Talk:Pure mathematics

Pure mathematics is not the opposite of applied math. Many results in "pure math" manage to find applications in other fields. Besides, results in "pure math" are often applied in other math fields and many fields in math are motivated by other fields. Critical

Yo
I recall that some mathematician/group or school of mathematicians was skeptical of the value of proof at all and instead desired to simply experiment with methods and find the best ones for physical modeling (and by this I do not mean the 18th century). If anybody knows anything about him/her/them/it, please add it before or after the entry on Hardy under "Purism" (which can be molded to fit the knowledge of those other persons). Diocles

About Users Critical and CStar
For the record, the user Critical ( talk,  contributions), who slapped the "disputed NPoV" sticker on this page, has made his or her first edits tonight (or today) and within less than two hours has attacked eight articles for PoV, including (ironically given the CStar example given on the Logical fallacy talk page), Physical law. These were the only "edits" (plus weak justifications on talk pages in the same vein as this one). I don't think the PoV claim has merit. We may ask if this series of attacks is to be taken seriously.

For the following reasons I am thinking that these pages has been the victim of a tiresome semi-sophisticated troll and the PoV sticker should be removed sooner rather than later, if not immediately. We may note that CStar ( talk,  contributions) after making edits, paused during the period user Critical made edits, and then CStar took up responding to these edits after the series of user Critical edits ends, as if there is only one user involved, and the user logged out, changed cookies and logged back in. Further, user CStar left a note on Charles Matthew's talk page,  Chalst's talk page, and   Angela's talk page pointing to a supposed PoV accusation placed on the Logical argument page, when in fact no such sticker has been placed. Perhaps the irony regarding the Physical law page is not so ironic. Hu 05:18, 2004 Dec 1 (UTC)


 * I have responded to this on the logical fallacy talk page, as well as on the pages of the above mentioned users. It does appear that these pages were as Hu suggests the victim of a tiresome semi-sophisticated troll.  But I wasn't the perpetrator.  This suggestion appears to have been an honest mistake, I consider the matter closed, and it appears that Hu does as well. CSTAR 01:36, 2 Dec 2004 (UTC)


 * Just because he hasn't been a registered user for very long doesn't mean he has no right to an opinion on the page. If you want to dispute his statement then do so, do not belittle his merit.
 * Having said that though, the article doesn't state that pure is the opposite of applied. And if it did, that is not POV, rather a simplification to the point of fallacy. Critical: Just change the wording next time.

A mathematician is walking through a carpark, late at night. Halfway to his car, he drops his keys. If he was an applied mathematician, he would drop to his knees and methodically search around his feet. If he was a pure mathematician, he would realise the probability of him finding his keys is greater in the lighted region 500m away, so he heads in that direction.

18/19th century
The introduction says the origin is in the 18th century, yet the history farther down begins at the 19th century. Which is it?

subfields of pure mathematics
The article says about number theory "It is perhaps the most accessible discipline in pure mathematics for the general public."

This is just wrong. If you are talking about statements of theorems, yes there are some hard theorems of number theory with elementary statements. But that is true in other subjects -- the isoperimetric inequality, the Poincare conjecture, can be stated in a way anyone can understand. But the real substance of the subject cannot be readily understood by the public ... things like factorization of ideals in a ring of algebraic integers, on to the Langlands program, or the Riemann hypothesis. 86.128.141.126 11:24, 4 March 2007 (UTC)

subjective?
The opening stagement "It is distinguished by its rigour, abstraction and beauty." seems to be a little subjective, what do people think?

Definitely. This should be removed; there's no way to prove it's something that's a matter of opinion, and if you wanted to prove that most people thought it beautiful, you'd need a citation. Verisimilarity (talk) 03:53, 1 December 2009 (UTC)

Mathematical logic
Perhaps mathematical logic should be listed as a subfield? --Quux0r 07:22, 16 April 2007 (UTC)

You also do probablity in Mathamatics its not only a little kid thing. Well there much other things that you can do with math that what you might of used in elementary school. SO GET USED TO IT. CUZ IT IS NOT FUNNY.

