Talk:Mathematics

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)

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)

• 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)

@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)

• 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.

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)

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 @D.Lazard: 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 Wikipedia: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'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.

Mathematics is the science of reasoning and consists of methods, theories and theorems that are developed and proved for the needs of empirical sciences (applied mathematics) or for the coherence and aesthetic of mathematics itself (pure mathematics). There are many subareas of mathematics with large overlapping; they 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, which is presently used for the foundation of all mathematics. Many definitions of mathematics have been proposed, but there is no general consensus among mathematicians for any of them, since either they cover only some aspects of mathematical activity, or they do not include new subareas of mathematics.

References

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, JSTOR 3029402:

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'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:

• 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.

--agr (talk) 18:08, 7 July 2024 (UTC)

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'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'easter (talk) 17:22, 12 July 2024 (UTC)

Elusive definition

G. F. J. Temple published his 100 Years of Mathematics attempting to grasp the substance of the subject. See Wikiquotes from Temple for a gloss of his summary. Rgdboer (talk) 22:15, 22 July 2024 (UTC)

Very interesting. These quotes are fully coherent with the spirit in which the page has been rewritten in 2022 (starting from 30 October 2021). They may be used to improve the article and its sourcing. D.Lazard (talk) 10:57, 23 July 2024 (UTC)

Field of study link

I've tried adding a link to the term "field of study" as used in the lead several times, but each time it got promptly reverted for different reasons: @Trovatore claimed it was unnecessary, while @Remsense claimed it was misleading.

Neither argument makes a whole lot of sense.

The link is clearly necessary to at least some readers. I say that with confidence because I'm one of those readers. When I first read the new lead, I was left wondering what exactly was meant by the term "field of study": does chess count? It's been rigorously studied for centuries, and theoretical study is a central part of chess, but it is rarely taught at university. Or what about game development? It's taught at university, but there is less research into it than e.g. into chess. The link clarifies most of the confusion: chess isn't a field of study because it generally isn't taught at university, while game development, because it is taught at university, is. Therefore, unless my experience is in some way invalid or unrepresentative, Trovatore's argument can be discarded.

As for Remsense's argument, if the link is misleading, surely that simply means that either the term "field of study" or the article on the term is inappropriate? Either way, the solution clearly shouldn't be to simply de-link the term in the lead, but rather to either rewrite the lead of this article or improve the accuracy of the "field of study" article.

Am I missing something? I'm genuinely confused as to why this is even controversial. Not only do the counter-arguments make no sense, but I can't imagine why editors would be so insistent on removing links that others might find useful in the first place. What are we even gaining by removing links from a lead that clearly isn't overlinked (only containing a couple of links)? Are people doing it out it of a rule-following principle, i.e. WP:WIKILAWYERING? My theory of mind is failing me here. Rhosnes (talk) 23:01, 18 August 2024 (UTC)

The immediate issue in my mind is the page Academic discipline is too limited in its scope: mathematics is not only done in an academic context. Moreover, I don't think there's a reason to think it's more restrictive in how it's being used here than you postulated above: one can study many different things, and that's not a problem for this article, where the history of study is very broad. Remsense ‥ 诉 23:04, 18 August 2024 (UTC)

None of the academic disciplines are done exclusively in an academic context; that doesn't, however, deprive them of their status as academic disciplines. The reason that the term "academic discipline" is useful is that most/all fields of systematic study with real-world relevance are also academic disciplines. This is likely why "field of study" redirects to "academic discipline".

If you don't think the redirect is appropriate, perhaps you should propose to remove the redirect. However, as things stand, the two concepts are treated as synonymous by Wikipedia. Rhosnes (talk) 23:20, 18 August 2024 (UTC)

Sure, but if we're giving the very first defining trait of mathematics, I would say "field of study" is appropriate while "academic discipline" is not. Not sure how to resolve this, as I think the redirect is generally appropriate. Really, I do think it's just better not to link it, or to link something else. Remsense ‥ 诉 23:29, 18 August 2024 (UTC)

You might personally think that, but most editors don't seem to agree, as highlighted by the broad support a previous version, which contained "academic discipline", had garnered. Unless you can provide a convincing argument as to why the term "academic discipline" is less appropriate for mathematics than it is for e.g. history, I don't see why the link isn't appropriate. Rhosnes (talk) 00:08, 19 August 2024 (UTC)

A very cursory inspection of previous discussions shows that no, there is actually no clear consensus on this specific phrasing. Remsense ‥ 诉 00:10, 19 August 2024 (UTC)

Rhosnes if you are confused about what is included in the phrase "field of study", a link is EXACTLY THE WRONG WAY to solve it. Links must, excuse me if I shout, NEVER EVER EVER EVER EVER EVER be used to make clear what is the meaning of text that is otherwise unclear.

Really really really really. That is not the purpose of links. At all. Trovatore (talk) 02:43, 19 August 2024 (UTC)

Mind citing actual policy which says as much? This doesn't make any sense to me. If this isn't the purpose of links, then what on Earth is? Rhosnes (talk) 10:05, 19 August 2024 (UTC)

Per Wikipedia:Summary style § Technique –

Each article on Wikipedia must be able to stand alone as a self-contained unit

Remsense ‥ 诉 10:21, 19 August 2024 (UTC)

My interpretation of that paragraph, especially given the context and the given example, is that every article should have all of its major claims sourced directly to RS rather than other Wikipedia articles.

This says nothing at all about whether or not links should be used to disambiguate vague notions or clarify terms which readers might not have a complete understanding of. Rhosnes (talk) 11:12, 19 August 2024 (UTC)

If links are being used to clarify what is otherwise unclear, then the full meaning is not available to a user who does not follow the links, and therefore the article is not a self-contained unit.

