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String theory (and related areas of purely theoretical quantum gravity, like loop quantum gravity) has a unique position within the academic physics community. Many academic physicists don't really consider string theory to be physics at all (due to its disconnect with any experimental evidence) and think that it should be more properly considered a branch of philosophy or mathematics. Other academics strongly hold the opposite opinion.

Are there any branches of academic mathematics for which there is a similar dispute as to whether those branches constitute math at all, as opposed to philosophy or some other field?

Let me clarify the scope of this question:

  1. It excludes the question of whether it's useful to separate pure math from applied math. Nor does it include the question of whether certain mathematical topics in applied math are so closely associated with an application field (e.g. computational biology) that they should be grouped within that topic (e.g. biology) rather than within mathematics. Instead, I'm focused on the boundary between pure math and (e.g.) philosophy.
  2. It also excludes the question of whether any specific mathematical axioms (e.g. the axiom of choice) "should" be included in the set of axioms that are typically assumed, or the question of which is the "best" mathematical axiom system.
  3. The actual question of whether string theory should be considered a branch of physics is out of scope. Similarly, the actual question of whether any given academic field of math should count as math is out of scope. Instead, I'm asking about whether there's consensus within the academic community that the field should count as math. This is a sociological question, which, while perhaps somewhat subjective regarding the term "consensus", is ultimately a factual question.
tparker
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    There is the question "do the natural numbers include zero, or not?" which some people think is a mathematical question with a mathematical answer. (They are wrong.) – Zach Teitler May 22 '23 at 01:17
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    As Timothy Chow says in his answer, there are fields such as combinatorics and graph theory that at least some mathematicians did not consider to be serious mathematics. I think, however, that these prejudices have disappeared. Theoretical computer science is also an area that many mathematicians do not consider to be real mathematics, but this prejudice has also diminished significantly. In every area of math, there are certain directions that are considered uninteresting by many working in the field. – Deane Yang May 22 '23 at 03:57
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    @DeaneYang I'm not asking whether that are branches of math (as defined by at least some mathematicians) that some (other) mathematicians don't consider to be "serious" math or "interesting" math. I'm asking about branches of math that some mathematicians don't consider to be math at all. I do not believe that combinatorics, graph theory, or even theoretical CS fall into that category. – tparker May 22 '23 at 04:10
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    @tparker, I think your question makes sense only if we can agree on what mathematics is. My minimalist definition of mathematics is "deriving new knowledge from old using deductive logic and abstraction". For me, any area that relies primarily on logic and abstraction to derive new knowledge is a candidate for being an area of mathematics. Do you have something like this in mind? – Deane Yang May 22 '23 at 04:33
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    @ZachTeitler: Hmm, I'm wondering whether I understand your comment correctly. Quite surely every mathematician will agree that it's a matter of definition whether the natural numbers include zero. Could you elaborate what you mean by "some people think [this] is a mathematical question with a mathematical answer"? – Jochen Glueck May 22 '23 at 07:26
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    I’m voting to close this question because (in my experience), those who mention areas that are truly controversial will get downvoted, so I do not see the point in this question. – Joseph Van Name May 22 '23 at 09:17
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    Is fuzzy mathematics a part of mathematics? – Kapil May 22 '23 at 14:33
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    As a physicist I would like to point out that the opinion "string theory is not physics" is mostly held by non-physicists, and any respectable physicist would find that statement quite laughable. – AccidentalFourierTransform May 23 '23 at 18:31
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    @JochenGlueck May I cite the discussion on this question. Anyway, I was not being entirely serious, forgive me for my lame attempt at humor. – Zach Teitler May 23 '23 at 20:09
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    Isn’t string theory an answer to your question by your own description and criteria? A considerable portion of academics (probably including some mathematicians and even string theorists) consider it mathematics and it’s not applied mathematics on account of lacking application. And of course, all of this is certainly controversial. – Wrzlprmft May 23 '23 at 20:18
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    @AccidentalFourierTransform I can state with confidence from firsthand experience that your claim is incorrect. I personally know many, many professional (and respectable) physicists who believe that string theory is not physics. Indeed, I even know several professional string theorists who believe that string theory is not physics. – tparker May 24 '23 at 00:34
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    @AccidentalFourierTransform If anything (and here I admit that this is a bit more subjective), I think that the zeitgeist within the professional physics community has swung against your claim over the past couple decades. 20 years ago, I think the proposition that "string theory is not physics" was a small minority position, if not fringe. Today, I think that this proposition is generally considered very reasonable, even if much (or even most) of the physics community disagrees with it. But anyway, this is all somewhat off-topic. – tparker May 24 '23 at 00:37
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    To enter philosophy into a discussion about mathematics or science is to me a total waste of time. – Wlod AA May 24 '23 at 00:39
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    @JosephVanName If I understand your reasoning correctly (please correct me if not), you are saying that this question is on-topic, but many Math Overflow voters will inappropriately discourage legitimate answers, so it is unlikely to lead to any correct answers being posted. That may well be an accurate prediction, but I don't think it's a legitimate reason to vote to close the question according to Math Overflow's guidelines. Indeed, it is preemptively surrendering to an anticipated heckler's veto. – tparker May 24 '23 at 00:41
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    @JosephVanName Moreover, the fact that the question has been made community wiki means that downvotes will not affect answer posters' reputation, so I don't think the threat of downvotes in a significant deterrent anyway. – tparker May 24 '23 at 00:45
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    Lol, it appears the subfield of mathematics which investigates which subfields of mathematics have their status as subfields of math disputed has its status as subfield of math disputed (the question was just closed as “offtopic”) – Moritz Schauer May 24 '23 at 03:18
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    @ZachTeitler: Oops. Some people who know me well would probably confirm that my irony detector malfunctions on a regular base... – Jochen Glueck May 24 '23 at 09:57
  • @tparker. I usually preemptively close soft questions like this; in the past I have tried to resist the heckler's veto, but now I have learned that the best way to convince someone of $\neg P$ is to use as much reason and evidence as possible in favor of $P$. There are plenty of technical questions and answers that need upvoting, so I would not want to use any upvotes on soft questions like this. – Joseph Van Name May 28 '23 at 13:52
  • One proposed definition says mathematics is the study of truths that are preserved by isomorphism. I used to think that that is self-evidently right. But if it could be shown that some tasks can be done and understood only by those who understand mathematics, as defined above, but do not fit into that definition, should those things be considered mathematics? – Michael Hardy May 28 '23 at 21:22
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    @Kapil It's a bit fuzzy. – Asaf Karagila Jun 06 '23 at 10:11
  • I'm having trouble understanding what the word "epistemic" is doing in the title of this question. It seems to make perfect sense with that word deleted, and less sense (to me) with it there. Do you mind clarifying? – Will Brian Jun 06 '23 at 14:35
  • @WillBrian I didn't mean anything terribly precise by the word "epistemic" - just that this question is somewhat philosophical and gets at the fundamental nature of mathematics, rather than some kind of convention-based taxonomy. It's fine to ignore the word "epistemic" in the title. – tparker Jun 07 '23 at 02:24