He's crackin' —Preceding unsigned comment added by 92.112.13.41 (talk) 06:07, 18 January 2009 (UTC)

Bridging Pure Mathematics and Philosophy of Mind/Body (x_0)

 * Suppose {1 | continuum(x)}
 * {o | o <- object}
 * { x <- o }
 * Run forever

--Vektor-k (talk) 18:43, 6 August 2013 (UTC)

"intrinsic to nature"
(said about automorphic forms) What does this mean, if anything?

And if it does mean something (e.g. implying some relevance to "nature"), how does the rest of the sentence make any sense? It seems like this needs to be removed or clarified. 211.31.63.48 (talk) 09:10, 25 June 2015 (UTC)

Euclidean geometry is the the most accessible pure math
We live in a universe of curved Einsteinian space-time. Neither Euclidean geometry nor Newtonian mechanics actually exist anyplace in our universe. That makes them pure math by the definition in this article. Before you get to the example of the Banach–Tarski paradox given in the article, it's fair to point out that there are no real spheres, they are paradoxical, but if you had a paradoxical sphere then you could see the increased paradoxicality of the Banach-Tarski proof 74.65.224.183 (talk) 17:50, 26 October 2017 (UTC)

Edits of the lead
I have restored the last version of the lead before the recent edit war. Please do not edit the lead again without getting first a WP:consensus in this talk page. Without consensus, every edit will be reverted, and if a reverted edit is restored, administrators intervention will be required for either protecting the article and/or blocking edit-warriors.

My opinion is that the version that I have restored is not good, but the reverted edits make it worse. I'll discuss the different points in separate threads for making easier the discussions toward a consensus. D.Lazard (talk) 15:11, 24 November 2018 (UTC)


 * I agree with all of the above.Paul August &#9742; 15:43, 24 November 2018 (UTC)

Fundamental mathematics
One of the edits in the recent edit-war was the introduction of "fundamental mathematics" as a synonymous for "pure mathematics". This is not true. The reference that has been added for supporting the claim is non-reliable source, as it is simply a course title, and does not establish any relationship between the two phrases. So the phrase "or fundamental mathematics" is original research (Wikipedia meaning), and its addition breaks Wikipedia policy WP:NOR.

The term "fundamental mathematics" is also confusing, because it seems to refer to foundations of mathematics, which is a completely different subject. D.Lazard (talk) 15:30, 24 November 2018 (UTC)


 * I removed "fundamental mathematics" as synonymous for "pure mathematics" (but was reverted), since I've never heard of that term being used this way. So even if it is used this way somewhere, I doubt that this use is common enough to be included here. Paul August &#9742; 15:51, 24 November 2018 (UTC)


 * I've never heard "fundamental mathematics" used as a synonym for "pure mathematics", either. To me, "fundamental mathematics" might refer to foundational studies, or to the basic contents of a primary school curriculum, depending on whether a philosopher or a schoolteacher is speaking. I agree: it doesn't belong here. XOR&#39;easter (talk) 15:58, 24 November 2018 (UTC)

Mathematics is not necessarily applied mathematics
Edit warriors removed "is not" from this sentence. The result is not grammatical. However the first sentence of this paragraph may be confusing, although the remainder of the paragraph is essentially correct. A more correct first sentence would be

In pure mathematics concepts are defined and studied independently of any application to the real word. However, these concepts may often be applied to the real world because, either they were introduced for modeling the real world (for example numbers and geometry), or after having been introduced independently of any application, they become widely used outside mathematics. An example is elliptic curves over a finite field, which are used for securing internet connexions (HTTPS protocol).

However, this is a definition of pure mathematics, which is better than the one that is given in the first paragraph. In fact it uses the concept of "entirely abstract concept", which is an oxymoron, as implying the existence of "partially abstract concepts".

