Talk:Mathematics
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Brainstorming / what does the article need?
Before editing the article in the coming year, I just wanted to ask what everyone thinks it needs most. I've always worked on more focused topics where improvements are relatively clear, but what specifically would move this one closer to FA status? Is there anything that stands out to the veterans in particular?
I've skimmed the article and come up with my own breakdown of possible changes, but that may not line up much with what everyone else agrees is important. I can discuss my proposed changes here first too since I imagine editing needs to be incremental here. Zar2gar1 (talk) 19:19, 8 December 2018 (UTC)
 Yes incremental changes, and discussing proposed changes here first, are both very good ideas. Paul August ☎ 19:49, 8 December 2018 (UTC)
 If nothing jumps out at anyone, I'll start by bringing up some smaller notes to discuss...
I really like mentioning Aristotle where the definition section starts, but I wonder if it should convey a little more continuity with what follows. Aristotle definitely focused on quantity & magnitude more than today, but he also commented on how it relied on a sort of abstraction ("separability"): Physics II.2(done) I noticed that the article (rightly, I feel) mentions the cognitive "moves" people use in math throughout (e.g. deduction, counting, etc.). However, synthesis & analysis in the mental/philosophical sense are never really brought up. If that would be a positive change, I don't know what the best way to massage them in would be, but I did a little experiment using find in my browser just to have some data...
 "synthesi(s/ze)", "assemble", and "combine" never appear in the article at all
 "build" only appears in historical discussions, never as an intuitive act
 "construct(ion)" does pop up several times, but only in this sense around the paragraph on intuitionism and maybe the discussion of Gödel's incompleteness theorem
 "analy(ze/sis)" obviously appears several times as part of subfield names. Otherwise, except for maybe two instances (one in footnote 59 for a source on set theory and the other near the mention of Newton), the word is just used as a synonym for study
 Other synonyms, like "divide/division", "split", "break down", and "decompose" never appear in this sense either, if it all; the one use of "reduce" near another discussion of Gödel's theorem may be the only exception.
 Finally, without overemphasizing it and biasing the article towards an intuitionist POV, it might be worthwhile to mention the role of intuition and creativity a little more directly. The "Inspiration..." section seems especially fitting for a couple sentences, but perhaps just namedropping & linking one of the words in the lead's second paragraph wouldn't hurt either. Zar2gar1 (talk) 14:23, 15 December 2018 (UTC)
 This is a digression, so I'm putting it in small type, but just so you know, you might want to be careful with the word intuitionism, which does not really mean what I think you think it means. The defining feature of intuitionists has little to do with intuition; it has to do with their rejection of excluded middle. The term "intuitionism" for this has a historical basis in Brouwer's personal intuition, but it does not have much to do with whether your foundations are based on intuition. Unfortunately the lead of our intuitionism article is misleading on this point and really ought to be corrected. Trovatore (talk) 21:52, 15 December 2018 (UTC)
 Oh, no worries. I'm aware intuitionist logic is pretty much what people usually mean by the word nowadays. My impression's always been that the different philosophical outlook is still there in the background though, even if nobody obsesses over it like Brouwer did, but I could be mistaken. Since the above changes all touch on more philosophical & mental things, I just wanted to let everyone know I was aware about it and wasn't going to use any edits to make math look like Romantic poetry or something. I really appreciate the feedback though and would be interested on what you think about the changes themselves. Zar2gar1 (talk) 00:45, 16 December 2018 (UTC)
 Oh, absolutely, the import is philosophical, not just formal. But the point is that intuitionists are not the only people who place a high value on intuition. For example Gödel (Platonist) proposed that truths about underlying mathematical reality were directly accessible to intuition. Trovatore (talk) 00:50, 16 December 2018 (UTC)
 I like Zar2gar1's suggestion that there be a small section discussing how mathematicians seem to think about how they obtain results. I feel I got good at math only when I learned how to make "cognitive moves", but I never know how to explain that to someone who does not know what it means. However, the terms "synthesize", "build", "construct", "analyze", "divide", "decompose", "split" "reduce" are all far more mechanical  that is what I do when I'm not particularly inspired or awake ... its necessary but sometimes tedious makework to split, construct, analyze, synthesize. Without these steps, you can't quite gain the required knowledge, but they're the leastfun part of math .. the "cognitive moves" being the most fun (for me). It seems that suitable articulations are necessarily philosophical, metaphysical, or appeal to psychology, sociology or have MD's place mathematicians into MRI machines. Here is a physics paper that asks: "how can assemblages of atoms (described by equations, taking the form of humans) come up with ideas and communicate them?": "Agent Above, Atom Below: How agents causally emerge from their underlying microphysics" by Erik P Hoel https://fqxi.org/community/forum/topic/2873 67.198.37.17 (talk) 19:43, 18 April 2019 (UTC)
 This is a continuation of the preceding discussion. However it has to do with the way of presenting mathematics, and this belongs to the answer to the initial question. So, I do not put it in small characters. What follows is somehow WP:OR, as I have never seen it published in an academic paper. However, it is based on a long career of professional mathematician, and I do not know any mathematician that will fondamentally disagree with me.