7 Answers7

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There are some speculative mathematical concepts that come to mind, such as the field of one element or motives, though perhaps these are more classifiable as "potential future mathematics" rather than "not mathematics at all", and certainly these speculative topics have at least inspired the creation of mainstream, commonly-accepted mathematics (rigorous theorems, applications to other fields of mathematics, precise conjectures, conceptual reworkings of existing theories, etc.). [And motives may be currently in transition from "potential future mathematics" to "actual mathematics"; I'll leave it to experts in the area to weigh in further on this.]

A more controversial example might be inter-universal Teichmüller theory, where there is genuine debate as to whether this is "actual mathematics", "potential future mathematics", or "not mathematics at all".

If one turns from subfields of mathematics to modalities of mathematics, then in the recent past there were some debates as to whether experimental mathematics or computer-assisted proofs counted as "real" mathematics, but I believe that the prevailing consensus nowadays (by which I mean in the last decade or so) is that these do broadly fall inside the realm of mathematics. (Though perhaps these debates may be re-ignited in coming years if AI-generated conjectures and/or AI-generated proofs of new mathematical theorems become commonplace.) Going back even further in time, we of course have some venerable debates about the use of non-constructive methods (cf. Gordan's quote on Hilbert's proof of his basis theorem being theology rather than mathematics), set-theoretic infinities, non-Euclidean geometry, complex numbers, etc., though again the modern consensus is very strongly in favor of classifying all of these methods and concepts as being part of the field of mathematics. (cf. Gordan's later quote - reported by Klein - on having convinced himself that theology has its advantages.)

