Talk:Mathematics

About 'Computational Mathematics'
I consider that in the areas of mathematics, Computational Mathematics should be eliminated, since it belongs, in any case, to an area of mathematics in conjunction with another science, such as Mathematical Physics or Mathematical Economics, and not to pure mathematics like the rest.

Alternatively, a section of applied mathematics could be incorporated where Computational Mathematics could be included.

Alex gnpi (talk) 09:12, 21 February 2024 (UTC)


 * Presently, section gives a misleading description of computational mathematics, and should be completely rewritten. Nevertheless, I strongly disagree with your suggestions.
 * You seem to give a strong importance to the distinction between pure and applied mathematics. There is presently a large consensus among mathematicians that this is not a classification of mathematics, but rather a point of view on mathematician motivations.
 * You seem also believe that most computational mathematics consist in applying mathematics to computations in another science. Ths is very much too restrictive. For example, a large part numerical analysis consist of elaborating tools for computing solutions of differential equations, which are applied to almost every science. Computational mathematics is not restricted to numerical analysis. It includes computation theory, cmputer assisted proofs such as the four color theorem, cryptography, the design of proof assistants, mathematical experimentation (computation for discoveintg and testing conjectures), etc.
 * In short, section deserves to be completely rewritten and expanded, not removed or dissolved in another section. D.Lazard (talk) 10:43, 21 February 2024 (UTC)

Lead
I'm in my mid-20s, and I remember reading the lead of this article as a kid and being happy with how elegant it was:

I think, broadly, this is significantly better than the present lead. There's a lot of 00s-isms there, we shouldn't consider copy-pasting it back, but would there be consensus to rewrite the lead based on a 2008 version, before the article got de-GAd? Remsense 诉  13:16, 22 February 2024 (UTC)


 * First paragraph: I think that the current version is better than the old version. It explicitly states how these major topics show up in current mathematics. It does not privilege Peirce's quotation.
 * Second paragraph: I don't love the current version. It seems overly long and detailed. The old version treats this logic/proof/axioms theme more concisely.
 * Third paragraph: You didn't mention this, but I hope that we agree that a paragraph about applications and utility is warranted.
 * Fourth paragraph: You didn't mention this. I don't love it, because it seems overly long and detailed. Mgnbar (talk) 13:50, 22 February 2024 (UTC)
 * I agree with your assessments of the second through fourth paragraphs, and your critique of the privileging of an individual person's quote in the first.
 * However, I think the important point for the first paragraph is it concretely—but not too concretely, this is math—broadly lays out the areas of experience that math usually touches. I think that's really important for an encyclopedia article on such a huge topic. The current first paragraph mentions [{em|things}}, which are for the moment undefined, but the old version deals with realms, if that makes any sense at all. It states the "purpose" of math first, before the means by which math gets there. Remsense  诉  13:56, 22 February 2024 (UTC)
 * The second paragraph is unnecessarily long.
 * Here is a proposed rewrite.
 * Rocfan275 (talk) 14:51, 22 February 2024 (UTC)
 * Here is an improved version: "Most mathematical activity involves the manipulation of abstract objects in view of proving statements called theorems. These objects may be abstractions of the natural world such as numbers and curves, or entities with no direct relation to reality such as rings, topologies and cryptographic protocols. A proof of a theorem is formed by applying a series of deductive rules starting from known properties, which may be either base assumptions known as axioms, or previously proven theorems." D.Lazard (talk) 15:27, 22 February 2024 (UTC)
 * Also, I suggest to remove the last sentence of the first paragraph ("There is no general consensus among mathematicians about a common definition for their academic discipline"). The reasons are
 * Such an assertion cannot be sourced
 * Such a negative assertion could be done about many sciences, and even about Science itself : there is no general consensus among scientists about a common definition for science.
 * If this sentence should be kept in the article, this should be in
 * There is a clear consensus among mathematicians that if there is no theorems or proofs, this is not mathematics, and that any subject where theorems are proven becomes mathematics.
 * I have no source attesting that this is a consensus, but this is an evidence for everybody that has participated to many editorial committees of mathematical journals and conferences). So, the second paragraph can be viewed as a definition of mathematics). D.Lazard (talk) 18:52, 22 February 2024 (UTC)
 * "Mathematics is the study of concepts such as number, structure, space, and change. These topics are broadly represented by the major mathematical disciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the manipulation of abstract objects in view of proving statements called theorems. These objects may be abstractions of the natural world such as numbers and curves, or entities with no direct relation to reality such as rings, topologies and cryptographic protocols. A proof of a theorem is formed by applying a series of deductive rules starting from known properties, which may be either base assumptions known as axioms, or previously proven theorems.
 * " is my synthesis of the first two paragraphs with the earlier version's opening sentence. Is this too vague? I also sense my simple use of "study" may sound too POV intuitionist for some? Though IMO describing math as a study does not imply that mathematical truths don't exist a priori.  Remsense  诉  02:11, 23 February 2024 (UTC)
 * "Mathematics is the study of concepts such as number, structure, space, and change. These topics are broadly represented by the major mathematical disciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the manipulation of abstract objects in view of proving statements called theorems. These objects may be abstractions of the natural world such as numbers and curves, or entities with no direct relation to reality such as rings, topologies and cryptographic protocols. A proof of a theorem is formed by applying a series of deductive rules starting from known properties, which may be either base assumptions known as axioms, or previously proven theorems.
 * " is my synthesis of the first two paragraphs with the earlier version's opening sentence. Is this too vague? I also sense my simple use of "study" may sound too POV intuitionist for some? Though IMO describing math as a study does not imply that mathematical truths don't exist a priori.  Remsense  诉  02:11, 23 February 2024 (UTC)

Info to add from a source to a section of the article
I have seen the section "Training and practice" in the article, to which some info could be added from the following source https://www.tandfonline.com/doi/full/10.1080/01425692.2023.2240530 178.138.99.208 (talk) 16:23, 15 March 2024 (UTC)


 * You must say which info you want to add. Moreover, this link is an original research paper, and Wikipedia policy WP:NOR implies that, for being acceptable in Wikipedia, every original research must have been discussed in other sources. Moreover, there are thousands of articles on mathematical education, and priviledging one of them contradicts anothe fundamental policy of Wikipedia, WP:NPOV. D.Lazard (talk) 16:05, 16 March 2024 (UTC)
 * If you want the article mathematical education (not this article mathematics) to discuss the varying results between students based on parental involvement/disposition, you should probably try to find a survey article or the like to use as your source, rather than a particular study. –jacobolus (t) 16:46, 16 March 2024 (UTC)

