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Wiles' proof initially involved reference to functional equivalents of inaccessible cardinals (here, Grothendieck universes). Rather than take this as evidence for the meaningfulness and usefulness of higher set theory (actually, not a much higher level of such a theory, all things considered), it seems as if many mathematicians were/are exercised to show that the proof can (or even "ought to") be reformulated without reference to inaccessible cardinals.

The "version of set theory" I'm working on depends heavily on the question of social constructivism in mathematics.V So I quote here Randall Holmes, from a discussion he was having back in 1998:

If one does not acknowledge that there is real formal structure in the world, then one cannot understand the objective character of mathematics. Formal structure (whether it exists Platonically or inheres in a more Aristotelean manner in objects, independently of us in either case) is what mathematics is about.

Now Fermat's vexatious proposition seems to be a structural question about ℕ. But if this structure is not socially constructed (if the question can be posed without direct/first-order reference to the social will of mathematicians, or agents generally acting in a mathematical capacity), why would it be so "upsetting" for Wiles' proof of that proposition to involve reference to inaccessible cardinals?


VOne of the keystones of the system is an "object" denoted as 𝔼, defined as the well-founded set of all sets-knowable-in-a-well-founded-way. Since this set is not an element of itself, it follows that it is (A) a specific set in the ascending hierarchy but (B) we cannot identify which set it is by the means of our ascent. Our awareness of 𝔼 is like being blind and facing the horizon at night, our nonfunctioning eyes set upon the point of the horizon where the sun will first rise in the morning. At any rate, 𝔼 is meant to be used in place of V (or proper classes more generally) to implement Zermelo's talk of V being an "unfinished totality." So not only is |𝔼| a specific large cardinal at any given time, it changes over time (or: which such cardinal it is, changes), as mathematical knowledge socially develops.

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    No, it is evidence of pragmatic minimalism of proof means and the aesthetic of matching proofs to results conceptually that are quite old and common in math. Minimizing the means of a proof allows for maximal generalization, one of the mathematicians' maxims that Maddy describes (which runs counter to constructivism, btw). The looking for algebraic proofs of algebraic theorems (history of FTA is an illustration), combinatorial proofs of combinatorial theorems, etc., has little to do with constructivism as well. But has much to do with the desire to stuff Wiles's proof into PA, if possible.
    – Conifold
    Feb 7, 2023 at 13:46
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    The MO post you have linked is very interesting and there are many answer, one with the following end note: "P.S.: If a mathematician of the caliber of Y.I. Manin made a point of asking in public whether the proof of the Weil conjectures depends in some essential way on inaccessible cardinals, is this not a sign that "Of course not; don't be stupid" may not be the most helpful reply?" Feb 7, 2023 at 14:44
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    IMO the "frustrating" aspect (if any) about Wiles' proof is due to the fact that "philosophically" is quite difficult to imagine why so simple an arithmetical fact must depends on so "far" a structure as Grot... universe. Maybe the "most natural" expectation is about an undecidability proof wrt PA, like Paris-Harrington and Goodstein theorems. Feb 7, 2023 at 14:49
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    Having said that, and I'm not a specialist, what is the real point about "social constructivism"? OBVIOUSLY, a math theorem is a theorem if it has been checked by the mathematical community; thus the original issue with Wiles' proof: very long and complex and the verification was initially slow. Feb 7, 2023 at 14:51
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    See also the post Is Fermat's last theorem provable in Peano arithmetic? Feb 7, 2023 at 15:02

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The discussions you cite may demonstrate that how mathematicians think about Wiles' proof may be a social construct, but those discussions do not speak to the question about whether mathematics is itself a social construct.

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  • My broader/deeper argument, here, would depend on a theory of mathematical epistemology. At this point, I myself believe that both ante rem and in re realism are true, and actually so are formalism, intuitionism, etc., all in their own domain, so to say. But at any rate, it seems like a quasi-empirical fact that mathematicians learn new things over time, so whatever kind of fact it is that they are engaged with is apprehended by a mutable process. Ultimately, I think pure mathematical objects are possible actions, e.g. countable infinity just is the transcendental will to count. Feb 8, 2023 at 7:02
  • But this will (understood in a neo/post-Kantian fashion) is conditioned by social factors, i.e. the evidentialism that is implicit in the ethics of mathematics (as a discipline, nevermind for now its applications) is partly social, and the pure will to count, to quantify, to refer, to encode, etc. would not be stable if it did not comply with the evidentialist form of the "moral law," which then is partly social (mathematicians ought to rigorously analyze their questions and answers, publicly). Feb 8, 2023 at 7:05
  • Actually, this was Cantor's assessment of the matter, too, albeit in a religious vein: one time when he talked about his deity over the alephs and omegas, he brought in the arcane doctrine of the "actus purus," the pure action. Anyway, your answer is brief but very plausible, and if I want to make my case more strongly, I'll have to cite more than this one controversy. Feb 8, 2023 at 7:15

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