Finally, in the 1990s, the topic of Bible codes / Torah codes did briefly attract some academic mathematical interest (and controversy), but it would be a stretch to consider it a "field of academic mathematics" currently.

EDIT: in the converse direction, there are certainly disciplines that are typically housed outside of academic mathematics departments that have a strong case of being considered to be primarily mathematical in nature. Theoretical computer science is one example that comes to mind; there may well be others.

SECOND EDIT: Section 19 (Mathematical Education and Popularization of Mathematics) and Section 20 (History of Mathematics) of the (2022) International Congress of Mathematicians are both devoted to fields which one could certainly argue do not have the epistemic status of mathematics, but are still perfectly valid fields of academic study, and which are the primary or secondary interests of a non-trivial number of faculty at mathematics departments. Whether they qualify as "fields of academic mathematics" depends on one's definitions, though.

THIRD EDIT: The Online Encyclopedia of Integer Sequences (OEIS) is not, strictly speaking, a field, but it does have an active community of both professional and amateur mathematicians contributing to it, and is widely used within the academic mathematical community. One could pose the philosophical question of whether contributing to the OEIS is an activity that can be ascribed the epistemic status of "mathematics". Similar questions could be asked for the communities centered around developing mathematical software, such as proof assistants. However, my personal view is to incline towards a "big tent" view of mathematics, and that excessive gatekeeping of what qualifies as "genuine" mathematics could be harmful towards achieving progress in the field.