Semi-protected edit request on 18 March 2024
where does the rules of math state that 1x0=0 add Jgomezbeyondpie (talk) 04:45, 18 March 2024 (UTC)
 * See Multiplication RudolfRed (talk) 05:02, 18 March 2024 (UTC)

Wiki Education assignment: 4A Wikipedia Assignment
— Assignment last updated by Ahlluhn (talk) 00:57, 31 May 2024 (UTC)


 * @Not Fidel I'd recommend against choosing such high level topics as Mathematics and Astrology for an introduction to working on Wikipedia, at least if you want your contributions to be valuable and stick around. Ideally you want to find an article which is at least moderately important but currently underdeveloped or in very poor shape, for example something with 'high' priority and 'start' quality' or 'mid' priority and 'start' quality (those links go to a list of all such articles within WikiProject Mathematics). To write an effective article you need to do quite a bit of book research about a topic, and it's pretty hard to wade into a topic as large as the ones you picked without quite a lot of reading, unless you intend to pick out a particular section that seems missing, undeveloped, or otherwise problematic to focus on. –jacobolus (t) 15:10, 18 March 2024 (UTC)

Edits of 2024/6/1
One editor has been making many edits today. I worry that they underestimate how contentious many parts of it are. For example, the opening paragraph has been argued heavily. This is why reliable sources were cited, even in the lede. I do not think that removing these citations is a good idea. It will exacerbate arguments later on.

In my opinion, the highly active editor should discuss on this talk page and build consensus before making more wide-spread changes. It would also be helpful if all, not just some, edits were accompanied by edit summaries. Mgnbar (talk) 22:14, 1 June 2024 (UTC)


 * I've reverted their changes. As someone who spends a lot of time trying to tighten up leads, this clearly meant well but went too far. Remsense  诉  23:21, 1 June 2024 (UTC)

Overlink issue in lede
An editor removed the link to Knowledge in the opening paragraph, citing MOS:OL. I reverted this edit, as I did not find a discussion on the talk page and I do not believe that OL applied in this case. My edit was then reverted by @D.Lazard, citing OL again and a consensus that (as far as I can tell) does not exist. Subsequently, the link was added back (by @Rhosnes) and then reverted yet again.

Regarding consensus, the original edit removing the link was made on 2024-06-27. No discussion occurred to justify the removal of the link. While several discussions on the lede do exist, it appears that the lede with the link was in fact the consensus version.

I strongly believe the concept of knowledge is directly related to mathematics and is not an instance of overlinking. A reader who arrives at this article would conceivably wish to know more about the topic, including any fundamental concepts relating to it, such as knowledge. To state that mathematics is an area of knowledge without then providing a link to what "knowledge" actually is appears to me as somewhat lacking. The fact that the concept of "knowledge" is supposedly common knowledge (no pun intended) is rather irrelevant, as it is not a passing reference, but rather a direct relation stated in the opening sentence.

Any application of MOS:OL against "common knowledge" in the lede would contradict established practice throughout the rest of the wiki. Take the article American football (randomly selected), for example. While "team sport" would conceivably be common knowledge, it is still linked, as it is directly relevant.

Additionally, I believe that the link itself should exist on "knowledge" and not "area of knowledge", as Area of knowledge does not exist, and as stated above, the concept of "knowledge" is directly related to this topic.

iczero (talk) 04:20, 2 July 2024 (UTC)


 * I would also add that the term "knowledge" is used rather idiosyncratically in the lede, since it's very questionable that mathematics describes, rather than merely models, objective reality. In fact, as far as I'm aware, most modern philosophers of mathematics favour the latter view. If so, that would make the term "knowledge" as used in the lede different from the common sense interpretation of the term. In that case, not linking to "knowledge" would be straight-up misleading. With all due respect, D.Lazard's edits are unproductive. Rhosnes (talk) 07:13, 2 July 2024 (UTC)
 * "area of knowledge" is frankly a very vague, ambiguous, and largely unhelpful description. I'd replace the phrase entirely. –jacobolus (t) 07:22, 2 July 2024 (UTC)
 * Britannica's article, by Wilbur Knorr and Craig Fraser, leads with "Mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter...." Their term "science" (in the plain English sense of the word) seems better than "area of knowledge". –jacobolus (t) 07:31, 2 July 2024 (UTC)
 * @Jacobolus Someone else in this discussion stated an issue with NPOV regarding "science". Otherwise, I would prefer that description as well.
 * For what it's worth, I argue that "area of knowledge" is a meaningful statement (see below). iczero (talk) 02:49, 3 July 2024 (UTC)

The present state of the first paragraph (including the phrase "area of knowledge") is the result of a compromise after many long discussions; see, in particular Talk:Mathematics/Archive 15. So, please, do not open this discussion again, unless you can propose something that has not been discussed before.

I would also be in favor of "science", but it is not possible to use it in the lead, because it is controversial, as there is no consensus whether mathematics is a science or not. Other terms have been proposed, which are all controversial either. As mathematics can be learnt, studied and taught, it is undoubtly a part of human knowledge, and "area of knowledge" is, up to date, the best phrase that has been found for refering to "a part of human knowledge".

About linking: Linking in the lead is useful only if the link can help for understanding. A link to Area of knowledge would be useful if the article would exist. The link to Knowledge is absolutely not useful since the link target does not contain anything that is useful here. More, it is disruptive, because of the time that the reader may spent for unsuccessfully trying to understand the relationship between mathematics and the present content of the article Knowledge. D.Lazard (talk) 10:56, 2 July 2024 (UTC)


