Talk:Mathematics

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 [1] 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)

Semi-protected edit request on 15 December 2024

Hello, Wikipedia editors. I would like to request access to edit this article because of my interest in mathematics and my desire to contribute constructively to its content. I am passionate about the subject and eager to help improve the quality of this page, ensuring it remains accurate and informative. Slavuska Shabliy (talk) 17:42, 15 December 2024 (UTC)

Pages get "semi-protected" when they see repeated vandalism, edit warring, or other disruption from IP editors and/or new accounts. On a semi-protected page, brand new accounts can only make requests on the talk page for someone else to implement. But if you wait a few days and make several edits (counted across all of Wikipedia) your account will become "confirmed" and you can then make edits to semi-protected pages, including this one. The goal of the "semi-protection" is to solve the steepest part of the maintenance burden while not excessively restricting people from editing.

In the mean time (i.e. as an "unconfirmed" editor), if you have a specific change that you want to make here, you can make a specific request and someone can make that change to the article. If you want to make a more extensive change you can work on your desired text somewhere else, for instance in your user namespace, at a page like User:Slavuska Shabliy/mathematics, then come back to this talk page when you are ready for someone to apply those.. –jacobolus (t) 18:51, 15 December 2024 (UTC)

High compliments

The first paragraph of the lede here is excellent. Extreme kudos to everyone who contributed. HiDrNick! 22:51, 22 December 2024 (UTC)