Terry Tao
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    Motives do have (currently multiple) definitions along with a number results; so in that sense, there is some actual mathematics there. On the other hand, there are also a huge number of conjectures compared to other fields, so maybe it's fair to place some of it in the "potential future mathematics" category. – Donu Arapura May 21 '23 at 18:21
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    I'd say that calling acceptance of computer-assisted proofs "prevailing" may be a bit exaggerated: almost every mathematician I know with whom I discussed that problem holds an opinion in the range from taking them as an "evidence in favour of conjecture" to outright calling them harmful. If you think that main goal of mathematics is providing understanding, and ability to prove theorems as evidence for your understanding being right, then it's hard to give those "proofs by coincidence" much weight. – Denis T May 21 '23 at 19:31
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    If I remember well, Departments of Statistics are often distinct from Math Departments, in the anglo-saxon area. Does this mean that there is no consensus about whether Statistics belongs to Mathematics ? – Denis Serre May 21 '23 at 19:55
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    @DenisT I'm surprised at your comment. Maybe we should distinguish between accepting that a computer-assisted proof establishes the truth the theorem being claimed, versus accepting that it is a "real proof" or a "satisfactory proof" or a "proof that conveys understanding"? Reuben Hersh, in his introduction to 18 Unconventional Essays on the Nature of Mathematics, said that he did not know anyone who either believed that Flyspeck would be completed or that, even if claimed to be complete, it would be universally accepted as definitively verifying the correctness of the proof. But... – Timothy Chow May 21 '23 at 21:18
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    I emailed him in 2014 and he retracted his claim, saying, "I now expect that the formal proof of Hales' theorem is or will soon become part of established mathematics." Do you think that Flyspeck is nothing more than "evidence in favor of" the Kepler conjecture, and that the Kepler conjecture should still be considered an open problem? – Timothy Chow May 21 '23 at 21:19
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    @DenisT There is certainly debate about whether computer-assisted proofs are preferable to entirely human-generated ones, but nowadays I don't see many people advocate that they are in fact invalid or otherwise unmathematical. In current practice, papers with significant numerical or other computer-assistant components routinely appear in top journals (particularly in the more applied areas of mathematics), and I have not seen significant recent dispute of the proofs of, say, the Four color theorem, Kepler's conjecture, or the odd Goldbach conjecture on computer-assisted grounds. – Terry Tao May 21 '23 at 21:21
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    @DenisSerre: Some statistics is mathematical, some is not. Ditto, physics, computer science, economics, etc. There is no controversy here. Separation of departments is mostly a result of local political decisions, that's all. – Moishe Kohan May 21 '23 at 21:22
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    ... Also, I would caution against conflating "computer assisted proof" with "a proof that does not provide understanding / is true by coincidence". For instance, the >10,000 page proof of the classification of finite simple groups is nearly 100% human-generated, but a hypothetical alternate proof consisting of 100 pages of human argument to verify the claim for simple groups of order $>10^{100}$, together with a clever computer calculation to handle the remaining cases, could be viewed as a more satisfactory proof in some respects (though not others). – Terry Tao May 21 '23 at 21:26
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    "Separation of departments is mostly a result of local political decisions, that's all." I agree. As a PhD student at Stanford, I took a course on coding theory in the Electrical Engineering Department, combinatorics in the Computer Science Department, random number generators in the Statistics Department, and game theory in I-forget-which department (but it wasn't the Math Department). – Gerry Myerson May 22 '23 at 04:44
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    The example of Inter-Universal Teichmuller Theory usefully illustrates that when people say something is “not mathematics”, they often mean just that it’s not acceptable quality mathematics — it’s aiming to be traditional rigorous mathematics, not anything else, but it’s not reached the standards of correctness/completeness/rigour to be accepted. I think this is a bit of a different sense of “not mathematics” than OP is after — at least, it’s much less interesting than the question of work that is only debatably mathematics but is clearly successful as something. – Peter LeFanu Lumsdaine May 22 '23 at 15:36
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    How does OEIS fit into this? It's like asking whether a conference is a journal, or whether Gauss's diaries are a book. You can ask whether claims made in the OEIS are reliable (IMHO they aren't without references, and the telegraphic style is a drain on the quality of the content). – darij grinberg May 22 '23 at 17:02
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    @darijgrinberg As I said, the OEIS is a mathematical database rather than a field, but arguably the subjects of "mathematical database development" and "mathematical software development" have at least some of the hallmarks of a field, even if they are not generally acknowledged to be such currently. As your reponse illustrates, the question of whether activities such as contributing to a mathematical database like the OEIS in the hope of finding a match, or adding to a formal math library, count as "doing mathematics", is currently controversial, but this may well change in the future. – Terry Tao May 22 '23 at 19:19
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    Ah, you're talking about the activity -- I should have been reading more carefully! From this viewpoint, however, there are even more basic questions around, such as whether editing a mathematical paper is "mathematics", or reading one is... – darij grinberg May 22 '23 at 19:31
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    @TerryTao instead of a hypothetical computer-assisted proof of the CFSG, one could instead point to the proof of the ternary Goldbach conjecture by Helfgott, which included computation checking it is true for the first $O(10^{30})$ cases, and by (mostly) by-hand proof above that. – David Roberts May 24 '23 at 00:46
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    Hello! You may be interested in including this as an example of "some academic mathematical interest in 'bible codes'". – user21820 May 24 '23 at 08:20
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    @PeterLeFanuLumsdaine: What exactly was it successful at? Note that it is easy to bury a fatal logical flaw in hundreds of pages of proof, in which every other line is correct, but it's hard to figure out the flaw if it is not formal enough. Worse still if the purported proof itself uses something that sounds like moving goalposts (""re-initialization" of (mathematical) objects, making their previous "history" inaccessible")... – user21820 May 24 '23 at 08:28
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    @user21820: Sorry if I was unclear — my point was, I agree that it’s not clear IUT was successful as anything. I think it’s an example of “not mathematics” just in the less-interesting sense, “failed mathematics”, not the interesting sense “not mathematics, but something else”. – Peter LeFanu Lumsdaine May 24 '23 at 11:49
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    @PeterLeFanuLumsdaine: Ah sorry I misinterpreted your comment. I had read it as "it { is much less interesting than the question of work that is only debatably mathematics } but { is clearly successful as something }". XD – user21820 May 24 '23 at 12:10
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    Another historical example worth mention are early approaches to infinitesimals (before the first rigorous development in Abraham Robinson's nonstandard analysis). One major foundational impediment here was logical - the lack of a rigorous language necessary to state essential facts like transfer principles etc. Such language/logic impediments occur throughout history, e.g. see Hankel's scathing critique of Cauchy's attempt to explain the construction of complex numbers via $\Bbb C\cong \Bbb R[x]/(x^2+1)$ long before quotient rings were invented. – Bill Dubuque May 24 '23 at 17:38
  • @MoisheKohan : "Separation of departments is mostly a result of local political decisions, that's all." Maybe that is correct up to and including the word "decisions", but saying "that's all" makes it sound wholly subjective. There are reasons for those decisions. Many---maybe most of those who administer math departments are not interested in understanding the different purposes that guide those whose interest is in statistics, and so cannot make informed decisions. – Michael Hardy May 28 '23 at 20:59
  • @PeterLeFanuLumsdaine : Either Teichmüller or Teichmueller can be considered a correct spelling, but if it were Teichmuller, the pronunciation would be different. – Michael Hardy May 28 '23 at 21:04
  • @MichaelHardy: Yes, and these are political, not scientific, decisions. ("Political" is not synonymous to "subjective.") As in your example, the fact that a pure mathematician running a department is not willing to understand issues related to research in statistics, has nothing to do with mathematical nature (or lack of thereof) of mathematical statistics. In the examples that I know, the main reason for split comes from statisticians having many people interested in things like policy recommendations. This side of statistics is non-mathematical. – Moishe Kohan May 28 '23 at 21:59
  • @MoisheKohan : I said "not interested in understanding the different purposes." The fact that the PURPOSES are different is not a matter of politics. – Michael Hardy May 29 '23 at 17:20
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There are several possible dimensions to the question, "Is it math?"

