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Mathematics was one of the Mathematics good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.

Article milestones Date Process Result January 22, 2006 Good article nominee Listed May 19, 2006 Peer review Reviewed April 3, 2007 Featured article candidate Not promoted September 8, 2007 Good article reassessment Kept August 3, 2009 Good article reassessment Delisted August 26, 2009 Good article reassessment Not listed This article was on the Article Collaboration and Improvement Drive for the week of May 23, 2006. Current status: Delisted good article

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Template:WP1.0Please add the quality rating to the {{WikiProject banner shell}} template instead of this project banner. See WP:PIQA for details. Mathematics Top‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Top This article has been rated as Top-priority on the project's priority scale.

Requests and metadata

Template:Outline of knowledge coverage There is a request, submitted by Sdkb, for an audio version of this article to be created. For further information, see WikiProject Spoken Misplaced Pages. The rationale behind the request is: "Level-1 vital article". To-do list for Mathematics: edit · history · watch · refresh · Updated 2022-11-29 Improve references and inline citations throughout Add distinct Introduction / Overview section? Meant to be a very accessible segue Can include Proposed definitions discussion Maybe also Etymology from the history section? Expand History section some More about math outside direct Greek -> Islamic -> Modern lineage A bit more on China, India, Japan, Pre-Columbian Mesoamerica Note universality & ability for ideas to transplant One or two sentences noting Hellenistic -> Byzantine -> Islamic continuity? Expand on connection to philosophy and liberal arts Probably want to keep it very brief Maybe a subsection under History? Suggestion: don't add a distinct philosophy of mathematics subsection Has previously led to scope-creep; just mention & link-out instead Add a section for mathematical technologies & tools Could also include converse discussion of "abstract machines" E.g. Archimedes' method, Carnot engine, Turing machine, etc. Add brief discussion about number sense & numerical cognition Also centralize discussion of other thought processes here? E.g. abstraction, analysis, synthesis, pattern-recognition Expand In society section further Add summary subsection for mathematics education Add subsection for collaboration & publication Brief sketch of math in academia, including publication process Also introduce examples / terms like Crelle's Journal, Nachlass, etc. A lot can be exported to the mathematician article Brief subsection for mathematicians as a subculture / in pop-culture?

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:

• 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

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)

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 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)

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").

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 — 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 ☎ 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

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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)

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 Misplaced Pages'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 § Discrete mathematics. On the opposite, the section § Combinatorics, 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 § Areas of mathematics 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 § Areas of mathematics was partly rewritten in this spirit in November 2021, but, for several reasons, this was not finished for sections § Discrete mathematics, § Applied mathematics, § Statistics and other decision sciences, and § Computational mathematics.

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 Misplaced Pages.

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

I'll try, in next days to fill the section § Discrete mathematics, and to merge in it the recently created section § Combinatorics. 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 § Discrete mathematics, and moved § Combinatorics as a subsection of it. Clearly my work requires many improvements, and § Combinatorics 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)

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 — 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 Misplaced Pages. 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 | contribs) 21:14, 20 May 2022 (UTC)

To editor Ancheta Wis: 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 | 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)

@D.Lazard: 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)

@D.Lazard: 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. Mgnbar'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 | contribs) 21:10, 22 May 2022 (UTC)

Ancheta Wis, 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 | 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 | 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)

@SilverMatsu: 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 | contribs) 00:29, 23 May 2022 (UTC)

@Trovatore: 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.

"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.

D.Lazard (talk) 10:00, 23 May 2022 (UTC)

Opening sentence, part II

I propose two changes to the current lede, from this:

Mathematics (from Ancient Greek μάθημα (máthēma) 'knowledge, study, learning') 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 (from Ancient Greek μάθημα (máthēma) 'knowledge, study, learning') 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 ☎ 21:29, 22 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)

@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 | contribs) 10:31, 23 May 2022 (UTC)

• Delisted good articles
• Old requests for peer review
• All unassessed articles
• B-Class mathematics articles
• Top-priority mathematics articles
• Spoken Misplaced Pages requests
• Misplaced Pages pages with to-do lists