 The modern mathematical discourse has two parts, an intuitive part (that is called either "intuitive explanation" or "context" in WP articles), and a "formal part". The formal part consists of formal deduction and proofs. If completely developed, it is, in principle, verifiable by a computer. The "intuitive part" of the discourse is the explanation of the relationship between the theory and the physical (or mathematical) experience of the audience. Until the end of the 19th century, the distinction between the two parts was clear only in geometry, where one needs a figure for understand a proof, and a proof is correct if one can verify it without referring to any figure. With the introduction, at the beginning of 20th century, of axiomatic theory, emphasis was given on formal rigor, and the intuitive discourse was often completely discarded. This culminated with Bourbaki's treatises and Dieudonné's presentation of Grothendiek's Éléments de géométrie algébrique. Meanwhile, some mathematicians tryed to develop some purely formal theories, and it appeared (at least to people in charge of evaluating them) that if a theory is developed without the aim of solving specific problems, and without any application outside itself, this theory reduces eventually to trivialities and problems that are impossible to solve. Both parts of the mathematical discourse are essential, and a mathematical theory cannot be really understood without mastering them. Using computer science terminology, one can say that the intuitive discourse it the semantics, and the formal discourse is the syntax.
 I know that what precedes is too WP:OR for being included directly in a WP article, but I thing that we must find a way for introducing these ideas in the article. In fact there is a consensus on them between professional mathematicians, but it seems that thing are much less clear for amateurs and teachers. D.Lazard (talk) 15:11, 6 November 2019 (UTC)
 I like Zar2gar1's suggestion that there be a small section discussing how mathematicians seem to think about how they obtain results. I feel I got good at math only when I learned how to make "cognitive moves", but I never know how to explain that to someone who does not know what it means. However, the terms "synthesize", "build", "construct", "analyze", "divide", "decompose", "split" "reduce" are all far more mechanical  that is what I do when I'm not particularly inspired or awake ... its necessary but sometimes tedious makework to split, construct, analyze, synthesize. Without these steps, you can't quite gain the required knowledge, but they're the leastfun part of math .. the "cognitive moves" being the most fun (for me). It seems that suitable articulations are necessarily philosophical, metaphysical, or appeal to psychology, sociology or have MD's place mathematicians into MRI machines. Here is a physics paper that asks: "how can assemblages of atoms (described by equations, taking the form of humans) come up with ideas and communicate them?": "Agent Above, Atom Below: How agents causally emerge from their underlying microphysics" by Erik P Hoel https://fqxi.org/community/forum/topic/2873 67.198.37.17 (talk) 19:43, 18 April 2019 (UTC)
 Oh, absolutely, the import is philosophical, not just formal. But the point is that intuitionists are not the only people who place a high value on intuition. For example Gödel (Platonist) proposed that truths about underlying mathematical reality were directly accessible to intuition. Trovatore (talk) 00:50, 16 December 2018 (UTC)
 Oh, no worries. I'm aware intuitionist logic is pretty much what people usually mean by the word nowadays. My impression's always been that the different philosophical outlook is still there in the background though, even if nobody obsesses over it like Brouwer did, but I could be mistaken. Since the above changes all touch on more philosophical & mental things, I just wanted to let everyone know I was aware about it and wasn't going to use any edits to make math look like Romantic poetry or something. I really appreciate the feedback though and would be interested on what you think about the changes themselves. Zar2gar1 (talk) 00:45, 16 December 2018 (UTC)
 If nothing jumps out at anyone, I'll start by bringing up some smaller notes to discuss...