The purpose of links is to provide a convenient resource for readers who want to know more about the topic being linked. They must not change the meaning of the text in any way whatsoever. Then the meaning would not be available to a user who doesn't follow the links (or doesn't have them; for example in a print copy).

No part of the purpose of links is to disambiguate the source text, whether this makes sense to you or not. --Trovatore (talk) 15:39, 19 August 2024 (UTC)

That's a take so outrageous I don't even know how to start addressing it.

First of all, I already explained that the page you linked does not use the word "self-contained" the way you are using it; the point of the policy is that every article should be sufficiently sourced in itself, without delegating some sources to linked articles.

Secondly, a sizeable chunk (I would estimate around 20%) of all links on Wikipedia clarify the meaning of terms that some readers might be familiar with. Using the example of this lead section, many, if not most, readers won't know exactly what terms like "mathematical analysis" or "set theory" mean, which is why they linked.

Thirdly, the link that I'm proposing wouldn't change the meaning of the text; it would only clarify it for those readers who aren't familiar with the precise meaning of the term "field of study".

Fourthly, your opinion is worth no more than mine. Just as my opinion that the purpose of links is partially to clarify meaning doesn't matter, neither is your opinion that this isn't the links' purpose. I honestly don't understand why you are so confident in your opinion. It doesn't make any sense and isn't supported by policy. But you do you. Rhosnes (talk) 22:53, 19 August 2024 (UTC)

Also, MOS:OL says explicitely that everyday words should not be linked. This is especially true when the everyday meaning is not exactly the same as the meaning discussed in the linked article; this is clearly the case here: the everyday meaning is not restricted to the academic world, while Field of study is a redirect to Academic discipline. D.Lazard (talk) 16:10, 19 August 2024 (UTC)

But this is confusing. I think we should change the opening sentence to "mathematics is the study of..." if we don't want to link "field of study". That's how articles on most other academic disciplines do it. Rhosnes (talk) 22:55, 19 August 2024 (UTC)

In this case, the main reason to not linking is that the word is really unimportnt in this sentence: the meaning of the sentence would not be really changed if "field of study" would be replaced with "something". IMO, the ambiguity of the term is deliberate, and linked it would change the meaning of the sentence. D.Lazard (talk) 08:40, 20 August 2024 (UTC)

Agreed. Mathematics is a lot of things to a lot of people. Any summary description we can give will necessarily be imprecise. XOR'easter (talk) 22:51, 21 August 2024 (UTC)

If an article contains a vague notion, that's a failing of the surrounding prose. A link isn't how to solve it. Remsense ‥ 诉 17:07, 19 August 2024 (UTC)

Developed and proved for the needs of empirical sciences and mathematics itself

The current opening sentence is "Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself."

The final clause of this sentence is badly phrased and, in my opinion, is quite misleading. Not only is the phrase "mathematics... discovers theorems... that developed and proved... for mathematics itself" self-referential and unclear (what does it mean for something to be developed "for mathematics itself"?), but putting the needs of empirical sciences first paints a false picture of mathematics: many mathematicians take pride in the fact that their work has no external applications since it makes mathematical knowledge an end in itself. Moreover, it seems implausible that all of mathematics is done exclusively for the empirical sciences or its own sake. Off the top of my head, I can think of mathematical humour, whereby which theorems (as trivial as they may be) are proved for their humourous value. I think what D. Lazard intended when he wrote up this version is that mathematics is generally partitioned into pure and applied. That's fair, but we should explicitly mention that to avoid confusion of the type that I'm describing.

My proposal is the following:

"'''Mathematics''' is a field of study that discovers and organizes abstract objects, methods, [[Mathematical theory|theories]] and [[theorem]]s that are developed and [[Mathematical proof|proved]] for general knowledge ([[pure mathematics]]) or the needs of [[empirical sciences]] and ([[applied mathematics]]).

@Remsense claims this version isn't an improvement. Once again, I don't understand why he would think that. In my eyes, it solves all of the problems that I describe in this comment - except possibly the issue of alternative purposes of mathematics, such as linguistics humour, although it certainly helps with it since general knowledge covers most of its other uses (including most of mathematical humour, which often hinges on the irony of mathematics being a tool of gaining general knowledge and the derived formulae being trivial).

Rhosnes (talk) 00:33, 19 August 2024 (UTC)

Your version is misleading: "general knowledge" is far to be reduced to "pure mathematics", and the distinction between pure and applied mathematics is not relevant here (see what is said about this in the article). Moreover, many mathematical theories were developed for the needs of preexisting parts of pure mathemetics. For example, Zermelo–Fraenkel set theory has been developed for solving Russel's paradox, and many theories were developed in view of proving Fermat's Last Theorem, which can certainly not be qualified as general knowledge or as applied mathematics. (It is a strange property of mathematics that many theories developed for unsuccessful proofs of Fermat's Last Theorem remain fundamental and widely used for other purposes, and that many theories used in the Wiles' proof of Fermat's Last Theorem were developed for very different purposes.) D.Lazard (talk) 09:26, 19 August 2024 (UTC)

If the distinction between applied and pure mathematics isn't relevant here, then you should get rid of the clause containing "for the needs of empirical sciences" altogether. Practically none of pure mathematics is developed for the needs of empirical sciences.

There also seems to be some confusion about instrumental goals vs terminal goals. The instrumental goals of many mathematical proofsmight well be answering mathematical questions, but the terminal goal of all of them is still general knowledge. To use your example, ZF(C) was developed to (very clumsily, but I digress) solve Russell's paradox, which was in turn necessary to provide a logically consistent foundation of mathematics, and you perhaps take this chain one step further and say a logically consistent foundation of mathematics was necessary to make existing, as well as future, mathematical theories more rigorous. But what's the goal of making mathematical theories more rigorous? General knowledge.