So the lead deserves to be completely rewritten. I'll propose later a new version of the lead. D.Lazard (talk) 16:27, 24 November 2018 (UTC)
 * I agree that the current lead is terrible, even if better than than what it replaced. I am looking forward to D. Lazard's new version and if it is in line with the above snippet I will heartily approve it. A few suggestions. I think that "elliptic curves over a finite field" may be a bit too esoteric for the lead and doesn't sit well being juxtaposed with the other examples ("numbers" and "geometry"). I would suggest that it be replaced by number theory or even more pointedly "factoring large numbers". This would also entail replacing "numbers" in the earlier example for fear of confusion. Maybe with something like "networks" or probability. Also, I have a friend (a pure mathematician working in an applied mathematics department) who was fond of making the distinction between applicable mathematics and applied mathematics, the latter being a field of study and the former referring to the mathematics that possibly could be used in some application. This may be a useful turn of phrase in this article. --Bill Cherowitzo (talk) 20:22, 24 November 2018 (UTC)

Draft for a new lead
I suggest the following for the lead. Feel free to improve the grammar and the style. However, this is a major modification of the lead. So we must base discussion on a rather stable version. Therefore I recommend that non-minor modifications should be discussed separately, before being incorporated in this draft. Also I have created subsections for distinguishing the discussions for improving the draft from the discussion on the choice between it and the present lead. These sections appear after a section (still not written) in which I'll explain and comment my choices. D.Lazard (talk) 11:31, 25 November 2018 (UTC)


 * The sentence "In particular, it is not uncommon that some members of a department of applied mathematics qualify themselves as pure mathematicians." is honestly bizarre and I removed it from the lead. It's not about what a mathematician identifies themselves as, it's about the particular fields, concepts, or ideas that are grouped as pure or applied mathematics.
 * Sidenote: I'm an undergraduate mathematics major. JohnAdams1800 (talk) 03:31, 4 January 2024 (UTC)

Original synthesis?
Both the draft and the present lead are WP:original synthesis. This seems unavoidable in such an article. In fact it always difficult to define an area of mathematics (and even mathematics them selves). But here, we are faced to the lack of consensus among mathematicians, not only on what should be understood as "pure mathematics", but also whether it is an area of mathematics. As we must keep this article, and it seems impossible to avoid original synthesis, we must find a consensus among editors, and then apply WP:IAR

Nevertheless, I have tried to keep a minima my personal opinions, and to support them with facts that cannot be disputed.

Here are two personal anecdotes illustrating the difficulties.

Around 1968 (I do not remember whether it was before of after), I participated to a working group, leaded by Pierre Samuel about Mathematics and society. At one of the sessions we discussed on what is applied mathematics, and, by contrast, what is pure mathematics. We has a long discussion, where it appeared that measure theory was a part of probability theory, and Sobolev spaces space belonged to numerical analysis. As, at least in France, probabilities and numerical analysis were considered as applied mathematics, and measure theory and Sobolev spaces are clearly pure mathematics, we had long discussion, which was concluded by one of us (a probabilist, I believe), who said that the only workable definition of applied mathematics is to define them as the mathematics that Bourbaki don't know of.

At the same period, Roger Godement claimed that he chosen to work on modular functions because they cannot be used for military purpose. Alas, a few years later, they become used in nuclear studies.

Body of the article
If my draft is accepted, all the body of the article must be rewritten for supporting it.

In particular, sections "Generality and abstraction" and "Subfields" should probably be suppressed as unsourced WP:OR.