Thanks for the feedback, everyone. It sounds like discussing mathematical intuition a bit more, possibly even in its own subsection, would be a popular addition. I didn't realize it before, but someone further below also mentioned that the article leans heavily towards mathematical research vs. common use and teaching. So that might be another topic that could use some expansion.
I went ahead & put in the note on Aristotle since it was straightforward. Apparently though, I cleaned out the draft notes I wrote back in 2018. Doh! I may have put them on a backup I left with family before a move though. Either way, I'll try to work a bit more on the article in my free time later this year. Zar2gar1 (talk) 02:11, 14 July 2020 (UTC)
Proposal for new structure (Sep 2020)
Hi everyone, I still plan to come around and work on the article, but in the meantime, I wanted to float a proposal to chew on for the rest of the year. It's a little farreaching but could lay groundwork to improve the article in several ways (add more content beyond math research, handle the "Mathematics as science" section more cleanly, etc.)
Essentially, I'd consider sections 24, as they are now, all different takes on what math is:
 The Definitions section first notes it's practically undefinable, but then throws in what are really 3 philosophies of math
 It then swings back to address the common notion that it's also a science (at least in the German sense)
 On top of that, the brief comment "Mathematics is what mathematicians do" arguably leans towards a 4th philosophy (instrumentalism)
 The next two sections describe wideranging properties of mathematics
 To me, these also seem to emphasize how math can't be reduced to any one of the previous aspects
I think all of this content could be harmonized though if we're willing to shuffle it a bit. Instead, we can just give each aspect/outlook a section & pull in from the old outline. I'm not suggesting using these exact names, but what about an outline something like this?
Mathematics: what is it?Aspects of mathematics Definitions ← Only the introductory paragraphs from the current Definitions... section (emphasizing it's undefinability)
 A way of reasoning ← The Logicist subsection and details on proof and abstraction from throughout the article
 A creative process ← The Intuitionist subsection, the Inspiration... section, the bits about aesthetics & "math as a liberal art" from other sections, and any new content related to how math / numeracy involves intuition or psychology
 A formal language ← The Formalist subsection, the Notation... section, and maybe some notes on how math is unusually good at jumping across time & place
 A body of knowledge ← Most of the current ...as Science section, especially the notes on empiricism, maybe a brief mention of platonism, plus new content on how mathematical knowledge accumulates and interconnects
 A skilled practice ← New content on how both mathematicians and other people use math in their lives, notes on math education and mathematical tools, and maybe a brief discussion of instrumentalism
What do you think? Zar2gar1 (talk) 23:30, 3 September 2020 (UTC)
 Ah, maybe I could have made my main point clearer, but moving away from narrow definitions to a full description of mathematics is the whole goal. You could even use "Description" as the main header instead of "Mathematics: what is it?".
 It just happens that the current arrangement partly casts some aspects of math as conflicting definitions. Under my proposal, any "Definitions" section would be reduced to at most what's in the lead of the current "Definition of mathematics" section.
 At the same time, moving towards describing distinct (but nonexclusive) aspects would allow consolidating at least sections 3 & 4 from the current arrangement. The descriptive schema also makes coverage gaps more recognizable; based on previous discussions, I think something like the "Skilled practice" section would be a particularly nice addition. Zar2gar1 (talk) 15:19, 6 September 2020 (UTC)
In that case, I'm afraid I'm opposed to your goal. That's not what this article is or should be about. Trovatore (talk) 15:35, 6 September 2020 (UTC)Hold on, maybe I haven't understood your point well enough yet. Trovatore (talk) 16:23, 6 September 2020 (UTC) I guess what I'm saying is I don't think we should focusing as much as your draft suggests on the question of "what mathematics is". That's more appropriate at the definitions of mathematics article, or to some extent at foundations of mathematics. I'm not opposed to expanding coverage of foundations in this article (why a "brief" mention of Platonsim, by the way? It's one of the main schools). But I'm not convinced that "what is mathematics?" is the right organizing principle for the discussion of foundations. Trovatore (talk) 16:29, 6 September 2020 (UTC)
Actually, that's a good point; I've seen in the archives how often arguments over definitions or the lede come up, so I don't blame you for keeping a wary eye out for that. I probably should have thought more about the main header because "what is it?" does typically imply a definition, doesn't it?