 * @D.Lazard I opened this topic strictly on the linking issue. The original change that removed the link had no consensus whatsoever. If the original "compromise" resulted in a link to Knowledge, that link should be kept.
 * I'm not exactly sure what you may consider to be a "useful" link target, but mathematics is surely closely related to the concept of knowledge. If a reader does not wish to see what the wiki has to say about knowledge, they do not need to click that link. However, the option should be there if they want, especially if it is described that way in the lede (even if that is a compromise). iczero (talk) 18:04, 2 July 2024 (UTC)
 * Is this perhaps something to do with Getting to Philosophy? –jacobolus (t) 18:33, 2 July 2024 (UTC)
 * @Jacobolus Completely irrelevant to my argument. iczero (talk) 19:54, 2 July 2024 (UTC)
 * Sure it's nothing you mentioned. I just wonder if it's part of the motive. If the first link of Mathematics is Algebra, and vice versa, then the entire part of the first-link graph of wiki articles that feeds into mathematics is going to get stuck into the Mathematics–Algebra 2-cycle instead of feeding into a cycle involving Philosophy. Instead of 95% or whatever of all articles feeding into Philosophy it will be cut by the double-digit (?) percentage that instead aim at Mathematics. [Edit: in fact, since Philosophy apparently points toward Mathematics, the new game will be "Getting to Algebra".]
 * This is a cute little wiki game that editors who work on math articles generally couldn't care less about, but some other people who don't ever work on math articles might think is important enough that they should come argue about what kind of links the lead of the Mathematics article should have. I'm not saying I know for sure that's what happened, it just seems like a plausible speculative hypothesis. (Notice that the Getting to Philosophy page talks about disputes in April 2024 where "there had been numerous attempts to switch the order of the links" to restore the previous graph, something we might anticipate seeing at other pages threatening to break the game.) –jacobolus (t) 06:29, 4 July 2024 (UTC)
 * I absolutely could not care less about whether The First Link Graph™ or whatever it's called points to Algebra or The Moon. It is, quite frankly, irrelevant. I, however, do believe that math and knowledge are closely related (being that math is one form of the pursuit of knowledge), and if knowledge is in the opening sentence, that it should be linked. Unfortunately, nobody else here seems to agree, which is why I currently think it's best to just swap it for "discipline" (without any link).
 * Or maybe we should just call it a science. Dissenters can go complain and hopefully propose a more meaningful definition. iczero (talk) 07:48, 4 July 2024 (UTC)

I essentially agree with Trovatore's comment. Nevertheless, it may be useful to recall the history of "area of knowledge". Some remarks: So, I'll restore the piped link on the article. D.Lazard (talk) 11:13, 3 July 2024 (UTC)
 * Delink "knowledge". While you can argue whether the word "knowledge" is being used in exactly its everyday sense here, one thing that is clear is that it is not being used in some particular specialized sense that one can expect to find explained at the article.  Also it does not strike me as especially likely that a reader who has looked up "mathematics" is suddenly interested in reading about knowledge in general. --Trovatore (talk) 00:39, 3 July 2024 (UTC)
 * @Trovatore I, for one, would be interested in reading Knowledge after Mathematics, especially since the former does frame the latter within its context. I would argue that links are essential to the wiki, and there's no good reason to remove it. The Web, after all, is named as such because of hyperlinks.
 * I see exactly no harm whatsoever in preserving the link. It was the previous state of the article by prior consensus. Links in openings are practically standard across the wiki. iczero (talk) 01:25, 3 July 2024 (UTC)
 * Look, we have to call mathematics something. "Science" is controversial (I would be personally fine with it but it's not NPOV).  Once upon a time I think we used "discipline", which I'd also be OK with, but people thought it sounded too grim.  "Area of knowledge" is basically a default option because we can't come up with anything else.  It isn't terrible but it's also not particularly meaningful (and is not meant to be).
 * To link it is to put too much emphasis on it, to make it seem like a substantive claim that mathematics is an area of knowledge, and that that means something in particular. Neither of those things is true.  We're not making any serious substantive claim about mathematics by saying it's an area of knowledge; we just need something to put in that part of the sentence. --Trovatore (talk) 02:11, 3 July 2024 (UTC)
 * @Trovatore I strongly disagree that linking provides emphasis. It's just a link. I particularly like links. Even if you were to replace "area of knowledge" with "academic discipline", I would still argue that it should be linked.
 * I personally prefer "area of knowledge" because I believe it to be a meaningful statement. (I am of the opinion that math is a science, but as you stated, NPOV.) You may view it as "simply a compromise", but I would disagree. Several articles, including Academic discipline and Science, directly include "knowledge" as part of their primary definition. Is math not also fundamentally related to knowledge? Or, perhaps, I'm just being a bit too idealistic. iczero (talk) 02:37, 3 July 2024 (UTC)
 * You may "particularly like links", but the consensus among Wikipedia editors is that they're best used sparingly. They definitely do come across as emphasis, whether you agree or not. --Trovatore (talk) 03:10, 3 July 2024 (UTC)
 * @Trovatore As far as I can tell from both MOS:LINK and simply reading the wiki, there is no consensus that links ought to be used sparingly and definitely no consensus that links ought to be removed from the lede. MOS:CONTEXTLINK seems to encourage contextual links in the opening sentence, and "knowledge" fits this. Even if this link in particular emphasizes knowledge, I don't regard that to be in any way harmful.
 * Regarding consensus (again), the link has been there for at least a year before it was removed a week or two ago with no discussion: iczero (talk) 03:56, 3 July 2024 (UTC)
 * On 23 January 2022 I introduced the phrase without link on 23 January 2022 with the edit summary "This seems a good way for avoiding the repeated complaint about the lack of definition".
 * On 1 January 2023 linked "knowledge" with the edit summary "'Knowledge' should be linked to the page for that word, embedding the topic 'mathematics' in a broader one".
 * The same day] I reverted them with the edit summary "Here this is area of knowledge that should be linked if such an article would exist".
 * The same day I italicized "area of knowledge" with the edit summary "italicizing for making clear that the phrase cannot be split into its components (see the previous reverted edit)".
 * On 15 january 2023] linked "area of knowledge" to "knowledge" through a piped link with the edit summary "A second attempt at embedding the subject, 'Mathematics', within a more general discipline, in the 1st sentence of the text. This is the usual practice in Wikipedia. I wonder if linking to articles like numbersis really a problem - readers would only click these links if they wanted to follow them up".
 * [https://en.wikipedia.org/w/index.php?title=Mathematics&diff=prev&oldid=1141676940 On 26 February 2023, putted "area" out the link, without edit summary.
 * On 30 June 2023 unlinked "knowledge" with the edit summary "Knowledge is everyday word, no link per WP:OL".
 * I made a mistake in my edit summary of 1 July 2023, by writing "this has been already discussed on the talk page". I should have written "this has been alresdy discussed through edit summaries".
 * Here, the "previous stable version" cannot be that of since it does not result of any consensus, and has not been explained in an edit summary. The fact that nobody took care of this edit does not mean that there was a consensus for it. So the  previous stable version should be that of 15 January 2023, with "area of knowledge" linked as  area of knoledge.
 * My opinion is that both this piped link and the unlinked version are acceptable. Linking "knowledge" alone is not, since this splits a phrase that should not be split. The advantage of the piped link is that it seems a good compromise between those who want a link and their opponents.