Chains of instrumental goals that you describe also exist in applied mathematics. One example is the Nash equilibrium, which was devised to solve the prisoner's dilemma, which was in turn formulated for the general needs of game theory, but that was ultimately formulated for the needs of social sciences.

You're not really making a strong case. If you don't like the term "general knowledge", propose an alternative. But we can't just leave the article as is; "for mathematics itself" is self-referential and nonsensical (I know the phrase "for its own sake" is quite common in colloquial speech, but it is technically meaningless; a more precise formulation is "as an ends in itself"). Rhosnes (talk) 10:22, 19 August 2024 (UTC)

The present state of the lead results of a consensus after long discussions in this talk page and its archives. So, any change requires a new consensus, and cannot result from the personal opinion of a single user. Note that, for the moment, the consensus is against you, since I am not the user that reverted your edit, and no user posted a support of your edit. D.Lazard (talk) 11:24, 19 August 2024 (UTC)

The current version of the lead isn't actually the version that gained consensus; @Tito Omburo simply implemented because he didn't think it would be met with much controversy. But I insist that it's inadequate.

Also, you aren't really helping by saying that my version hasn't yet garnered consensus. I know, and I'm trying to change that by starting this discussion. Rhosnes (talk) 23:02, 19 August 2024 (UTC)

While I don't love the phrasing "for the needs of mathematics itself", I don't think "general knowledge" is apt. Also, I think the needs of the empirical sciences are extremely important, because this constitutes most of mathematics. Tito Omburo (talk) 12:43, 19 August 2024 (UTC)

Why not? And if not, what's the alternative? My phrasing clearly isn't worse than the existing one, and I don't see why mentioning the distinction between pure and applied mathematics isn't appropriate when this is exactly what the latter clause of the opening sentence is indirectly and ineloquently referring to. Rhosnes (talk) 23:05, 19 August 2024 (UTC)

I think the lede is fine as it is. Your version pushes the pov that there is some distinction between pure and applied mathematics, when in fact mathematics was developed hand-in-glove with the empirical sciences. Tito Omburo (talk) 00:21, 20 August 2024 (UTC)

I agree that the lede is fine as is. Trying to squeeze a statement about pure versus applied mathematics into the first sentence is not going to end well. ("Pure" subjects can become "applied" and vice versa.) For that matter, I don't think that "for mathematics itself" is self-referential in a way that is actually confusing. It's like saying a celebrity "is famous for being famous". XOR'easter (talk) 22:40, 21 August 2024 (UTC)

‎THIS ARTICLE HAS GOTTEN SO MUCH WORSE

When I was in college and read this article it was inspiring now it's fucking middling bullshit that's like trying to be like PC or something. "Mathematics is the study of patterns of quantity, structure, change, and space." is a way better first sentence. 2A02:FE1:E179:1C00:A524:A608:9B18:BE91 (talk) 16:59, 22 August 2024 (UTC)

This is true. What should we do about it? Danecjensen (talk) 17:01, 22 August 2024 (UTC)

As one of the crafters of that sentence I have to say I like it too. Paul August ☎ 20:41, 22 August 2024 (UTC)

This sentence is elegant and seems easy to understand. Its problem is that it is biased, misleading and wrong: It is presented as a definition of mathematics, when it is blatant and well sourced that there is no consensus about a definition of mathematics. It excludes a large part of mathematics such as mathematical logic, set theory, combinatorics, probability theory,computational complexity theory, and many other areas. The study of space and quantities belongs more to physics than to mathematics. Article Structure has 7 sections and 6 of them are devoted to the study of non-mathematical structures. If mathematics would include the study of patterns and change, we should have an article Change theory, and Pattern theory should belong to Category:Mathematics. So, there is nothing that is worth to be kept in this sentence. D.Lazard (talk) 21:49, 22 August 2024 (UTC)

As someone who came in roughly a year ago making more or less this same line of inquiry—I appreciate that you're here to give this answer again. Remsense ‥ 论 22:34, 22 August 2024 (UTC)

At various points there have been versions of that sentence that did not come across as definitions. There was one I thought was pretty OK that said something like [m]athematics includes such topics as... and then listed the quantity, space, structure, change items, but did not state them in such a way as to appear to claim that these were exhaustive.

That said, I think the current version is also pretty OK, and possibly a little better, though it's a bit wordy.

On a different note, it's pretty extreme to say the article has gotten "so much worse" and then give the first sentence as the only concrete complaint. Not that this is new — a huge fraction of the discussion on this talk page over the years has been about the first sentence — but it's revealing. This is a long article that covers a lot of ground. The first sentence is not that important; it can harm the article but can't really help it much. The important thing is to get it over with without doing any damage, and then move on to the substance. --Trovatore (talk) 02:00, 23 August 2024 (UTC)

I spend much more time on first sentences; leads; infoboxes than I'd like. Remsense ‥ 论 02:20, 23 August 2024 (UTC)

To be clear, I meant the first sentence of this article in particular is not that important. It mostly needs to avoid trying to make things look neater than they are. There are articles on smaller, more well-specified topics where you can accomplish a lot in the first sentence.

The original poster's comment made me think of someone saying, "boy, California has gotten so much worse than it used to be. I don't like the new 'Welcome to California' sign at all." --Trovatore (talk) 20:51, 23 August 2024 (UTC)

I'm not too fond of the study of patterns of quantity line, either. It's more evocative than it is explanatory. For example, the only way that the mention of structure makes sense is if the reader already understands the word structure in the way that a mathematician does.

