Take Kant and Gödel, for example. Kant was neither just an intuitionist nor just a formalist, nor even absolutely a non-realist (the forms of space and time are, after all, empirically real and transcendentally ideal, or variously provide for such statuses). Gödel was a strong realist with abiding sympathies for constructivism, intuitionism, and perhaps even definabilism (where this is a distinct enough position). Hamkins has spaces in the logical flux of the set-theoretic multiverse to envision worlds based on different axiomatizations of various schools of philosophy-of-math: a world where the inscription axioms are all literally inscriptive (semiotic formalism) but so where we are still, ultimately, talking about abstract forms of symbols; and so on and on.

So it doesn't seem quite accurate to try to pigeonhole any eminent philosopher of mathematics into a strictly exclusionary standpoint, but we might redescribe these standpoints as ones such as if Kant, say, were to rank intuitionism highest, formalism next, and realism third or last, with some fourth or other if we had to keep going along, here.

Do philosophies of mathematics have to be advanced in a strongly exclusionary way as such, in the end, or instead is it epistemologically stable to imagine a system of epistemic weights and ranks? One can also think about a system where some methodologies are assigned nonzero weights while other methodologies do have no weight in such a system, although unless we were talking about inchoate numerology, I would not be minded to consider ruling out any specific methodology in advance.

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    I think it totally depends. Like Haim Gaifman is pretty one-minded that mathematics is “what mathematicians do”. And Ed Nelson was a staunch formalist. And even Hamkin’s thinks we know abstract objects with more confidence than physical. However, which abstract objects to think about and what inspires thinking about which ones he is fluid on. And again for the at-least-slightly-fluid side there is Clarke-Doane’s there are no absolute certainties in philosophy (and even math I think he’s said) so he seems at least partially open to any epistemology.
    – J Kusin
    Commented Sep 25, 2023 at 1:16
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    If by "different epistemologies" you mean the three round holes of logicism/intuitionism/formalism of early 20th century then yes, many epistemologies out there will be square pegs. SEP gives a longer list including square holes like naturalism, fictionalism, structuralism and varieties of "intuitionism" very distinct from Brouwer's and constructivism generally. Realism is not really an epistemology, and Gödel's epistemiology was very different from Plato's, close to Husserl's "ideal intuitionism". But Kant, Gödel and Husserl are prime examples of overcommitting to one epistemology - theirs.
    – Conifold
    Commented Sep 25, 2023 at 2:52
  • Epicurus's famous principle of multiple explanations... Commented Sep 25, 2023 at 5:07

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You use the terminology of "epistemology" but in fact what you are talking about is closer to "ontology" than epistemology. Benacerraf famously distinguished between ontology and practice. His distinction tends to "relativize" (as they say in French) the importance of ontology in doing the philosophy of mathematics. To take the example of Hamkins that you mentioned: while his multiverse approach tends to undermine realism about sets, he insists that he is a realist - but at the higher multiverse level. It is hard to know what this means exactly, beyond the fact that, since many mathematicians are realists (or Platonists) about the mathematical entities they happen to be working on at the moment, it is wise to avoid challenging realism too directly.

  • You're right that I need to track the epistemology/ontology distinction better, here. I suppose to some extent I take them to be reciprocal in this case, like what Kant thought about free will being the ratio essendi of the moral law, which is the ratio cognoscendi of free will in turn. I do ultimately think that the underdeterminacy of the mathematical universe "as a whole" (a la Hamkins) is constitutive of our free will's own openness, no less, or rather that our free mathematical (abstract) will is "where" pure mathematical objects "reside," which affects my own epistemology, then. Commented Oct 3, 2023 at 15:18
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    By bringing up Benacerraf's use of a distinction between ontology and practice, which I was not fully/directly familiar with (although I appreciate that his identification and epistemological problems for at-the-time set theory reflect such a distinction), you have provided a useful point of departure for reflecting on modern philosophers of mathematics as such. I also appreciate how you point out that Hamkins might count as sort of both a realist and a non-realist, if not in an incoherent way, yet in an interesting one. Commented Oct 3, 2023 at 22:25

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