 * I seems that domain of knowledge is more colloquial than "area of knowledge" and has the same meaning. Moreover, the linked article is more appropriate than the too general Knowledge. So, I suggest to replace the latter phrase with the former. D.Lazard (talk) 13:27, 3 July 2024 (UTC)
 * @D.Lazard I'm not really sure that "domain knowledge" is appropriate here. Various sources ("For example, in software engineering, domain knowledge can apply to specific knowledge about a particular environment in which the target system operates.", "[...] in a specific domain" ) state that it is for more specialized fields, and "mathematics" seems a bit too general for that. The Domain knowledge article also seems to state the same but lacks inline citations.
 * The piped link on "area of knowledge" is fine by me. Sorry for the misunderstanding regarding the consensus on that. iczero (talk) 14:50, 3 July 2024 (UTC)
 * The phrase "domain of knowledge" sounds very formal and somewhat awkward to me. YMMV. (A Google scholar search suggests it is about half as common as "area of knowledge", though I did no investigation of what context those words are being used in.) If we're going for this general type of phrase, how about something like "field of study" instead? (This is more than an order of magnitude more common than area/domain of knowledge.) –jacobolus (t) 17:02, 3 July 2024 (UTC)
 * @Jacobolus "Field of study" sounds a bit restrictive to me, as if it were mostly an academic thing. Same with Academic discipline. Math is used extensively in applied fields (computing, for one) and I don't think that implication should be there. iczero (talk) 21:32, 3 July 2024 (UTC)
 * Something being a "field of study" doesn't mean it can't be applied or studied by non-specialists. For example, history is a field of study, but non-historians apply its lessons all the time, e.g. in law or politics. Another common alternative is to describe mathematics as a 'discipline'. –jacobolus (t) 21:38, 3 July 2024 (UTC)
 * I would be happy with "discipline" (unlinked, of course). I think the previous objection was that it made math sound too much like punishment (someone asked something like "if math is a discipline does it hurt?" which I had to admit was a funny line). --Trovatore (talk) 00:17, 4 July 2024 (UTC)
 * Yes, discipline would be much better than area of knowledge, which is much more passive. Tito Omburo (talk) 00:26, 4 July 2024 (UTC)
 * @Trovatore I personally prefer "area of knowledge" but I'm fine with unlinked "discipline" as well (especially since it's linked later on anyways). Notably, most other articles with "discipline" in the opening sentence do not link it either. iczero (talk) 02:10, 4 July 2024 (UTC)
 * To clarify, I want to link "knowledge" for a few reasons: (1) pretty much every other page does, (2) MOS:CONTEXTLINK says we should, and (3) it is absolutely beneficial to the article if that's how the article starts. iczero (talk) 02:15, 4 July 2024 (UTC)
 * So CONTEXTLINK is a guideline that should be observed in most articles. Most articles need to be contextualized.  Mathematics really does not.  Everyone knows what math is, more or less, and it's about as broad as contexts get.
 * As for point (3), I just completely disagree. "Area of knowledge" is deliberately vague.  If we link it, it looks like we're reifying it, which is the exact opposite of being deliberately vague. --Trovatore (talk) 04:20, 4 July 2024 (UTC)
 * Presumably if everyone knew what math was, they wouldn't visit the article. If they don't want to know more about knowledge, they don't need to click the link. Arguably, Philosophy is the ultimate example here as it links 6 entire "everyone would know" articles. There is no harm in providing more context, especially in this case.
 * "Area of knowledge" may be vague, but it sure is a statement. It's already emphasized by being both 3 words into the first sentence and the "short description" for the article. I personally think it's perfectly fine because mathematics is directly related to knowledge anyways. It's like the "science" definition but without science. Again, no harm in linking, which seems to be an acceptable compromise between everyone involved unless we go the "discipline" route. iczero (talk) 05:19, 4 July 2024 (UTC)
 * But the point is that it's as little of a statement as possible, and that's on purpose. There is absolutely harm in linking, because it tends to defeat that purpose.
 * I really do think we should consider rephrasing to avoid "mathematics is" entirely, but if we have to have a "mathematics is" statement, the predicate nominative should absolutely not be linked. --Trovatore (talk) 05:45, 4 July 2024 (UTC)
 * What do you think about the following?
 * "Mathematics concerns numbers, formulas and related structures, shapes and the spaces in which they are contained, quantities and their changes, and other related topics. [...]"
 * There's an argument to be made that this is just a non-definition filler, but if you interpret "area of knowledge" to be an empty statement, the current opening would be as well. Statistics and Science both use "discipline", so perhaps we should just use that instead as was previously proposed. iczero (talk) 08:00, 4 July 2024 (UTC)
 * This sentence is factually wrong: it excludes many important parts of mathematics such as mathematical logic, set theory, group theory, homological algebra, probability theory, etc. It includes the study of "quantities and their changes, which is not mathematics but physics (calculus is not the study of quantities and their changes"; it is a tool for this study and many other purposes). D.Lazard (talk) 08:22, 4 July 2024 (UTC)
 * I'm happy with formal study of quantities and their changes as mathematical analysis, fwiw. Perhaps just "study of quantities", if its "changes" that invokes physics.  Homological algebra and group theory are broadly speaking "algebra".  Probability theory is notably missing, but it is not universally accepted that probability theory "is" mathematics.  Some probabilists have the opinion that probability is really its own sort of science.  Set theory and foundations are missing from the first sentence, but described in more detail in later paragraphs, such that it seems unnecessary to squeeze them into the first sentence.  Tito Omburo (talk) 16:38, 4 July 2024 (UTC)
 * This is the current opening modified to eliminate "area of knowledge". That is all it attempts to be. iczero (talk) 21:47, 4 July 2024 (UTC)
 * Also, Domain of knowledge should absolutely not redirect to Domain knowledge. These mean entirely different things! –jacobolus (t) 17:13, 3 July 2024 (UTC)
 * I strongly prefer that the first sentence not attempt to link to anything more general than "mathematics". Mathematics is already an extremely general thing.  We don't really have to put it in context.  Attempting to put it in context is actually an actively bad idea, because different people have different ideas about what the context should be, and the less we say about it that early, the better.
 * What we could consider is moving away from the "mathematics is..." model to a more active verb. Something like "mathematics studies topics such as..." would be a possibility. --Trovatore (talk) 19:56, 3 July 2024 (UTC)
 * @Trovatore That would go against MOS:CONTEXTLINK. Pretty much every page, including arguably more general pages like Science, link context in such a manner. Context doesn't need to be "more general", it just needs to position the topic in context, which I believe the current opening does a great job at.
 * WP:EGG does not apply. Knowledge clearly covers constituent areas, even if it is not directly mentioned in the first paragraph.
 * Also, please do not revert prior consensus without new consensus. iczero (talk) 21:49, 3 July 2024 (UTC)
 * It's not exactly EGG. It's a similar issue.  When "area of knowledge" appears as a single link in blue, the natural expectation is that it points to an article about areas of knowledge, rather than about knowledge.  Thus it violates the least surprise principle, which is the same problem as easter-egg links. --Trovatore (talk) 22:00, 3 July 2024 (UTC)
 * @Trovatore I completely fail to see the problem with the link "area of knowledge" linking to a page which discusses knowledge and its areas. There is no surprise here. iczero (talk) 23:44, 3 July 2024 (UTC)
 * There certainly is. A link to "area of knowledge" should point to an article about areas of knowledge, not about knowledge.  --Trovatore (talk) 23:52, 3 July 2024 (UTC)
 * @Trovatore Is there truly a significant difference between an article about areas of knowledge and an article discussing both knowledge and its areas? That feels like nitpicking more than anything. iczero (talk) 02:03, 4 July 2024 (UTC)
 * There is a significant difference between the content you would expect to find at an article called "knowledge" and one called "area of knowledge". --Trovatore (talk) 04:16, 4 July 2024 (UTC)
 * On a page titled "Area of knowledge", I would expect to find information on areas of knowledge. On a page titled "Knowledge", I would expect to find information on knowledge itself and areas of knowledge. In either case, I get what I'm looking for, no astonishment necessary. (This, of course, ignores the fact that a hypothetical article named "Area of knowledge" would not exactly be useful anyways.) iczero (talk) 05:06, 4 July 2024 (UTC)