  1. Does it belong in the mathematics department? I think you mostly want to exclude this dimension, because of your comment about pure versus applied math. But you mentioned philosophy, so what about logic? In some universities, logicians are placed in the philosophy department, while in other universities, logicians are placed in the mathematics department. In many cases, the decision is based on considerations similar to those that are used to draw the line between pure and applied mathematics. The same can be said of several other subjects such as statistics, computer science, or operations research.

  2. Does something have to meet certain standards of professionalism to count as math? The field of recreational mathematics is considered by some to be "not real math" because of its perceived lack of seriousness or scholarliness. It is not really controversial nowadays to say that graph theory and combinatorics are part of math, but for example, Euler did not think that the famous problem of the bridges of Königsberg was really a mathematical question, and Gian-Carlo Rota used to complain that combinatorics was long considered to be a "Mickey Mouse subject." In a similar vein, some mathematicians will say that the content of elementary school classes that drill students in the mechanics of arithmetical algorithms "isn't really math." Of course, they are not saying that said content should instead be taught in English class or music class; rather, they are saying that until the content crosses some threshold of sophistication, it does not count as "real math." These debates can become heated and can have significant real-world consequences, but I suspect that this isn't the dimension you're primarily interested in.

  3. Does something have to satisfy certain standards of rigor to count as math? I like to cite Jaffe and Quinn's article on theoretical mathematics as an example of a debate about rigor. This seems closest to what you're asking about. When Witten won a Fields Medal, some people raised questions about whether Witten's work counted as mathematics, not just because it was grounded in theoretical physics, but because many of his arguments did not obey the usual canons of rigor in mathematics. Nevertheless, note that Jaffe and Quinn are not arguing that "theoretical mathematics" is not mathematics; they acknowledge that it is most definitely mathematics, and are just raising questions about the role of rigor in mathematics. Similarly, when Zeilberger argues in favor of semi-rigorous mathematics, he is primarily concerned with what constitutes a satisfactory proof and not what constitutes mathematics. On the flip side, people who complain about computer-assisted proof usually do not say that computer assistance causes something to no longer count as mathematics.

