in a single sentence one of the three major revolutions of mathematics does not give a WP:DUE weight to this revolution. IMO (this may be considered as WP:OR) there are three major revolutions in the history of mathematics: the systematization of proofs with Euclid's Elements, the introduction of mathematical notation, algebrization of geometry and calculus around the 16th century, and the systematization of axiomatic method and set theory, at the beginning of the 20th century (after the foundational crisis of mathematics). None of these three revolutions receive their due weight in this history section. D.Lazard (talk) 12:55, 19 May 2023 (UTC)

Misspelling
The caption for the second illustration "On the surface of a sphere ..." misspells the word "Euclidean" (as "Euclidian"). I hope someone will fix this.
 * . D.Lazard (talk) 15:24, 4 July 2023 (UTC)

Bad caption for Cauchy sequence
The caption for the illustration of a Cauchy sequence reads as follows:

"A Cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right)"

Because this can be interpreted in several different ways, it does not clearly exclude something like the sequence {sn} of partial sums of the harmonic series sn = 1 + 1/2 + 1/3 + ... + 1/n, which is of course not a Cauchy sequence.

I hope someone can rewrite the caption so that it is unambiguous and accurate.
 * . D.Lazard (talk) 15:36, 4 July 2023 (UTC)

"Number transformation" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Number_transformation&redirect=no Number transformation] 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  until a consensus is reached. Skarmory  (talk •   contribs)  03:43, 14 August 2023 (UTC)

Semi-protected edit request on 9 September 2023
Please can I have an edit about the branch which was announced at the second conference of astronomic sciences, Astromathematics? This is my dream Hinduistic (talk) 14:51, 9 September 2023 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate.  — Paper9oll  (🔔 • 📝)  15:47, 9 September 2023 (UTC)
 * Draft:Astromathematics
 * Astromathematics is a branch of physics, astrophysics and cosmology which deals with mathematics.
 * Instead of studying this with the existing contents in Physics (Astromath – bits), one needs to study Astromathematics to understand the relationships between Astronomy and Mathematics better.
 * Even while physics provides the majority of the mathematics required to comprehend the data obtained through astronomical observation, there are some unique scenarios that involve mathematics and phenomena that may not yet have enough physics to fully explain the observations. Due to these two applications, astromathematics continues to take this as a challenge.
 * Math is constantly used by astronomers. When using a telescope to see celestial objects, one application of it is this. The camera that is mounted to the telescope, in particular its charge-coupled device (CCD) detector, essentially converts or counts photons or electrons and records a series of numbers (the counts); these counts may indicate how much light various celestial objects are emitting, what kind of light, etc. We need to utilise arithmetic and statistics to interpret these figures in order to be able to comprehend the information they convey.
 * Astromathematics is a term that combines astronomy (the study of celestial objects and phenomena beyond Earth's atmosphere) with mathematics. It refers to the use of mathematical techniques and principles in the field of astronomy. This can include a wide range of applications, such as: Hinduistic (talk) 12:46, 10 September 2023 (UTC)
 * This is just a neologism for a form of applied mathematics, it has no place on this article. MrOllie (talk) 13:59, 10 September 2023 (UTC)
 * No its not, https://www.researchgate.net/publication/355960814_Astromathematics
 * This is a research gate article by my friend Syed Arif Kamal.
 * It has proof.
 * I really need to make a edit, plsss.
 * MrOlli, it isnt neologist, even wikidictionary have the word and its meaning. Hinduistic (talk) 13:28, 11 September 2023 (UTC)
 * We used to have an article on it. It was considered at WP:AFD and found to be a neologism and deleted there. It has no place here. MrOllie (talk) 13:32, 11 September 2023 (UTC)
 * Proof please!! Hinduistic (talk) 13:34, 11 September 2023 (UTC)
 * Articles for deletion/Astromathematics MrOllie (talk) 13:39, 11 September 2023 (UTC)
 * https://en.wikiversity.org/wiki/Mathematics/Astronomy
 * What about this? Hinduistic (talk) 13:43, 11 September 2023 (UTC)
 * What goes on at other projects (who have very different standards) has no bearing on the English Wikipedia. MrOllie (talk) 13:45, 11 September 2023 (UTC)
 * Like MrOllie I am skeptical. Wikiversity is not considered one of the Reliable sources. Please consider making more edits to Wikipedia that do not involve your friends' work. Mgnbar (talk) 14:17, 11 September 2023 (UTC)

Why is there a section on astrology?
This is a crackpot footnote clearly unworthy of an entire section, let alone sentence 120.18.236.250 (talk) 14:55, 14 September 2023 (UTC)