Here "scientific discipline" is I think a sufficiently vague grammatical necessity that more accurately conveys that mathematics is a "science" without committing to any particular criteria for what makes a science (like "Science" might). Note that we *don't* pipe a link to science, which could be read as suggesting some particular definition of "scientific discipline". However, we do pipe a link to formal science, which there is universal agreement (particularly when paired with "includes the study"). Also, it does not commit any more to what mathematics "is" than the previous version. In fact, we have sharpened the "topics" which belong to mathematics (see Lazard's comment earlier about "quantity and change" possibly being physics - the addition of "formal" should help). Tito Omburo (talk) 10:55, 4 July 2024 (UTC)
 * Surely mathematics is a science, not just a vague "area of knowledge". It would make more sense to write that, and link science. Linking knowledge is unhelpful to the reader. Tito Omburo (talk) 22:38, 3 July 2024 (UTC)
 * Unfortunately there's a strong current of opinion that (1) "science" applies only to disciplines that follow Popper's criterion of falsifiability and (2) mathematics does not meet that criterion. Both of these propositions can be criticized, but there are enough serious workers who hold to this line of thinking that I don't think we can contradict them in Wikivoice, especially in the first sentence.  The question can certainly be discussed in the body.
 * I completely agree that linking "knowledge" is unhelpful. --Trovatore (talk) 22:53, 3 July 2024 (UTC)
 * Fair enough, but it strikes me that "area of knowledge" is not really appropriate for other reasons. Mathematics is certainly closer to a science than it is an "area of knowledge." This framing seems to confuse mathematics with something like the collection of theorems. Tito Omburo (talk) 23:19, 3 July 2024 (UTC)
 * Well, as I said above, I think it's a throwaway term, intended to satisfy a grammatical function while saying as little as possible. That's actually a reasonable goal (because anything we do say here is going to be contentious and rightfully objected to) but I think it makes it particularly inappropriate to link it.
 * That's why I think a first sentence that doesn't use the word "is" is something we should consider. If we made it something like [m]athematics studies topics such as quantity, structure, change..., it might mitigate this problem.  I might not even object to some links on the things mathematics studies (provided we can do it without using pipes, or at least without using pipes in a way that violates least surprise).  --Trovatore (talk) 23:29, 3 July 2024 (UTC)
 * Hmmm. The first paragraph read as a whole really advances this "area of knowledge" thesis. Mathematics is presented somehow as a collection of topics rather than a systematic method. That strikes me as misleading. Tito Omburo (talk) 23:52, 3 July 2024 (UTC)
 * At this point it would be worth looking back in the archives to see the old discussions on this. I think it was around '06 or '07.  Anyway we made the decision long ago to avoid trying to characterize mathematics too precisely in the opening paragraph.  In my opinion that was a necessary choice, and the reasons it was necessary have not changed.  Search for my remarks on "definition" versus "demarcation". --Trovatore (talk) 23:57, 3 July 2024 (UTC)
 * Ok fair enough. Tito Omburo (talk) 00:10, 4 July 2024 (UTC)
 * @Trovatore Perhaps we should simply keep the opening sentence the way it was. Reading the archives, this is about the millionth time this discussion has occurred. I would personally propose "science" and then later clarify the contention. For what it's worth, it seems most mathematicians and scientists are too busy with their actual research to debate the definition of math.
 * I still think that "area of knowledge" is a reasonable description. Is mathematics not also a pursuit of knowledge? iczero (talk) 01:59, 4 July 2024 (UTC)
 * I am fine with "area of knowledge", as long as it is not linked. --Trovatore (talk) 02:02, 4 July 2024 (UTC)
 * "Area of knowledge" is too passive. Mathematics is more than a collection of results or topics.  Might I suggest "scientific discipline"?  Tito Omburo (talk) 10:41, 4 July 2024 (UTC)
 * I would expect "scientific discipline" to be even more likely than "science" to create a misleading impression that the subject should involve falsifiability as a criterion or follow the "scientific method". –jacobolus (t) 15:53, 4 July 2024 (UTC)
 * The discussion has become quite rarified. In the interests of moving things along constructively, I have committed this:
 * Mathematics is a scientific discipline that includes the formal study of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.