Timothy Chow
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    A short answer might be that individual papers might be described (by critics) as "not even wrong," but not entire subfields of mathematics. – Timothy Chow May 22 '23 at 12:23
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    There is a difference between logic as a philosophical field and logic as a mathematical field. Therefore I think the premise that the placement of logicians in the mathematics department or the philosophy department describes some particular epistemic position of the institution is not quite correct. – Jannik Pitt May 24 '23 at 08:47
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    About rigour: there are interesting observations to be made around the works of Ramanujan and Hardy. Hardy tried to help Ramanujan become more rigorous; and after Ramanujan's death, Hardy said something along the lines of "I might have given too much importance to rigour. Perhaps I should have left Ramanujan do all his crazy things. There would always have been time later to proofread it and make it more rigorous." – Stef May 24 '23 at 09:08
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    Also about rigour: Fourier's initial publication was technically false, since he claimed all continuous functions had a Fourier transform, and actually stronger assumptions were required for his theorem to hold. So his work definitely lacked rigour, and he was criticised a lot for it. But his work was amazing, and there have been many mathematicians after him to make it more rigorous. So, I think there is an argument to be made that sometimes, it's okay to drop rigour if it gives more freedom to our imagination and allows us to discover new things. We can always make them rigorous later. – Stef May 24 '23 at 09:10
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    @Stef, often when I think of Hardy and Ramanujan, I think of an encounter between Nadia Boulanger and George Gershwin (see https://en.wikipedia.org/wiki/Nadia_Boulanger}. – Tom Copeland May 24 '23 at 17:28
  • I doubt anybody will say that Euler was not a mathematician. – Michael Hardy May 29 '23 at 17:23
  • @MichaelHardy I'm not sure if your remark about Euler was a response to my remark about Euler, but if so, I want to clarify that I'm not suggesting that anyone would say that Euler was not a mathematician. Rather, I was saying that Euler thought that graph theory (or at least, the specific question about the bridges of Königsberg) was not mathematics. He answered the question anyway, but not (in his own eyes) in his capacity as a mathematician. – Timothy Chow May 29 '23 at 18:32
  • @TimothyChow : I wasn't responding to your remark; I was just suggesting that some standards of rigor are not necessary to calling someone a mathematician. – Michael Hardy May 29 '23 at 18:49
  • As far as "certain standards of rigor" are concerned: Imagine a right circular cylinder whose height equals the diameter of its base. Inside it is a double cone whose apex is right at the center of the cylinder and whose two bases are those of the cylinder, and a sphere whose diameter is that of the cylinder's base. Every plane parallel to the base intersects the part of the interior of the cylinder that is outside the cone in a ring whose area is the same as the area of the corresponding cross-section of the sphere. THEREFORE, the volume of the sphere$,\ldots\qquad$ – Michael Hardy May 29 '23 at 18:52
  • $\ldots,$equals the volume of the part of the interior of the cylinder that is outside the cone. That is a use of "Cavalieri's principle" (which it seems Archimedes thought of long before Cavalieri). Perhaps every attempt to make this principle conform to standards of logical rigor makes it unduly complicated and excludes some obvious cases to which is it applied. – Michael Hardy May 29 '23 at 18:54
  • The "boundary rule", as I call it, says the size of a moving boundary times the rate at which it moves equals the rate of change of size of the bounded region. For example: (1) the derivative of the volume of a sphere with respect to the radius is the surface area, and (2) the derivative of an $n$-dimensional cube with respect to its side length is the sum of $(n-1)$-dimensional volumes of moving faces faces, and (3) doing the same with a rectangle with two moving sides yields the product rule, and there are yet other applications. Now the point will be that$,\ldots\qquad$ – Michael Hardy May 29 '23 at 18:58
  • $\ldots,$attempts to make the "boundary rule" precise have the effect of making it appear more complicated than it is and also of excluding some of its applications. – Michael Hardy May 29 '23 at 18:58
  • Economists find that the "indifference curves" saying you are indifferent between having this many guns and this much butter, and on the other hand, having this many guns and this much butter, can have any of many shapes. But suppose two points are on such a curve. If you are indifferent between them, then you don't mind if I choose which one you get randomly. But that means you are indifferent between a bunch of Bernoulli distributions---infinitely many. There you have an indifference curve that is a straight line. So the question is whether$,\ldots\qquad$ – Michael Hardy May 29 '23 at 19:03
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    $\ldots,$that argument can be made rigorous and still be that argument, and then whether it can be understood other than by understanding mathematics. If something like this can be understood only by understanding mathematics, and cannot be made logically rigorous, what should we conclude? – Michael Hardy May 29 '23 at 19:04
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It is my impression that the relationship between statistics and mathematics is definitely not completely agreed upon (e.g. a quick Google for "is statistics math" gives both emphatic "statistics is branch of math" and "no, statistics is not mathematics", Wikipedia has "Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. ").

To clarify, I think few people would argue that what is usually called "mathematical statistics" is not math. The discussion is AFAIK on the status of things like experimental design that provide the basis for connecting mathematical statistics to the real world - there is a level of formality and mathematical thinking involved, but there are also clearly are non-mathematical considerations.