A snappy definition of mathematics may be impossible in principle. It is certainly impossible in practice here on Wikipedia, since all we can do is reflect the lack of consensus in the literature. XOR'easter (talk) 14:01, 24 August 2024 (UTC)

Aside about the discussion title: the original title of this discussion was "‎THIS ARTICLE HAS GOTTEN SO MUCH WORSE", which Remsense changed to "Another chat about the lead section" on the grounds that the all caps and vagueness were unhelpful, along with adding a gratuitously snarky "robot" message at the top. I'm replacing the message with this one, no longer at the top of the section, and changing the title yet again, to "Definition of mathematics seems uninspiring compared to the previous version", which seems more substantively descriptive of the complaint. –jacobolus (t) 02:52, 24 August 2024 (UTC)

In any case, I didn't intend to excise any meaning from the OP's message, so if I did I appreciate you rectifying that. Remsense ‥ 论 04:18, 24 August 2024 (UTC)

I've restored the original section title. In general we should not modify other peoples comments. Paul August ☎ 18:26, 26 August 2024 (UTC)

In regards to the definition of mathematics, I don't think that the first sentence: Mathematics is the study of patterns of quantity, structure, change, and space mentioned by the original poster (and its various versions) were intended to be a definition of mathematics. Paul August ☎ 18:37, 26 August 2024 (UTC)

Fair enough, though cf. WP:TALKHEADPOV and WP:SHOWN. –jacobolus (t) 22:05, 26 August 2024 (UTC)

Thanks for pointing out those links. In this case though, I don't think the section title, needs changing. Paul August ☎ 23:36, 26 August 2024 (UTC)

Intended or not, that's the tone it gives me — too far in the direction of saying that mathematics is just that and nothing else. XOR'easter (talk) 03:28, 27 August 2024 (UTC)

Yes, well there have been, as Trovatore has pointed out, versions of that sentence that did not come across as definitions. Paul August ☎ 20:07, 27 August 2024 (UTC)

I don't personally object to describing mathematics in this way, but it does strike me as a point of view that is ripe for nitpicking. In such circumstances, we try to assert facts, which is usually drier and less punchy, although perhaps more informative. Tito Omburo (talk) 20:30, 27 August 2024 (UTC)

"Math facts" listed at Redirects for discussion

The redirect Math facts has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 September 1 § Math facts until a consensus is reached. 1234qwer1234qwer4 01:51, 1 September 2024 (UTC)

Semi-protected edit request on 8 september 2024, remove double comma

There is a double comma in the second sentence of the chapter Symbolic notation and terminology: "This notation consists of symbols used for representing operations,, unspecified numbers". /Arimetat Arimetat (talk) 09:15, 8 September 2024 (UTC)

Done. Mgnbar (talk) 12:57, 8 September 2024 (UTC)

Changes to the article

I'm thinking about implementing changes to this article with the hope of moving it in the direction of GA status. Currently, the article uses some unreliable or low-quality sources such as arXiv, GeeksforGeeks, Online Etymology Dictionary, and Byju's. Additionally, it gives undue weight to certain topics such as having a main section dedicated to the "Relationship with astrology and esotericism". In the philosophy section, "Rigor" is not one of the principal topics in the philosophy of mathematics, and some of the Wikivoiced claims made in this section present just one among several competing views. I'm also not happy about having separate main sections for "Etymology" and "Awards and prize problems". It might be better to include this information elsewhere. For example, the "Etymology" section could be replaced with a "Definitions" section, which would cover some of the information from the later subsection "Proposed definitions" and include a paragraph or two on the etymology.

Another thing that caught my eye is that many sections on specific subtopics focus more on the history of their topic than the topic itself, such as the subsections "Algebra", "Pure and applied mathematics", and "Education". While this isn't necessarily wrong, given the vastness of mathematics and the limited space in a single article, I think it would be better to concentrate on the subtopics themselves. The main discussion of historical aspects could be reserved for the history section or for articles on more narrow topics.

I was thinking about adding a section on basic concepts to cover topics like number, operation/function, set, variable, mathematical expression/statement/equation, proof, etc. Some of this content is currently addressed in the section "Symbolic notation and terminology" so these sections could be combined.

Currently, the problem of the foundation of mathematics is mainly discussed in the subsection "Mathematical logic and set theory". Unfortunately, this subsection does not really explain what mathematical logic and set theory are. Another approach would be to focus this subsection on providing a more basic explanation and create a separate section dedicated to a simple overview of the foundation of mathematics. This new section could include approaches based on mathematical logic and set theory and mention other approaches as well.

I tried to break the issues down into separate points that can be addressed individually. While there are more points to discuss, including some of the things mentioned in the talk page todo list above, I fear that the ones raised so far are already quite extensive. I'm curious to hear what others think about these suggestions and further ways to improve the article. Phlsph7 (talk) 08:07, 25 September 2024 (UTC)

About sources, I mostly agree. As this is not my main competence, I'll focus on your comments on the content.

