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.



Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?
There are some speculative mathematical concepts that come to mind, such as the [field of one element](https://en.wikipedia.org/wiki/Field_with_one_element) or [motives](https://en.wikipedia.org/wiki/Motive_(algebraic_geometry)), 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](https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_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](https://en.wikipedia.org/wiki/Experimental_mathematics) or [computer-assisted proofs](https://en.wikipedia.org/wiki/Computer-assisted_proof) 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](https://academic.oup.com/princeton-scholarship-online/book/23470/chapter-abstract/184556786) 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](https://en.wikipedia.org/wiki/Bible_code) 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](https://en.wikipedia.org/wiki/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](https://www.mathunion.org/fileadmin/IMU/Report/SC/2019/structure_committee_final.pdf) 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)](https://oeis.org/) 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](https://en.wikipedia.org/wiki/Proof_assistant). 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.