Section "History" should be adapted for given more details on the "mathematical revolution" of the end of 19th century that I have sketched in the draft, and its influence on the view

The section "Purism" is mainly focused on Hardy's opinion, which is 78 years old. Even at that time, this opinion was not a consensus among mathematicians. It should thus be adapted for reflecting other opinions and giving more modern point of views. Probably, one could find one in Cedric Villani writings, as the results (of pure mathematics) for which he got the Fields Medal were related to some problems of physics, and this relation was a part of his inspiration. His book about his discovery is probably a source of quotations on the relationship between pure mathematics and applications. D.Lazard (talk) 17:31, 25 November 2018 (UTC)

Discussion for improving the daft
I have qualms about the proposed definition of pure mathematics. More later, maybe.... Michael Hardy (talk) 17:57, 25 November 2018 (UTC)
 * Overall I am quite happy with this draft. I have made some minor grammatical changes directly in the draft and I am sure that there will be some tweaks suggested by various editors (myself included), but I think the tone is correct and it says what should be said about pure mathematics. As it is the lead, I am not very concerned about the OR issue as long as the points are reliably justified in the body of the article. As to my tweaks, there are only two. In the first paragraph it is stated that pure mathematicians do not care about applications. I am a pure mathematician and I do not care about applications, but that statement seems a bit harsh and doesn't really capture the situation. The fact is that I do care about applications, but they do not dictate what or why I study what I do. Applications are useful to communicate the essence of what I study to the non-specialists that I interact with, and if they don't exist I am forced to use some far-fetched analogies to make this point. I would be much happier if the statement was that pure mathematicians are not motivated by applications. My second tweak is very technical. In the reference to Einstein's Theory of Relativity a link is made to Non-Euclidean geometry and I think the connection should be made to "a geometry that is not Euclidean" since traditionally the term "non-Eulidean" is restricted to either elliptic or hyperbolic geometry and Minkowski space doesn't appear on that page. --Bill Cherowitzo (talk) 22:06, 25 November 2018 (UTC)


 * The second sentence piles up a lot of negatives. This does not mean that these concepts do not originate in the real world nor that the obtained results cannot be useful for modeling the real world, but rather that the pure mathematicians do not care about applications. Can we make the same point more directly? How about this: These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but the pure mathematicians are not primarily motivated by such applications. XOR&#39;easter (talk) 00:17, 26 November 2018 (UTC)


 * Additionally, mathematics for itself does not sound quite idiomatic. How about mathematics for its own sake? XOR&#39;easter (talk) 22:49, 26 November 2018 (UTC)


 * I like both of these suggestions. Pushing the logic behind the first suggestion, should we perhaps make a statement about what does motivate a pure mathematician? I realize that this may open a can of worms that we do not wish to open.--Bill Cherowitzo (talk) 23:29, 26 November 2018 (UTC)

I would really like to be able to cite some reliable sources, for a definition. Perhaps we can make use of Browder's definition: "that part of mathematical activity that is done without explicit or immediate consideration of direct application"? Also something perhaps from Hibert's (Hardy's of course), A Mathematician's Apology? I think this quote must be from there: "We are often told that pure and applied mathematics are hostile to each other. This is not true. Pure and applied mathematics are not hostile to each other. Pure and applied mathematics have never been hostile to each other. Pure and applied mathematics cannot be hostile to each other because, in fact, there is absolutely nothing in common between them."Paul August &#9742; 00:22, 27 November 2018 (UTC)
 * I believe that should be G.H. Hardy rather than Hilbert. The sentiment of the quote is clearly in line with Hardy, but I can not find any mention of hostility in A Mathematician's Apology. On the other hand, section 22 starts with

"It is quite natural to suppose that there is a great difference in utility between 'pure' and 'applied' mathematics. This is a delusion: there is a sharp distinction between the two kinds of mathematics, which I will explain in a moment, but this hardly affects their utility."


