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 § Computational mathematics 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 § Computational mathematics 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:

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".

Other practitioners of mathematics maintain that mathematics is the science of pattern, that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

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?

References

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.

Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.

Here is a proposed rewrite.

Most mathematical activity involves statements about abstract objects, known as theorems, and the use of reason to prove them. These objects may be abstractions of the natural world or entities with no relation to reality. A mathematical proof of a new theorem is formed by applying a series of deductive rules to these objects, using their known properties, which come from base assumptions known as axioms as well as previously proven theorems.

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 § Proposed definitions
• 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)

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

This article was the subject of a Wiki Education Foundation-supported course assignment, between 12 February 2024 and 14 June 2024. Further details are available on the course page. Student editor(s): Not Fidel (article contribs). Peer reviewers: GabrielleMatalaTala.

— 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 Wikipedia: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)

• 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: [1] iczero (talk) 03:56, 3 July 2024 (UTC)

I essentially agree with Trovatore's comment. Nevertheless, it may be useful to recall the history of "area of knowledge".

• 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 John Gibbons 3 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] John Gibbons 3 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, Treetoes023 putted "area" out the link, without edit summary.
• On 30 June 2023 Closetside unlinked "knowledge" with the edit summary "Knowledge is everyday word, no link per WP:OL".

Some remarks:

• 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 Treetoes023 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 [[knowledge | 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.

So, I'll restore the piped link on the article. D.Lazard (talk) 11:13, 3 July 2024 (UTC)

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." [2], "[...] in a specific domain" [3]) 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)

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)

• 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)