Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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?

Answer 6

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.

Answer 7

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.

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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.