 * His explanation of the difference goes on for two sections and would be hard to distill into a pithy quote.--Bill Cherowitzo (talk) 19:09, 27 November 2018 (UTC)
 * Oops, yes of course Hardy;-) Paul August &#9742; 19:57, 27 November 2018 (UTC)

I think the level of the proposed lead is too high. We should aim the lead, and a good deal of the start of the article, so it can be understood by people who might want to know about the topic. I would put in there principally high school students who might be thinking of what they want to do. There is no point in aiming it at people who already know about Hilbert's basis theorem. Perhaps the picture of one ball becoming two is okay as an object of wonder in how pure mathematics has no relationship to the real world, but I think perhaps that is just a bit too ivory tower right at the beginning of the article. Dmcq (talk) 13:37, 5 December 2018 (UTC)

Discussion for choosing the version
Incorporating the comments above, I get the following:

How unhappy are we with this? XOR&#39;easter (talk) 18:32, 5 December 2018 (UTC)
 * Fine for me. D.Lazard (talk) 18:52, 5 December 2018 (UTC)


 * Certainly better. The distinction started with the Greeks like the article says and I'd say an earlier applied mathematics use of pure mathematics was when Newton used the theory of conics from Apollonius to show how gravitation produced elliptical orbits. Dmcq (talk) 23:29, 5 December 2018 (UTC)


 * I think it would be desirable to say what does motivate the pure mathematician in addition to saying what doesn't. Perhaps something like: "... but the pure mathematicians are not primarily motivated by such applications, but rather by the intellectual challenge and esthetic beauty of working out the logical consequences of basic principles."--agr (talk) 03:14, 6 December 2018 (UTC)


 * I think that referring to motivation is at best reporting of retold hearsay (relata ... relata refero). There is no chance for evidenced motivation. That pure mathematicians often tell tales about the challenge and beauty of their profession might have a foundation in their abundant social competence. Sometimes even I (no pure mathematician) claim to see beauty in proofs from the book. Purgy (talk) 07:23, 6 December 2018 (UTC)

Revised accordingly:

XOR&#39;easter (talk) 03:48, 6 December 2018 (UTC)

engels quote
just in the off-chance and  engage in a dispute over the quote, i want to say that i like it.

and while applied mathematics isn't explicitly mentioned by engels, he does specifically state 'pure mathematics'. the question i guess would be: was there any distinction at that time?

since engels was born in 1820 it would be fair to say we had a few titans who were active in this time period, Lords rayleigh and kelvin immediately coming to mind.

what do others think? i will admit i'm not a huge hardy fan, but that in no way affects my opinion on the relevance of the engels quote.

what do you think dr ?

maybe some context around the quote would help, but it seems to stand well on its own. 198.53.108.48 (talk) 20:13, 12 June 2021 (UTC)


 * Keep - as i said before: i like it. it sums up my opinion on what pure mathematics is as well. it may be 'theoretical', but the foundations are certainly grounded in reality. 198.53.108.48 (talk) 20:13, 12 June 2021 (UTC)


 * Keep as it provides a good summary of that philosophical position. However, the introduction to the quote needs to be written in proper English. "Friedrich Engels argued in Anti-Dühring that pure mathematics owe to reality for its existence" is very awkward. I suggest something like "Friedrich Engels argued in Anti-Dühring that the concepts of pure mathematics are based on physical reality". Burrobert (talk) 01:30, 13 June 2021 (UTC)
 * Keep—agree with above: the quote is fine, but the English is stilted (presumably in translation). How about: '...that pure mathematics owes its existence to reality, as "it is not true that...'? [NB I think the singular use of 'its' here correctly reflects 'pure mathematics' as being a 'non-count' noun, i.e. a singular concept, and as such takes 'owes' (3rd person singular) rather than 'owe' (3rd person plural!)]Kitb (talk) 10:24, 13 June 2021 (UTC)