I have since added a few brief clauses to the end of the first paragraph on mathematics' status as a Science, and its general lack of characterization. This seems like important information to give our readers up front. Tito Omburo (talk) 11:42, 4 July 2024 (UTC)


 * I have it as "scientific discipline", although perhaps "systematic discipline" might be better? Tito Omburo (talk) 14:18, 4 July 2024 (UTC)
 * IMO, this change is not an improvement. In particular, I am against the use of "scientific discipline" that suggests that mathematics does not exist outside the academic world, and of the use of"formal" (linked to formal science), since most mathematicians are certainly against reducing mathematics to its formal aspect. Also, this edit does not fix a blatant inaccuracy that is here for many years: the "study of quantities and their changes" is not mathematics; it is physics.
 * Per WP:BRD, I'll revert this change, bur it may be restored if there is a consensus for it. D.Lazard (talk) 16:08, 4 July 2024 (UTC)
 * I've put my proposal in below. The main point is that mathematics involves its own rigorous methods and ideas, and is not just a fixed "area of knowledge".  Also, I included a sentence that not only is there no consensus on the academic discipline, but what mathematics itself is (including whether it is a science) is not settled by consensus.  I'm puzzled by the objection to the word "formal".  Surely most of mathematics is a formal science, at least to a first approximation? And the formal study of quantities and their changes is (supposed to be) mathematical analysis, which also seems right to a first approximation.  Tito Omburo (talk) 16:26, 4 July 2024 (UTC)

Proposed lede paragraph v 2.0
Here is my proposed lede. It is largely identical to the lede that reverted, but note that "scientific discipline" has been replaced by "systematic discipline". I felt that this change is important because mathematics is not just an "area of knowledge", it's a discipline (like any science) which involves its own rigorous methods and ideas. So, I would like to propose the following: "Mathematics is a systematic discipline that includes the formal study of 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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline; nor in the philosophy of science as to a characterization of mathematics, nor whether it should be regarded as a science." For reference, here is the current version: "Mathematics is an area of knowledge that includes the topics of 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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline."
 * Support. As nom. Mathematics is not an "area of knowledge".  This term is insufficiently broad to accommodate all of mathematics.  It is a systematic discipline.  There is some objection to the word "formal" here, in saying "[Mathematics] includes the formal study of..."  The concern is that the lede sentence might suggest to someone that mathematics is reduced to formalism.  However, the language is not exclusive and, rather obviously, mathematics is both formal and systematic (although it is not reducible to a single formal system).  I have also included an important clause at the end.  Not only does the academic discipline of mathematics somewhat defy characterization (e.g., which department gets to hire the Deep Learning candidate?) but the difficulty in characterizing mathematics is significantly deeper.  There is no agreement in philosophy about what mathematics is, nor whether it is regarded as a science.  These seem like important points to have in a lede paragraph whose purpose is to define the subject.  Finally, Lazard's concern over "quantities and their changes" seems overwrought.  To a first approximation, this is mathematical analysis.  (And often it's true that it is hard to separate analytical reasoning from physical.)  But I'm not wedded to this wording if someone has a better idea.  Tito Omburo (talk) 16:45, 4 July 2024 (UTC)
 *  Oppose  per my above post of 16:08. The new version is a disimprovement, because of the use of the undefined neologism "systematic discipline". D.Lazard (talk) 18:04, 4 July 2024 (UTC)
 * The article science begins with "Science is a strict systematic discipline". So our use is not a neologism.  Tito Omburo (talk) 19:24, 4 July 2024 (UTC)
 * Support. It's at least a lot better than "area of knowledge" and "discipline" alone, which are both (intentionally) vague and are arguably a non-definition. With a different interpretation, the current definition is circular.
 * As for NPOV, there's already a disclaimer at the end of the current "definition" anyways. I highly doubt there will ever be a universally agreed upon definition (or even description) of mathematics. This is not to say that NPOV is not necessary, but rather that a definition accepted by most (see Britannica et al.) is better than no definition. iczero (talk) 21:39, 4 July 2024 (UTC)


 * Support with modification: Remove "systematic" (which borders on WEASEL), "formal" (which is freighted with unintended meaning, per discussion above), and "academic" (per above). Mgnbar (talk) 23:21, 4 July 2024 (UTC)
 * I don't think "formal" has unintended meaning here ("based on rules"). If you think the link to Formal science could be misinterpreted, perhaps we can link to Formal system instead? I agree that "systematic" and "academic" probably aren't necessary. iczero (talk) 02:20, 5 July 2024 (UTC)
 * Support the version with "systematic discipline" as mathematics isn't a science. Otherwise, I feel that this is about as general of a compromise as we can hope for. Rhosnes (talk) 10:25, 5 July 2024 (UTC)
 * Is it fair to say we've reached a consensus on this proposal? Rhosnes (talk) 08:40, 6 July 2024 (UTC)
 * Support There's no way we can get a one-line summary of what mathematics is absolutely right, but this is good enough. XOR&#39;easter (talk) 18:12, 6 July 2024 (UTC)
 * Comment At a quick read I think it's mostly fine, as prose. The link of "formal" to formal science has to go, though.  That's a WP:EGG; the reader does not have notice to expect that target.  --Trovatore (talk) 20:37, 12 July 2024 (UTC)  UPDATE:  I changed "fine" to "mostly fine" -- "formulas and related structures" doesn't really work.  Formulas are syntactic; structures are "semantic" or "ontic" or whatever word you prefer. --Trovatore (talk) 20:40, 12 July 2024 (UTC)
 * Arguably the article formal science is not currently very useful, but come on, this is nowhere close to an "easter egg" link. –jacobolus (t) 04:21, 13 July 2024 (UTC)

New proposal for the first paragraph
I have a feeling of déjà vu with the preceding section: most of its content has already been discussed in Talk:Mathematics/Archive 15, and I do not see in it any new idea for improving the first paragraph. Saying that mathematics is the "science of reasoning" seems a new compromise, for two reasons. Firstly it emphasizes the fundamental role of theorems and proofs (when a new theory is submitted to a mathematical journal, a common reason of rejection is "this is not mathematics, since there is no theorem, and this cannot lead to theorems"). Secondly, the main argument against the use of "science" is the lack of falsifiability. Since wrong proofs are falsifiable (by finding a gap in the reasoning or by providing a counterexample), using "science of reasoning" is correct from Popper's point of view. A major example of falsifiability in mathematics is Russel's paradox and the work on set theory that was needed for resolving it. Here is the new first paragraph that I submit to the discussion.