• § Relationship with astrology and esotericism I am in favor to remove this section. I suggest also to remove section § Specific sciences: It consists of expanding the first sentence of {{alink]Relationship with sciences}} by providing technical details that are irrelevant here and belong specialized articles.
• § Etymology: I agree that this does not deserve to be a first level section. I suggest to move it as a the first subsection of § History.
• § Awards and prize problems: I moved it as a subsection of § Popular impact (as this consisted only id adding two "=", I guess that it was a typo). D.Lazard (talk) 11:39, 25 September 2024 (UTC)
• § Proposed definitions. I am against to remove "proposed", since it must be clear that none of these definitions is commonly accepted.
• History in topic descriptions: This article is not the place for technical descriptions of the areas of mathematics. This belongs to the corresponding {{main article}}. What belongs here is the explanation on the object of an area and its relation with the remainder of mathematics. Since all mathematics has deeply changed during the last 150 years, I do not seed any way to avoid misunderstanding that refering to history. The only other way is to reduce the description of areas to bulleted lists of subareas. There are too many such bulleted lists in the article.
• adding a section on basic concepts to cover topics like number, operation/function, set, variable, mathematical expression/statement/equation, proof, etc. The technical definition of these concepts does not belong here, and for each concept, these is already a link to the relevant article. So, readers that come here for learning basic mathematics can find easily the relevant article (Wikipedia is not a textbook).
• Currently, the problem of the foundation of mathematics is mainly discussed in the subsection "Mathematical logic and set theory". Unfortunately, this subsection does not really explain what mathematical logic and set theory are. What mathematical logic and set theory are, is or should be explained in the linked main articles; IMO, this cannot be explained reasonably in a few non-tchnical lines in this article. In fact, this section is about foundations of mathematics, but it cannot be so called, because foundations of mathematics is not presently an area of mathematics, while mathematical logic and set theory are active areas of mathematics.

D.Lazard (talk) 11:39, 25 September 2024 (UTC)

Hello D.Lazard and thanks for your detailed response to the different points. I removed the section "Relationship with astrology and esotericism" for now. I'll see if its ideas can be included somewhere else as I go along. Turning the section "Awards and prize problems" into a subsection is an elegant solution. You are right that the proposed definitions of mathematics are a touchy topic. I'll try to come up with a draft that includes etymology in the next few days. If that doesn't work, your idea of moving etymology to the "History" section could be a viable alternative.

If our choice for presenting different topics and concepts is between historical descriptions, technical definitions, and bullet lists, then I'm all for historical descriptions. But I hope these are not our only options. At least in some cases, it may be possible to give a direct and accessible explanation of the topic itself. We can outsource details to child articles, but to comply with broadness/comprehensiveness, we need to struggle to make the important points accessible.

I agree that the subsection "Specific sciences" is far from ideal in its current form. The article should cover somewhere that mathematics affects both the natural and the social sciences but does not hold the same importance in the social sciences. But we don't need 5 subsubsections to cover that point. Phlsph7 (talk) 16:19, 25 September 2024 (UTC)

Change of the section on the definitions of mathematics

Phlsph7 has completely changed the section § Proposed definitions. I reverted this change twice. The two main reasons of my revert are:

• The previous version seems correct and Phlsph7 never explained their reason for changing its structure
• Both versions asserts that there is not consensus on a definition, and Phlsph7's version starts with an unattributed and controversial definition.

Also almost every sentence is controversial. Here are some examples:

• what is taught in mathematics classes. Such a definition of mathematics is so ridiculous that if it has really been proposed (I am unable to verify in the provided sources), it does not deserve to be mentioned. It is ridiculous since it would imply that Wiles' proof of Fermat's Last Theorem would not be mathematics, since it has never been taught.
• Mathematics studies absract patterns: this is an opinion presented as a fact.
• Precise definitions of mathematics are controversial: no, all proposed definitions are controversial.
• Some definitions emphasize .... Too vague, since the reader cannot know what are these definitions without searching in the references.

I could continue, but the previous version is definitively better, and every change of the section must be incremental. D.Lazard (talk) 18:13, 4 October 2024 (UTC)

In general I agree. Paul August ☎ 20:09, 4 October 2024 (UTC)

I tried to provide an explanation in my earlier posts and the edit summaries, but maybe I should have gone more into detail. The basic ideas were to include the main points of the section "Etymology" in the section "Proposed definitions", to provide better sourcing, and to better showcase the variety of definitions.

Concerning the "controversial sentences": there may not be consensus on how to define mathematics, but there seems to be consensus about certain general characteristics of mathematics, like that numbers and shapes are among the things studied in mathematics or that mathematics is used by the natural sciences. Saying these things is not the same as defining mathematics. My idea was to start the section with some general characteristics and then move on to the more controversial definitions. This way, we give the reader a basic idea of the discipline before we confront them with all the difficulties and disagreements. Phlsph7 (talk) 15:19, 5 October 2024 (UTC)

If moved, the place of § Etymology is as a first subsection of § History, since the etymology is the history of the word.

The general characteristic of mathematics are already discussed in other parts of this article. "It studies abstract patterns" and "it is a form of inquiry" are not general characteristics of mathematics, and are blatanly wrong assertions. "It is connected to the empirical world ..." is developed else in the article and the connexion is much more complex than asserted in your version: How the classification of finite simple groups is connected with the empirical world? So, all your general characteristics are controversial.

Also, your version removes the fact that the proposed definitions evolve with the evolution of mathematics.

"I tried to provide an explanation in my earlier posts ...": You never stated clearly your intention of rewriting completely § Proposed definitions, and you never explained why you disagree with the current version. D.Lazard (talk) 16:16, 5 October 2024 (UTC)

I followed your suggestion and moved the "Etymology" section. I have some concerns about the current section and I would be interested to hear your opinions on them.

• There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities. I'm not sure why the epistemological status is mentioned here and why the epistemological status of mathematics is equated with "its place among other human activities". "Epistemological status" usually refers to the way knowledge is obtained and justified, like the contrast between knowledge a priori and knowledge a posteriori. My suggestion would be to remove that part.
• This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.[172] Thanks for adding the section locations, but I don't see how these sections directly support the statement. The section "What Is Mathematics?" discusses a few definitions but does not say that there is strong consensus or what most definitions agree on. The section "What Is Mathematics, Really?" says that there is no real answer and talks instead of "many mathematics" depending on the purpose for which the term is used.
• With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task. This sentence is not supported by the following source. We could use [4] instead. I would suggest using a weaker formulation since "impossible" is a strong word.
• Is there a specific reason why Saunders Mac Lane's "Mathematics, form and function" is explicitly discussed? It's a good source but I don't think that it is important enough.
• So, an area of study can be qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction.[176] I'm not sure that this is supported by the source. One definition is given in the section "What is the nature of mathematics?". It talks about "the study of pattern and structure and the logical analysis and calculation with patterns and structure". It does not talk about proofs and deduction. The article discusses proofs at other points, but, as far as I'm aware, does not propose to define mathematics this way.