 * Keep - Agree with the comments above and also think it would improve the article to paraphrase the quote to something less awkward and more easily understandable. I like what Burrobert suggested: "Friedrich Engels argued in Anti-Dühring that the concepts of pure mathematics are based on physical reality." TrueQuantum (talk) 18:18, 13 June 2021 (UTC)
 * comment I think it's better to stand in the same line of sight as Hardy or ride on Hardy's shoulders. To explain the positional relationship between the two lines, I think it is necessary to look beyond the horizon from a high position. I think the signature and ping are out of sync. So I didn't get any notification either. So --SilverMatsu (talk) 06:00, 14 June 2021 (UTC)
 * As you may already know, I add because I checked the discussion on the talk page and the article newer 50.--SilverMatsu (talk) 08:52, 14 June 2021 (UTC)
 * I don't see why my opinion on this burning issue is so central to have been pinged twice. I'm not familiar with the quote, and don't necessarily agree with it, but that's no matter. I don't see a good reason to exclude it. I am a little concerned, however, with its placement, as if we are saying with Wikipedia's voice that Engels is a complete answer to Hardy and had the final and correct word on this subject. Is there some way to make it more clear that it is merely one more opinion among many? Surely we have a significant body of more recent work in the philosophy of mathematics to draw on. —David Eppstein (talk) 06:12, 14 June 2021 (UTC)
 * Perhaps like a critical reception section of a media article? "Hardy argued X, Engels disagreed, arguing Y." Would just require a change of wording of the quote.Tazerenix (talk) 08:30, 14 June 2021 (UTC)
 * Thank you for your reply. I also think there is more new reference. Maybe move to the 19th century section? And I will act carefully so as not to get burned. I simply pinged everyone who seemed already pinged in this discussion because I didn't get the notification, but I'm sorry if the notification was duplicated.--SilverMatsu (talk) 08:43, 14 June 2021 (UTC)
 * Thank you for your reply. It's to be fair because I think it accurately represents the content added this time. But considering this addition as Hardy's counterargument, the citation is weak. In the current reference, I don't think we can tell that Engels' theory argued against Hardy after looking at modern mathematical theory (In the era when Hardy was alive), just to see that Engels did say so.--SilverMatsu (talk) 14:25, 15 June 2021 (UTC)


 * Keep - However, one should verify that the quote is a correct translation of the German original. Also, if a paraphrasing is added, it must not change Engels' thought: Engels used "reality", and above paraphrase uses "physical reality", which is definitively not the same, and Engels did know well the difference. D.Lazard (talk) 15:14, 15 June 2021 (UTC)
 * Thank you for your reply. Regarding the correctness of the translation, I found the following preprint.--SilverMatsu (talk) 15:47, 15 June 2021 (UTC)
 * Comment I don't see a reason to exclude the Engels quotes, though I can see an argument for paraphrasing them. I trimmed the lead-in to the quoted passages, as that was more awkward English than the translated quotations themselves. Overall, that section looks like a collection of bits and pieces that people happened to like, rather than even an attempt at a systematic overview. In other words, it's Wikipedia prose, hooray. XOR&#39;easter (talk) 17:07, 15 June 2021 (UTC)
 * Thank you for your reply. Apparently, misunderstood the big picture of the section, and from that point of view, it seems to be keep.--SilverMatsu (talk) 02:57, 16 June 2021 (UTC)

Maybe rewrite the article with school(s) of thoughts in mind ??
I know that this is none of my business, cause I do physics, but I noticed the addition of Essay-like tag, it was added and removed multiple times anonymously. I do agree on the tag actually.