D.Lazard (talk) 17:56, 4 July 2024 (UTC)


 * Looks good to me. Tito Omburo (talk) 19:41, 4 July 2024 (UTC)
 * I like it except for the first sentence. The first sentence is overly complicated. It includes the contentious word "science". The second half could be misconstrued as a definition, that unfortunately encompasses the other formal sciences (statistics, computer science, maybe linguistics). Mgnbar (talk) 23:26, 4 July 2024 (UTC)
 * Some nitpicking:
 * "subareas of mathematics with large overlapping; they include" I believe this should be "large overlap".
 * The period before ref 3 should be a comma.
 * The description on "set theory" is inconsistent with prior syntax, perhaps it should also be parenthesized?
 * Do we need the part after "but there is no general consensus among mathematicians for any of them"? It seems a bit restrictive, as if those were the only two reasons for there being no "standard" definition.
 * (edit) No longer directly support. This definition could probably be confused with Logic. 01:19, 5 July 2024 (UTC)
 * iczero (talk) 01:05, 5 July 2024 (UTC)
 * First of all, the most general science of reasoning is logic, not maths; whether mathematics is even based on rigorous logic is a matter of contention (see logicism) ─ personally, I think the vast majority of mathematics isn't based on rigorous logic, but I digress. Secondly, maths isn't based on the scientific method (there is no "observation" in maths), so it's, at best, weird to call it a science.
 * I like Tito's suggestion much more. I think calling maths a "systematic discipline" and a "formal study" is about as general of a compromise as we can hope for. Rhosnes (talk) 10:16, 5 July 2024 (UTC)
 * The word "science" is not limited to natural science. (Clicking that shows we have an article about branches of science and an article about formal science.) –jacobolus (t) 10:23, 5 July 2024 (UTC)
 * Even formal sciences, such as artificial intelligence, are still based at least partially on the scientific method. Mathematics isn't based on the scientific method, so it's weird to call it a science. It is more accurately characterised as a branch of philosophy since it studies the most general structures or patterns of reality. It is closely related to logic, but is less rigorous since it is based on a greater number of unverifiable axioms and undefined primitive notions. It sits right at home with all the other branches of analytic philosophy. Calling it a science but not the rest of analytic philosophy seems arbitrary to me. Rhosnes (talk) 10:43, 5 July 2024 (UTC)
 * "Science" is a 1000 year old English word which comes from a 2000 year old Latin word. The name "scientific method" is from the 19th century, and the weird misconception that mathematics isn't a "science" in the ordinary sense of the word is only a few decades old, at most, based on people taking the content of their 7th grade science class too literally.
 * A quick search turned up this nice letter by Alexander Ebin to Mathematics Magazine, 1953, :
 * Unfortunately, Mr. Foster hangs his argument upon a total misconception, amounting to an inversion, of the meaning and history of the word "science". By historical tradition and continued usage, mathematics is not only "The Queen of the Sciences", a title used by Eric T. Bell; this subject [..] is also the Queen Mother of the Sciences for all the exact sciences, and the name of science itself, are derived from mathematics. This point was brought out by the late Charles S. Slichter, one of America's most beloved teachers of pure and applied mathematics, in a delightful essay, Pilymaths: Technicians, Specialists, and Genius, where he explains: "In ancient times, of course, all engineers were scientists, and all scientists were masters of many fields. The etymology of the word mathematics illustrates this. Mathematics does not mean mathematics; it means science or, more accurately, general science or all science."  Thus, the original name for science was mathematics, and to declare that mathematics is not a science is, etymologically, an inversion of facts. It would be more appropriate if the natural sciences were deprived of the use of the name, science, which came into being through mathematics.  This is purely a verbal point. A more important point is that, from ancient to modern times, exact science has evolved out of mathematics. As Slichter points out in his essay, many modern engineers earn their living from the theory of moments defined by Archimedes in, be it noted, a purely mathematical paper. Archimedes, like other ancient founders of exact science, conceived that the order of nature was to be understood through pure mathematical ideas, and he performed physical experiments only to fix or toverify these ideas.  Newton worked in the same way to found the theory of universalmechanics. He did not use mathematics as a "serving maid" but as the door to universal knowledge. As he said in the preface to his Principia, he proposed "to sub- ject the phenomena of nature to the laws of mathematics." As far as Newton was concerned, science was primarily and dominantly mathematics, and only secondarily experimental and observational.  [...] Mathematics is a uni- versal science, the original and foundational exact science, and the door to all exact science, Calculation or reckoning is now, as it has been from ancient times, a practical language of quantity which is properly subordinate to the sciences, industries, etc., that it serves. [...]  The very identity of exact science, in all the basic sciences of nature, issues from and is defined by mathematics. Thus, to say that mathematics is not a science, in the sense that physics or astronomy are sciences, is to say that science is not science, a manifest absurdity.
 * –jacobolus (t) 11:31, 5 July 2024 (UTC)
 * Standards change and the meanings of words change. Before the 19th century, science wasn't as rigorous as it is today; many disciplines that are now almost universally regarded as pseudoscience ─ such as phrenology ─ used to be considered "science". If you accept the standards by which we label such disciplines "pseudoscience", you must also accept that mathematics isn't a science. You can't have your cake and eat it, too.
 * Anyway, I feel that we are going off at a bit of a tangent here; the bottom line is that the characterisation of mathematics as a science is controversial, as is admitted by the lede of the article on "science", so we shouldn't claim that mathematics is a science in Wikipedia voice. Rhosnes (talk) 14:57, 5 July 2024 (UTC)
 * People are trying to do too much with the initial paragraph, working in controversies that don't belong in the lede. To begin with, lawyers, philosophers and others might dispute "the science of reasoning." We don't need a precise pigeon hole either. I think the vague "area of knowledge" is fine, without the link to knowledge. I couldn't find anything helpful in that target. Nor do we have to debate esthetics vs practicality. The target of this article should not be professionals, with their axes that need grinding, but the general public that wants a better understanding of the topic. The more arcane issues can be addressed later in the article. Also I don't understand why number theory is the first topic listed. I'd start with Geometry and Algebra, which most people have heard of before, and maybe include the word calculus in the description of Analysis.--agr (talk) 21:56, 5 July 2024 (UTC)
 * My main problem with "area of knowledge" is that it's just wrong. Mathematics isn't an "area of knowledge".  Sciences aren't "areas of knowledge".  They are patterns of human activity, that certainly include knowledge but other things too.  Mathematics has its own kind of methods and forms like any of the sciences.  "Discipline" is a much more accurate word.  Tito Omburo (talk) 15:47, 6 July 2024 (UTC)
 * I was going to say much the same thing; "area of knowledge" is just too inert. For your entertainment, here is a passage from the Afterword to Do Not Erase (Princeton University Press, 2021): Mathematicians know what mathematics is but have difficulty saying it. I have heard: Mathematics is the craft of creating new knowledge from old, using deductive logic and abstraction. The theory of formal patterns. Mathematics is the study of quantity. A discipline that includes the natural numbers and plane and solid geometry. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A science involving things you can't see whose existence is confined to the imagination. A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined. The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. Laboriously constructed intuition. The largest coherent artifact that's been built by our civilization. The thing that all science, as it grows toward perfection, becomes. An ideal reality. No more than a formal game. What mathematicians do, the way musicians do music. XOR&#39;easter (talk) 18:22, 6 July 2024 (UTC)
 * I think all areas of knowledge, even theology, implicitly include the process of gaining new or improved knowledge, but whether I'm right or wrong, this discussion doesn't belong in the first paragraph of Mathematics. The first paragraph should (according to WP guidelines) be simple and clear to a person who has at best a vague idea of mathematics.  Subtleties can be discussed later in the article. Zaslav (talk) 20:53, 6 July 2024 (UTC)