There are different ways to address these points, but I think they should be addressed. Phlsph7 (talk) 08:46, 6 October 2024 (UTC)

You have two sorts of concerns: some are related on the phrasing, some are related on citations.

If you think that a citation does not support the preceding sentence or paragraph, you can either search for a better citation, or, if you do not find a better citation, you can tag it with {{better citation needed}}. Also, do not forget that single citation may support.

If you have good reasons to challenge a specific wording, then fix it boldy, and, if you do not know how to fix it or if you are reverted, then open a specific thread on the talk page. Note that the lack of an adequate citation is not by itself a reason for changing wording.

Here are some answer to your concerns.

• Epistemology: It seems that you have a restricted view of epistemology. The question whether mathematics is a science or not is epistemology, as well as the analysis of the relation of mathematics with other sciences.
• Citation [172]]: This citation present several proposed definitions that are all based on the object of study of mathematics. So the citation is correct.
• Strong consensus: I agree that this is not sourced, but this is true. Are you able to find a source, or, if impossible, to say the same thing in a way that can be sourced?
• Impossible: This is the correct word since new domains of mathematics appear every year. For a citation, one can use the already existing citations that mention the size of Zentralblatt and Math Review.
• Mac Lane quotation: you may open a discussion for deciding whether it must be kept, replaced by a better quotation or simply removed.
• Citation [176]]: It refers to the whole preceding part of the paragraph, which includes "Another approach for defining mathematics is to use its methods". As a large part of the citation is about methods, the citation is well suited. Moreover, the definition quoted in your post talks of "logical analysis", which is the same as the "purely-logical deductions" of your article. In any case, you are free to find better sources.

D.Lazard (talk) 14:18, 6 October 2024 (UTC)

• The part about epistemology is not supported by the current sources. An easy solution to avoid the problem would be to just remove that expression, leaving us with: "There is no general consensus about the definition of mathematics or its place among other human activities." Do you think it is important that we additionally say that "epistemological status" means "its place among other human activities"? As I see it, this is not the standard meaning of the expression "epistemological status". The sources on the definition of mathematics that I'm aware of don't use that expression.
• I removed the part about the strong consensus since we currently don't have a source for it. I'm not sure that it is true. If there is no consensus on how to define it, it would be surprising if there was strong consensus on what it is.
• I added a source for the claim about the new areas and used a weaker formulation since the term "impossible" is not supported by this source. Various approaches define mathematics by its object of study, like saying that it is the study of abstract patterns or of formal systems. So it's not obvious that it is impossible in a strict sense. Phlsph7 (talk) 08:46, 7 October 2024 (UTC)

  OK for the last two points.

  For the first point: the readers of this article are not supposed to have an expertise in philosophy. This is the reason for the explanation "that is, its place among other human activities" is there. The mention of epistemology is here for emphasizing that this place among human activities is to be considered from the point of view of the theory of knowlege. This is a case where sourcing is not formally required. Indeed, WP:NOCITE says that no cite is required for "General common knowledge: Statements that the average adult recognizes as true." Here, this is the general common knowledge on epistemology that is used. Morever, only statements and assertions require a citation.

  Reading again the paragraph, I see "There is not even consensus on whether mathematics is an art or a science", which is clearly about the epistemological status of mathematics. Also the paragraph contains too many citations to 170 and 171, which go against the guideline. So, since all sentences of the paragraph, but the last one have a common source, I'll move the 2 citations just before the last sentence, and removing their other occurences. D.Lazard (talk) 10:12, 7 October 2024 (UTC)

  I changed "that is, its place among other human activities" into "that is, its place inside knowledge". I hope that resolves your concerns with the previous version. D.Lazard (talk) 12:42, 7 October 2024 (UTC)

  Thanks, that looks better. Phlsph7 (talk) 08:13, 8 October 2024 (UTC)

Section "Proposed definitions"

I reverted two changes in § proposed definitions for the following reasons:

• A first change consists of expanding "A common approach is to define mathematics by its object of study" by a description of the nature of this object of study. As there are many such definitions, summarizing them in a single sentence is either WP:original synthesis, or WP:POV (as omitting the definitions that are not represented by this short sentence).
• The second change replaces "So, an area of study can be qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction" with "According to this view, mathematics examines its object of study by following high standards of precision and relying on deductive reasoning, logical analysis, and the application of general rules." The proposed version is, at best, a wrong vague definition of the concept of a proof (the correctness of a proof does not rely on "high standards of precision", but of a correct application of the used logic, generally a higher order logic). Nevertheless I'll improve the previous formulation by replacing "so" with "for example" and "can" with "is often".

D.Lazard (talk) 09:14, 8 October 2024 (UTC)

• Sources for the first change:
  • Colyvan 2012: Mathematics seems to be the study of mathematical entities - such as numbers, sets, and functions...
  • Mura 1993: The study of formal systems ... The study of patterns ... [Definition by] Reference to specific mathematical topics (number, quantity, shape, space, algebra, etc.)
  • Brown & Porter 1995: ...the study of pattern and structure...