Since that's ultimately the opposite of what I was hoping to go for, I've struck & replaced the header to emphasize describing facets of the field, not defining. If the reorganization did go forward, we could also add a sentence somewhere in the new section intro stating the subsections are mutually reinforcing, not exclusive.
why a "brief" mention of Platonsim, by the way? It's one of the main schools
Oh, definitely not to downplay Platonism. On the contrary, if these changes happen & work somewhat like I'm picturing, I think we could outlink & pare back most of what's already there about the other "isms". So discussions on all the philosophical schools would be equally brief.
Each subsection would still have a corresponding philosophy or 2 to mention. By moving the emphasis from semantics to description though, there's less need to linger on those ideas that belong more in the other articles you mentioned. Zar2gar1 (talk) 22:05, 6 September 2020 (UTC)
 I understand a little better now. It seems worth discussing at least. I'd have to look through the article and try to figure out how and where it would apply before I could offer a clear opinion. Maybe I'll try to do that tomorrow. Trovatore (talk) 06:01, 7 September 2020 (UTC)
 I appreciate that you're trying to improve the clarity of the article. But I'm not sure that your plan doesn't also encompass Section 5 on the Fields of mathematics. Aren't these the "body of knowledge" actually? And then the only sections that your plan doesn't encompass are the comparatively trivial Etymology and Awards sections.
 In other words, your plan seems to be pretty close to rewriting the article. And that's not a crime  surely a better version of this article exists in some Platonic realm  but the article as it stands is the result of compromise among many editors, and there are many devils in the details. It would help me to see a more detailed draft of your rewrite. Mgnbar (talk) 13:55, 9 September 2020 (UTC)
 +1. Paul August ☎ 17:25, 9 September 2020 (UTC)
First off, I really like the draft suggestion; it may not happen until the winter, but if everyone is at least willing to consider the changes, I can definitely whip one up.
But I'm not sure that your plan doesn't also encompass Section 5 on the Fields of mathematics. Aren't these the "body of knowledge" actually?
Yes I suppose, looking at it very maximally, you could include section 5 under that if you wanted to. Besides spacing out the sections though (pure copyediting), I think this schema could emphasize how section 5 is distinct: "aspects" vs. "parts", form vs. subject matter, the activity vs. the result, that sort of thing.
In other words, your plan seems to be pretty close to rewriting the article. And that's not a crime....
A draft will definitely be good to clarify for everyone & provide a sanitycheck on the idea in my head. In advance though, I'd reassure you that for immediate changes, I'm hoping for minimal rewriting (just sentence & paragraph transitions). Looking back, my list mixes the more immediate splicing with what I think the new schema could grow into.
If everyone resolves to go forward, my intent is only to reorganize what's already in sections 24 at first. After that, I would honestly prefer to give other editors several months to work their magic before I touch those sections to add or subtract more. Zar2gar1 (talk) 03:26, 10 September 2020 (UTC)
Changes in regard to Fundamental Thm. of Algebra
(a) Sorry, the statement of the FTA is such a nonsense that it needs to be removed immediately. Take the equation in one complex variable:
exp(z)=0
That is an equation in one complex variable, but it cannot be solved since exp(z) is always non zero. The FTA is about such equations as
z^5 + b z^4 + c z^3 + d z^2 + e z + f = 0
a polynomial equation for which, in general, no formula for the solution exist, but a complex solution IS always guaranteed by the FTA. That's a very strong statement about IC. (algebraic closure)
(b) IR is actually invented to have limits of rational approximations of Sqrt[2]. Or similarly, Pi as the area of the unit circle. Continuous properties can also be studied in IQ which has a metric (distance). This part of the article is only semicorrect.
(c) I moved the statement about FTA closer in the writing where IC is introduced. In the previous version, it seemed lost contextwise where it was.