Complete misunderstanding of economic rationality
Economics do not assume that people only seek to "maximise profit". We assume that they have consistent preferences. Absent mind-reading, this is not actually a testable theory beyond observing one's own preferences, because it can be retro-fitted to any data. Really, when we say rationality is useful, we mean that tractable preferences can be used to reasonably predict behaviour. Preferences do not have to do with money. In the most common economic models, preferences are over bundles of goods. Preferences can include altruistic motivations, and indeed any motivations. 120.18.236.250 (talk) 14:58, 14 September 2023 (UTC)
 * Is this in response to the second paragraph of the subsection "Social sciences"? To me it all hinges on how narrowly one defines "profit". What if we replace "profit" with "profit (as defined by that individual)" or "self-interest"? Mgnbar (talk) 16:53, 14 September 2023 (UTC)
 * Profit has a specific meaning in economics, and also most peoples heads. I'm sure you'd agree that it would be strange to hear "I went to the petting zoo and made a lot of profit". I'm happy with self-interest, even though it suggests that rational choice precludes altruism to the uninitiated. Perhaps it would be a good idea to link the word to the rational choice theory page. 120.17.128.99 (talk) 02:15, 17 September 2023 (UTC)

Math works?
Why does math work at all? Pretty important question that is totally ignored. 2601:5C6:4180:3D20:6458:84CF:EA31:E39B (talk) 13:59, 11 November 2023 (UTC)


 * See, for example, and, which are about this question. D.Lazard (talk) 15:11, 11 November 2023 (UTC)

Possible addition to the History of Mathematics Section
Could the idea of the “discovery” of complex or imaginary numbers be mentioned? As it is a significant turning point for mathematics. A possible edit could go as follows:

“…with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603),the first examples of complex numbers by Gerolamo Cardano (1501-1576) which led to the conception of the fundamental theorem of algebra, the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation…” 98.97.177.87 (talk) 05:09, 23 November 2023 (UTC)


 * In my opinion, the "discovery" of complex numbers is an important result, but this did not revolutionized mathematics. D.Lazard (talk) 09:06, 23 November 2023 (UTC)


 * Like D.Lazard, I'm not sure the History section here needs that specificity. But I'd definitely think there's a place for it on the History of mathematics article.
 * A sentence or two about how complex numbers actually work might be a nice little addition here too. The idea that "imaginary numbers aren't 'imaginary', they're rotational" is pretty intuitive and accessible but hasn't seemed to reach popular awareness yet. Zar2gar1 (talk) 09:53, 23 November 2023 (UTC)
 * Thank you both for your opinions. I will note briefly that my “example edit” included content around the edit of the page; my apologies if this was unclear. In regards to Zar2gar1, how would you propose such an edit be made? Because in the history of mathematics section there is no mention of “imaginary” numbers. I think a quality transition could be made following the mention of René Descartes, as it was his original criticism of complex numbers that led to the more popularized name of “imaginary numbers.” 98.97.177.87 (talk) 00:14, 24 November 2023 (UTC)
 * Just to clarify, for the extra detail on imaginaries, I wasn't thinking of something in the history section. I was picturing just a sentence or even part-sentence to add some detail, probably where we discuss complex analysis (Areas of Math), or maybe Symbolic Notation? It would probably be sourced from an undergrad diff-eq or complex analysis book, or maybe a detailed pop-sci book.
 * Looking closer at your edit though (I didn't realize much of it's already on the page), adding a mention of the complex numbers might not hurt. My one tweak though would be that you would probably want to mention Rafael Bombelli, alongside or instead of Cardano. IIRC Cardano recognized non-real roots of the general polynomial formulas but still treated them as degenerate solutions; Bombelli was the one that built out the early theory and argued they were valid. Zar2gar1 (talk) 06:33, 24 November 2023 (UTC)
 * Thank you so much! I will take a look at the complex analysis section to see what you mean. I agree, I think Bombelli was the more influential figure in the history of the “imaginary” numbers, so I am more than happy to add his name into the edit. R2214 (talk) 16:20, 24 November 2023 (UTC)

The Topic about '0'
The concept of zero is believed to have originated in the Hindu cultural and spiritual space around the 5th century CE. In Sanskrit, the word for zero is śūnya which refers to nothingness. In scientific history, astronomer and mathematician Aryabhata is often associated with inventing the number '0'. Manveermg (talk) 15:59, 28 December 2023 (UTC)


 * There are several concepts of zero: zero as a number, zero as a digit, zero as a placeholder in decimal representation, etc. So, it is impossible to give to zero a unique origin. D.Lazard (talk) 17:09, 28 December 2023 (UTC)