  The point of the edit is mainly that the expression "define mathematics by its object of study" may not be very enlightening to the reader without a clarification. We can try to workshop something similar, if you want. What about A common approach is to define mathematics by its object of study, for example, as a study of abstract patterns or topics such as numbers, shapes, sets, and functions. We can also mention other items if you prefer.

• For the second point, my suggestion did not mention the term "proof", so I'm not sure how it can be a wrong vague definition of the concept of a proof

  our article currently says: For example, an area of study is often qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction. This would probably mean that all formal sciences are mathematics, including logic.

  The source for this claim has one paragraph dedicated specifically to the definition of mathematics. It starts with Mathematics is about the study of pattern and structure, and the logical analysis and calculation with patterns and structures. I don't see how that and the remaining part of that paragraph support our sentence. Phlsph7 (talk) 11:34, 8 October 2024 (UTC)

  The three above sources define mathematics by some (alleged) objects of study. The problem is that "study of pattern" is a highly controversial term when applied to mathematics in general so, it must not be used without making clear that many people do not agree with this term. The article begins with the long section § Areas of mathematics, which describes with some details (but not enough) the objects of study of mathematics. So, no further explanation is needed. However, if you think that more enlightening is needed, you may add an explanatory footnote such as "see Areas of mathematics for a description of the main objects of study of mathematics".

  About the second point: firstly, if the source does not support well the the sentence, one must first search a better source, before changing the sentence. About logic, I intend to add a footnote explaining that logic does not belong to mathematics, but mathematical logic became an area of mathematics more or less with the proof of Gödel's theorems. D.Lazard (talk) 16:57, 8 October 2024 (UTC)

  Why is the term "patterns" highly controversial? Alternatives from the quotes above would be "structures", "systems", and "mathematical entities". Do you consider them less controversial?

  Your footnote seems to imply that everything in logic associated with proving theorems belongs to mathematics. This is not generally accepted. The current reference supports neither the sentence nor the footnote. I added a "failed verification" tag to the source. Phlsph7 (talk) 15:40, 9 October 2024 (UTC)

• You removed a sentence about a set-theoretic definition, saying that the source does not support it. The source says Throughout the twentieth century many mathematicians went a step further in claiming that ultimately mathematics is set theory. This seems to support the sentence. Was there a problem with the specific formulation of our sentence? Phlsph7 (talk) 11:34, 8 October 2024 (UTC)

  This sentence seems a pure WP:original research of its author. During my career, I attended to many mathematical conferences, and never heard any mathematician saying something like that. To verify this assertion, one needs at least a source authored by a mathematician that says something like that. So, the source does not allows us to verify the assertion, and this goes against the Wikipedia policy of WP:verifiability. D.Lazard (talk) 17:14, 8 October 2024 (UTC)

  WP:ORIGINALRESEARCH applies to statements in Wikipedia articles, not to statements in reliable sources, meaning that we don't have to provide additional reliable sources for statements made in reliable sources. The source itself quotes several examples, right after the sentence I quoted. For another example, see Buium 2014 p. 67: "Mathematics is a particular theory $T_{set}$ (called set theory)". Phlsph7 (talk) 15:44, 9 October 2024 (UTC)

  Firstly, the claim "ultimately mathematics is set theory" describes mathematics by its object of study (set theory), not by its methods. So, its placement is controversial.

  But is is not my main objection. The main issue with this sentence is that it reports a WP:FRINGE theory that is not even supported by the provided sources:

  Buium begins its introduction with "In this course, we view mathematics as a chapter of logic". This means that the given definitions are related to this particular book, must not be viewed as a general definition of mathematics.

  Strauss states "many mathematicians went a step further in claiming that ultimately mathematics is set theory" and "Most of the time the general and concise statement simply is: 'mathematics is (axiomatic) set theory'." For supporting these assertions he provides two quotations asserting that set theory is commonly accepted as foundation of mathematics. So, its fringe theory is that mathematics is defined by its foundations. It is as reliable as an assertion such as "Molecular biology is chemestry, since chemestry is at the basis of molecular biology".

  Note also that the assertion "set theory is commonly accepted as foundation of mathematics" is developped in § Mathematical logic and set theory.

  So, I'll remove this sentence again. D.Lazard (talk) 10:38, 11 October 2024 (UTC)

I agree that defining mathematics as axiomatic set theory is fringe. Very few mathematicians would define mathematics this way (and mathematics was around for thousands of years more than set theory). Tito Omburo (talk) 10:58, 11 October 2024 (UTC)

Indeed. Paul August ☎ 15:34, 11 October 2024 (UTC)

Speaking as a set theorist by training, I agree with Tito and Paul. Set theory can encode virtually all of mathematics, but the claim that mathematics therefore simply is set theory doesn't stand up to the mildest critical thought. That said, there may be enough sources (even mathematical sources) that make this silly claim that we might have to represent it somehow. --Trovatore (talk) 19:42, 11 October 2024 (UTC)

One might as well define mathematics as category theory. Paul August ☎ 20:16, 11 October 2024 (UTC)

Mathematics is just the image of the Curry-Howard isomorphism. Tito Omburo (talk) 20:31, 11 October 2024 (UTC)

Definition source

I am trying to find the origin or the author of

"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"

...but my search was fruitless so far. Any ideas? 217.77.54.213 (talk) 17:06, 12 October 2024 (UTC)

Mathematics Article Problems.

I am not yet an auto-confirmed user( Hellow Hellow i am here 19:01, 23 October 2024 (UTC) )but I spotted some problems. It shows the types of numbers, but is missing real and complex numbers, as well as imaginary numbers.