For what it's worth, "discipline" is frequently defined as "a branch of knowledge" or "field of study," often tied to higher education. Some examples:


 * Oxford languages: "a branch of knowledge, typically one studied in higher education"
 * collinsdictionary: "a branch of knowledge or learning"
 * dictionary.cambridge.org: "a particular area of study, especially a subject studied at a college or university"
 * britannica.com/dictionary/discipline: "a field of study : a subject that is taught"
 * www.merriam-webster.com/dictionary/discipline: "a field of study"
 * our article academic discipline: "An academic discipline or academic field is a subdivision of knowledge that is taught and researched at the college or university level."

Given Math's origins in ancient times, the higher ed implication may be anachronistic. "Field of study" might work. I also looked at the lede of our Science article:


 * Science is a strict systematic discipline that builds and organizes knowledge in the form of testable hypotheses and predictions about the world. [where "systematic" links to Scientific method.]

A parallel lede might be: --agr (talk) 18:08, 7 July 2024 (UTC)
 * Mathematics is a field of study that discovers and organizes knowledge about numbers, shapes and patterns using logical proofs based on stated assumptions and previously demonstrated results.


 * I like the "field of study" construction. But I think this excludes applications too much.  Mathematics is closely wed to the sciences, because of those applications.  Also, I would eliminate the word "logical".  Most mathematical proofs aren't "logical"; "rigorous" would be a better word; but even non-rigorous arguments have a place in mathematics.  How about
 * Mathematics is a field of study that discovers and organizes knowledge about numbers, shapes and patterns, often using proofs based on stated assumptions and previously demonstrated results, and applies this knowledge to quantitative aspects of the sciences.
 * Tito Omburo (talk) 21:28, 8 July 2024 (UTC)
 * I like this one, too. I think both of your suggestions are a big improvement on the status quo. I do agree that the current opening sentence is just wrong. Whether mathematical "knowledge" even constitutes knowledge in an epistemological sense is a matter of contention, and iirc most contemporary philosophers of mathematics think that mathematics models, rather than describes, reality. Rhosnes (talk) 10:23, 9 July 2024 (UTC)
 * I may accept the first part of the sentence, although I am dubious with the grammatical structure ("field of study" as a subject for "discovers" and "organizes").
 * I oppose strongly to the second part of the sentence that reduces mathematics to a very small part of it; moreover the use of "patterm" is controversial, since the term is rarely used in mathematics, and certainly not for qualifying theorems and calculus (which is not about numbers and shapes).
 * So, I propose to merge this last suggestion with my initial one to give:
 * Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
 * For the remainding of the first paragraph, I suggest a slight modification of my first proposal:
 * There are many areas of mathematics that include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics).
 * I suggest also to remove from the lead the sentence about the lack of consensus for a definition. D.Lazard (talk) 11:17, 9 July 2024 (UTC)
 * Yes, this seems like an improvement over all suggestions so far, although perhaps geometry could be made into "geometry and topology". Tito Omburo (talk) 12:36, 9 July 2024 (UTC)
 * IMO, one must consider here (for the lead of a general-audience article) that topology is included in geometry. In any case, the list of areas presented here is not supposed to be complete ("include") and we must limit the the list to areas whose names are known by everybody (this seems not the case of topology). D.Lazard (talk) 14:43, 9 July 2024 (UTC)

I've gone ahead and put the latest proposed revision in, since it seems unlikely to cause much contention. But something is slightly off about analysis described as the study of continuous changes. It seems to me more accurate to state that analysis is the study of approximations and limits? Tito Omburo (talk) 21:14, 11 July 2024 (UTC)
 * I haven't been following this latest discussion closely, but traditionally the logic has been: We have sources saying that math is the study of change, shapes, etc. Then we translate those topics into modern research terminology. Change becomes calculus, which becomes analysis.
 * In other words, you'll need sources backing up your description (with which I agree, by the way). Mgnbar (talk) 07:42, 12 July 2024 (UTC)
 * If one would be very accurate, one could say that analysis is the study of continuous and differential functions; this includes limits, and somehow approximations (a differential is a best linear approximation). However, talking of limits and functions seems too WP:TECHNICAL here, while "continuous change" seems understandable to almost everybody. Some older versions of the article used "change" but this is confusing as this includes also discrete changes. D.Lazard (talk) 09:38, 12 July 2024 (UTC)
 * This seems to me even further from the mark. Analysis is the study of approximations, in the sense of inequalities.  This includes continuous and differentiable functions, but also other things which are not functions, like integrals, sequences and series.  It includes things like measure theory and ergodic theory, estimates (such as those related to the distribution of prime numbers, abelian and tauberian theorems, etc.)  Tito Omburo (talk) 10:13, 12 July 2024 (UTC)
 * I agree that "limit" is too technical for this spot. XOR&#39;easter (talk) 17:23, 12 July 2024 (UTC)
 * This is just my own off-the-cuff take, but I'd say that analysis uses "approximations and limits" to understand what "continuous change" is. In other words, the latter is what we want to get a handle on when we study the subject, and the former is the tool used to get there. XOR&#39;easter (talk) 17:22, 12 July 2024 (UTC)