I do not agree on the proposed new lead by the way, few comments on the lead:
 * Every single existing mathematical concept has a nearby or faraway application, or concrete intuition at least, outside mathematics (at least in physics).
 * "but the pure mathematicians are not primarily motivated by such applications. Instead, the appeal lies in is attributed to the intellectual challenge and esthetic beauty of working out the logical consequences of basic principles.": essay style - sales pitch - beauty is not objective - working out logical consequences can be done by a computer. Seems a religion, where the pure mathematicians are the priests. See "Athya is math invented or discovered ?". Much of it is subjective.
 * Greeks had not whatsoever clue of pure math, they mostly do geometry and there is not much distinction between geometry and physics either. I believe this a historical gross invention, you shall prove this based on original greek sources only actually.
 * Non euclidean geometry was deeply rooted in geometry if you look at works of Gauss, Lobachesky and Poincare, and in general if you use embeddings.
 * Cantor was mangling with paradoxes, infinities and fractals in the scope of infinite sets (which are all rooted in geometry).
 * Nowadays Mathematical rigor is questionable outside automatic theorem proving
 * Again on rigor to be relevant nowadays one shall include type theory, higher order categories, isomorphisms between hierarchies of types, hott etc. One may argue that category theory in that sense is trendy but not rigorous.
 * On rigor: maybe is worth mentioning the bankrupcy of the algebraic geometry school of Italy, or author like elie Cartan which often do not separate out or formalize proofs
 * On rigor: two words on education are worth while, rigor is subjective, every time there is novelty one way or another there is less rigor.
 * The lead is too long, it shall be 5 lines max. This proves the kitchen sink approach of the article.
 * Philosophical approach: there is a need of full departments called pure mathematics to be funded. This is a biased opinion.
 * Philosophy of science and mathematics is not objective, it is relative and it changes over time.
 * Last Abel prize: Michel Talagrand was also a physicist, isn't this debate pure vs applied obsolete ?

If you want to leave in the article each of these statements, they deserve one citation for each, and eventually also counter arguments somewhere else in the article. I would argue that the best way is to rewrite the article in terms of school(s) of thought.

In regards to the historical section
 * Most of the examples in the greek section are related to geometry, maybe useless geometry at that time, but pretty far from the definitions of "pure/abstract" of nowadays.
 * The number theory example comes from Plato. Plato was selling his idea of the cave, so is a source with a distinct "pure" and biased opinion. Aristoteles for example would go the opposite way.
 * Number theory can manage to be very ugly if you look at the amount of special cases. Number theory has applications also, there is no boundary between pure and applied again.

In regards to the generalization/abstraction section:


 * Same essay-style/ sales pitch style
 * Most arguments are repetitive
 * Short vs long proof again subjective
 * Each of the 400 proofs of pythagora's theorem may give you a different insight
 * You can argue that all abstractions are just type theory alike constructions, a ring is a type and so forth, they shall belong to math on the same footing of the rest (they are not special).
 * Again following Arnold Abstract algebra does not exist
 * On abstraction: "two words on the constructivist approach and the example as the testimony" are worth it
 * On abstraction: "proof by negation" does it belong to the axioms of mathematics or not ? do we always need a testimony ?
 * "A steep rise in abstraction was seen mid 20th century". This was a French prob to fix the Italian bankrupcy of algebraic geometry. Is this hiding bourbaki under the carpet ? It does not belong to Russian math for quite long time really. Again subjective. If I like graph theory I may not have such a problem.

I see a kinda funny debate here, it remembers these bourbaki style discussions + restart from scratch approach they used to do with books. The debate around the engels quote also shows the same trouble with this essay-style article, again either embrace in full this philosophy of math approach showing pros / cons or don't include it at all. The discussion may become obsolete by the time is complete.

I am actually writing this comment cause this morning I found this wonderful piece from Arnold which states mathematics is part of physics. Arnold also states that limiting mathematics to this definition + axioms + theorem + proof is not enough. You cannot promote axioms and definitions out of the hat. A small variation in the definitions may create a huge discrepancy in the consequences (i.e. this is actual chaotic behaviour - where is the rigor ?). The scope of mathematics is much wider, it includes this finding the best approach to definitions and axioms, it includes creating general tools (e.g. Gelfand), it includes experiments, induction and so forth and in some shape of way mathematics is at least part of natural sciences.

There was another quote from Von Neumann stating that mathematics without physics is at best baroque, plus you can mention also Gelfand, Kolmogorov, Klein, Poincare (who considered set theory a disease) and a few others.

Maybe a "School of thoughts" style approach in both the lead and the article, with a good amount of citations, would help a bit to make the article looking more objective, unbiased and professional. Flyredeagle (talk) — Preceding undated comment added 17:34, 15 April 2024 (UTC)