The article does discuss each of these topics. Remsense ‥ 论 19:07, 23 October 2024 (UTC)

Oh, sorry. I must have missed them. Hellow Hellow i am here 14:00, 25 October 2024 (UTC)

Number theory topline definition

@D.Lazard: Daniel: 90% of number theory is about the properties of algebraic numbers, and saying "numbers" in general is very misleading, since most study of real numbers occurs in analysis. "Whole numbers and fractions" are the main interest of number theory, and algebraic numbers appear as generalizations of them.

I wouldn't use "algebraic number" in the lead, but do you really think "whole number" and "fraction" are too technical, that they will confuse readers who are curious about mathematics, but do not know what whole numbers are?

Magyar25 (talk) 21:50, 5 November 2024 (UTC)

My take: If we need to gloss the term "number theory" at all, I would prefer "the theory of the natural numbers", which is accurate in spite of the fact that arithmeticians consider other sorts of numbers. Rationals are ratios of natural numbers. Algebraic numbers are algebraic over the natural numbers. Et cetera.

As to the term whole number, my preference would be that we should never use (as opposed to mention) it at all, especially in a math article. Mathematicians essentially never use the term. --Trovatore (talk) 22:18, 5 November 2024 (UTC)

The lead of the article Mathematics is not the place for an accurate definition of number theory. This sentence is here to explain what mathematics is about, and the way readers understand "number" does not really matter. Moreover, restricting number theory to some sort of numbers would go against a common consensus: the facts "every real number has an infinite decimal expansion" and "π is not an algebraic number" are properties of real numbers that everyone considers as belonging to number theory. If there is something that is misleading in this sentence it is the definition of analysis as "the study of continuous changes", since the phrase in rarely used in analysis, except for explaining one of several motivations of analysis. Nevertheless, after many discussions on this talk page, nobody has found a better phrase. D.Lazard (talk) 01:49, 6 November 2024 (UTC)

Sorry, I completely disagree with the claim the facts "every real number has an infinite decimal expansion" and "π is not an algebraic number" are properties of real numbers that everyone considers as belonging to number theory. I think that's just absolutely wrong. The first one is definitely not part of number theory. The one about π is a little closer but I still think it's unlikely to be called number theory. --Trovatore (talk) 19:01, 6 November 2024 (UTC)

Transcendental number theory and diophantine approximation are both part of number theory, fwiw. Tito Omburo (talk) 19:26, 6 November 2024 (UTC)

But π is not. --Trovatore (talk) 19:31, 6 November 2024 (UTC)

I mean, of course you use π in number theory, to give approximations and so forth. But you don't really study π. --Trovatore (talk) 19:32, 6 November 2024 (UTC)

The Borweins would disagree. Tito Omburo (talk) 19:34, 6 November 2024 (UTC)

Ref? --Trovatore (talk) 19:42, 6 November 2024 (UTC)

Pi and the AGM: a study in analytic number theory and computational complexity, Jonathan and Peter Borwein, 1987. Tito Omburo (talk) 19:45, 6 November 2024 (UTC)

Well. Categorizing branches is always fraught. I did say π was "a little closer". My general take is that nothing that involves the completed infinite is part of the subject matter of number theory, though it might be part of the methods.

Anyway the best solution might be just not to gloss "number theory" at all in the lead. I don't see that a gloss saying it's the "theory of numbers" adds anything at all; it just sounds like the natural meaning of the words. If we are to have a gloss I continue to think the "theory of natural numbers" is better wording. --Trovatore (talk) 20:12, 6 November 2024 (UTC)

p-adic numbers and adeles are unambiguously a part of number theory, and certainly involve completion. Tito Omburo (talk) 20:39, 6 November 2024 (UTC)

OK. I was never a number theorist, and maybe the field has moved on since I took my one class in it as an undergrad (we used Apostol's Introduction to Analytic Number Theory). I still don't find the current gloss useful. (Note that p-adic numbers and adele rings are not likely to be evoked by the phrase "the study of numbers".) Do you agree with just removing the gloss? --Trovatore (talk) 21:29, 6 November 2024 (UTC)

(edit conflict)The gloss for number theory is here for the balance of the sentence. If you can propose a better gloss, please do.

The first sentence of Transcendental number theory is "Transcendental number theory is a branch of number theory that investigates transcendental numbers". If you read the article, you will learn that a major result of this branch of number theory is Gelfond–Schneider theorem, which implies that ⁠$e^{\pi }$⁠ is trancendental, and that a major open question is whether ⁠$e+\pi$⁠ is transcendental.

About "completed infinity" in number theory: I never saw anybody writing that Fermat's Last Theorem and Wiles's proof of Fermat's Last Theorem do not belong to number theory, although the proof makes a fundamental implicit use of the axiom of infinity, and even (in the original proof) of a much stronger axiom. So, your opinion on the subject matter of number theory goes against a consensus of number theorists. D.Lazard (talk) 21:35, 6 November 2024 (UTC)

On the other hand, the first sentence of number theory says that it's about natural numbers and arithmetic functions. So there's a bit of a conflict there. Myself, I would not have counted transcendental number theory as part of number theory, but I don't know how workers in the field think about it.

My proposal is simply to have no gloss at all.

As to your second paragraph, you're talking about the proofs, not the subject matter. Fermat's last theorem is about natural numbers. Its proof uses completed infinite objects, but that is not what it is about. --Trovatore (talk) 22:30, 6 November 2024 (UTC)

Small change suggestion

Change the begining of the article to be: Mathematics is a branch of knowledge and a field of study... Linking knowledge to not break the philosphy game Moondarkside01 (talk) 16:46, 6 November 2024 (UTC)

Thanks for your suggestion, but maintaining the philosophy game is not one of our goals. (If it were, then we would maintain the philosophy game, and it would be unsurprising that the philosophy game held, and then there would be no point to the philosophy game). Mgnbar (talk) 18:32, 6 November 2024 (UTC)