Martin Modrák
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    I do think that in some countries i know about statistics would be in a better shape if it was considered mathematics, and such taught in math departments! – kjetil b halvorsen May 22 '23 at 16:49
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    I encountered a statistician who claims that all of mathematics is a special branch of statistics. I doubt this is a popular view. I have already heard that all of mathematics is a branch of string theory, or a branch of geometric algebra. – Ben McKay May 22 '23 at 17:27
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    "connecting mathematics to the real world" has "clearly non-mathematical considerations" is insightful for me. That's shared by Statistics, Theoretical Physics, Error Analysis (in Numerical Analysis), ..., Logic?, ... etc. – Pablo H May 22 '23 at 17:54
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    E. Hopf once said: "ergodic theory is statistics and statistics is measure theory." – User1865345 May 23 '23 at 15:25
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    @User1865345 Oh well, in some sense, yes, at least for the second part. Whether it’s useful to think it that way is another question… – Massimo Ortolano May 23 '23 at 21:02
  • Much like mathematical statistics is clearly math, there are things in statistics that are pretty clearly not mathematics. For example, there are statistics papers about what kind of graphs and what color schemes are most easily understood. – Michael Greinecker Jun 13 '23 at 23:29
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There are certainly subdisciplines that are sometimes grouped into academic mathematics departments, or studied by academic mathematicians, and that require some level of mathematical understanding to study, but whose nature does not involve mathematical proof or mathematical modeling and therefore might reasonably be thought of as "not really mathematics". Mathematics education, the philosophy of mathematics, and the history of mathematics all come to mind.

David Eppstein
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Decision theory fits the description well. On the Wikipedia page, it's listed as a subfield of mathematics, and there are mathematical venues for decision theory.

Nevertheless, the key issues are fundamentally not mathematical in nature, at least in the sense of today's mathematics. Accepted papers in the field would for sure offend at least some mathematician sensibilities; see e.g. the Discussion section in [Everitt et al., 2015].

For an example of decision theoretic research being clearly not considered admissible by the math community, see this MO answer. The "mathematical" probabilistic argument yields a Dutch book vulnerable agent; I think this is evidence that questions of how to handle causality and counterfactuals are not in the realm of today's mathematics.

And that's only the "mainstream" decision theories; what about the less prominent, such as Non-Nashian or functional decision theories? Or, look at this poll over Newcomb's problem; I don't think there is a mathematical field which splits over a problem like this.

On another note, I think causality research might be controversial too. Some related "agency" research in AI might also be fuzzy: is Discovering Agents (DeepMind, 2022) mathematics?

Daniel Paleka
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Suppose $X_1,\ldots, X_n$ are independent observations from a normally distributed population, and $Y_1,\ldots,Y_m$ from another (and $m,n$ may differ). What can be inferred about the difference between the means of the two populations?

That's the Behrens–Fisher problem. That, as stated, is not a precisely defined math problem. But it is definitely a statistics problem. Is "precisely defined" an essential characteristic of a math problem?

The physicist Edwin Jaynes posed this problem: A very limp string of length $\ell$ is thrown very unskillfully onto the floor. What is the probability distribution of the distance between its two ends? That is, by comparison to the Behrens–Fisher problem, a precisely defined math problem. You're supposed to figure out precisely what he had in mind. But the Behrens–Fisher problem can be modeled in any of a variety of ways, and any claim that one math problem used to model it is the "right" one is a philosophical claim.

I hesitate to post this example because I'm not prepared to explain what those approached are. What is recommended in recently published textbooks for the use of non-mathematically inclined users of statistics involves estimating the two variances separately and using critical values from Student's t-distribution with a non-integer number of degrees of freedom that is determined by the data—i.e. by $X_1,\ldots,X_n,Y_1,\ldots,Y_m.$ And others say one should use a conditional distribution of the difference between means given the data, thuse requiring possibly very vague a prior distribution. And there are yet other proposals. And each of them leads to a solution that is found by relying heavily on mathematics. But the question of which of those math problems is the right one is a statistics problem but not a math problem.

Moreover, the purposes of statistics are different from those of pure mathematics.

Michael Hardy
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  • I'm not sure quite I understand the intent of this answer. This seems like a specific problem in statistics where there is ongoing academic debate. But is it meant to support the claim that statistics has a different epistemic status than math? Or perhaps that all applied mathematics, where there is legitimate debate about how to model various real-world phenomena that cannot be resolved according to the abstract, logical rules of pure math, has a different status? – Sam Hopkins May 28 '23 at 20:50
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    @SamHopkins : Part of the claim is that the question of how to model this is arguably philosophy or something like that, rather than mathematics, but is a question that obviously belongs within statistics, so that some things in statistics are not mathematics. And this is far from the only one. – Michael Hardy May 28 '23 at 21:26
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Set theory and category theory fit the bill for 'fields of academic mathematics whose epistemic status as mathematics is up for debate'.