(f) I find it good that the symbols IN...IR..IC are introduced even for laymen. I introduced them to resume the discussion after the statement of FTA.
LMSchmitt 21:58, 30 October 2020 (UTC)
 As to point (f), please see Wikipedia:Manual of Style/Mathematics#Blackboard_bold. My view is that blackboard bold is primarily for blackboards (though I have used a blackboard bold numeral 1 in a published paper). Bbb is not banned, but introducing it de novo in a highprofile article has the potential to be contentious, particularly when it requires using inline
<math>
tags, which still do not render very gracefully. Trovatore (talk) 22:21, 30 October 2020 (UTC)
Scutoid § Appearance in nature
The mathematical shape Scutoid#Appearance in nature occurs not only for biological lifeforms, but also for geological shapes.
The Devils Postpile in California includes not just hexagonal prisms, crystals that are up to 100 feet long, and pentagonal prisms, but also scutoids, on the face of it. A glacier polished the top of the Postpile, exposing the geometrical crosssections of the prisms. I bring a question up on this talk page because this group may have an opinion whether some illustrations of the Postpile are appropriate in the scutoid article, Scutoid § Appearance in nature.
Or do we wait for a naturalist to provide an observation of a scutoid at the Postpile. Or perhaps a beekeeper will present a picture of a curved honeycomb showing that scutoids might also appear, as well as prismatoids, and the familiar hexagonal prisms.
Ancheta Wis (talk  contribs) 04:28, 21 December 2020 (UTC)
Maths
I recently edited this article saying that "maths" is a common term for this subject, but it was reverted and dubbed as incorrect. Can someone explain this to me, please? Maths even redirects to this article. GOLDIEM J (talk) 01:12, 15 January 2021 (UTC)
 This is a convention enforced by the editors. No other reason. Ancheta Wis (talk  contribs) 01:15, 15 January 2021 (UTC)
 There have been a few discussion, viewable in the archives here. John (User:Jwy/talk) 04:28, 15 January 2021 (UTC)
 I agree that this does not belong in the wp:lead, let alone in the wp:first sentence, as this fact(oid) is hardly ever discussed or even mentioned in the literature. It is casually mentioned at the end of the Mathematics#Etymology subsection, and I think that is just about sufficient.  DVdm (talk) 14:01, 15 January 2021 (UTC)
 As far as I know, the abbreviations are rarely used for refering to the subject of the article (an area of knowledge). They are generally used in education for referring to the courses (and their content) about mathematics. So, a mention could be worth in a section about mathematical education, if such a section would exist. D.Lazard (talk) 14:56, 15 January 2021 (UTC)
 "Math" is used in the USA (and elsewhere) and "maths" is used in the UK (and elsewhere). The question is whether either one should be explicitly mentioned in the lede. Personally I agree with DVdm that the treatment in etymology is sufficient.
 However, do we need to mention them in the lede for the sake of obeying Wikipedia:Redirect? My opinion is "no", because almost all readers searching for "math" or "maths" will know/realize that they are short forms of "mathematics". No serious risk of astonishment. Mgnbar (talk) 16:51, 15 January 2021 (UTC)
My thoughts: math and maths should be mentioned early partly due to WP:BOLDSYN, but also because someone who enters the term unfamiliar to them (like Anita if she entered 'maths') will see immediately the two are equivalent. Restricting discussion to things "in the literature" seems to me overly formal for what Wikipedia is. I don't quite understand the vehemence of those who do not want the terms in the lead, but long ago decided that my interest in the topic was not up to defending my position beyond a statement like this. John (User:Jwy/talk) 08:44, 16 January 2021 (UTC)
Semiprotected edit request on 16 July 2021
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Algebra was way before from the Greeks, the arabs just pirated and stole as china then published and sold gyp gypsy "books" claiming their "idea" , pirated rights claimed infringement , as usual for china arab ghandi gyp gypsies intellectual copyright , pirating and infringement as usual 2600:1702:FA0:B360:B47C:34A2:ADB0:DE81 (talk) 18:28, 16 July 2021 (UTC)
 Not done: please provide reliable sources that support the change you want to be made. ScottishFinnishRadish (talk) 18:38, 16 July 2021 (UTC)