As evidence, consider the preponderance of set theory/category theory papers published in philosophy journals, or the lack of a preponderance of people with phd's in category theory/set theory who go on to be tenured professors, or the excellent set theorists who are hired into philosophy departments at major universities.

I have never personally encountered any mathematicians who will say out loud that they don't think pure set theory or category theory are mathematics, but it's one of those things like the gender pay gap where the proof is in the pudding even if nobody wants to say it out loud/feels sexist.

Wow! I knew I was stirring the pot here, but the level of response to this post is genuinely surprising to me -- I feel obliged to clarify what I mean.

In my personal opinion, category theory and set theory are some of the most wonderful and interesting mathematical subjects in existence, and close to the core of whatever may be called 'mathematics proper'.

The opinion that set theory and category theory are somehow not 'mathematics proper' is not mine; I assert that it exists elsewhere in the established mathematical world, as evidenced by the data points mentioned above (and some personal anecdotes from my time as an undergrad, but these are less convincing than hard data).

I am well aware that I didn't provide any actual examples; this is because all three are phenomenon I have encountered frequently enough that I expect others have too, and that if anyone desires hard evidence I will be able to produce it with minimal effort.

In accordance with my personal opinion mentioned above, the idea that any professional mathematician thinks these things infuriates me to no end, and is part of why I decided not to stay in academia. I suspect it is an opinion held by people in positions of career power throughout the mathematical world, which results in the stymying of set theorists and category theorists mathematical careers in the ways mentioned above. I sincerely hope that I am wrong, or that if I am right something changes in the near future.

Alec Rhea
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    This answer has nothing to do with epistemic status in my opinion (both areas are universally accepted as rigorous mathematics by the general community). It may be deleted soon. – Todd Trimble May 24 '23 at 15:18
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    Indeed, I don't think there is much question as to whether these subjects, and other topics related to logic, are mathematics. The "lack of preponderance" you mention seems to me to be merely an effect of there being relatively few people who primarily work with those topics (though this does seem to change recently). Getting tenure in philosophy departments seems to be more related to the hiring structure - just because the work is also considered relevant to philosophy doesn't make it "not mathematics". – Wojowu May 27 '23 at 19:40
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    Again, I think you may be misreading the question, which I think OP tried to head off at the pass. Yes, there are many, including many in positions of political power in the community, who would question whether e.g. category theory is a worthwhile thing to be doing. That's something different from whether the field uses the recognized traditional standards of how we know or accept something is true in mathematics, or mathematical methods of precise definitions. The Jaffe-Quinn paper is all about that, for example. For my money, the answers by Tao and Chow are hitting it on the head. – Todd Trimble May 27 '23 at 19:49
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    @ToddTrimble Having re-read the question, I honestly think this is exactly what the OP was looking for -- they specifically ask for entire fields of mathematics, not subfields as addressed by Terry (I think answers like that were meant to be 'headed off at the pass'). I don't see any possible answer besides this one to the question I read at the top of the page, but I do think Terry and Timothy wrote excellent answers on less controversial topics. I am of course open to being corrected; if tparker wants to chime in and tell me I'm way off base, I am fine deleting this answer manually. – Alec Rhea May 29 '23 at 02:32
  • This will be my last comment. The actual word was "branches". Whether this can mean "fields" or "subfields" or something else, who's to say. Maybe tparker can say. But we then disagree over whether category theory and set theory are correct answers, i.e., enjoy the same epistemic status as other branches like, say, algebraic geometry, according to consensus. They do, as I have argued. As opposed to History of Mathematics or Mathematics Education, mentioned by Terence. These are not deductive sciences in the same way; a consensus view is these are not mathematics proper. – Todd Trimble May 29 '23 at 11:54
  • @ToddTrimble I was referring to the title of the question, but I agree that this is starting to feel like a waste of both our time. I regret that this ordeal has caused any tension between us -- I respect your work greatly and like you very much as a person based on what I've seen on-site. I seem to have struck a nerve I did not intend to strike, and I apologize. – Alec Rhea May 29 '23 at 13:21
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    No apology is necessary! And it's not striking a nerve, so don't worry. I share your frustration about the way "old prejudices die hard"; I think we're just disagreeing about the aptness of your answer to this question, but in the final analysis, that's no big deal. – Todd Trimble May 29 '23 at 14:48