I read Asvin’s post. I generally agree with it. I am a bit shaky on some examples – I think there is room for a balance. It is certainly disturbing to imagine that certain empirical predictions will fail. What Asvin seems to be saying is that it is at least conceptually possible that the following empirical facts may be true simultaneously:
1. ZFC is consistent; a computer searching for a proof of a contradiction from the axioms of set theory
2. The proof of Fermat’s last theorem can be integrated into ZFC
3. A computer can demonstrate a triple of positive integers violating Fermat’s last theorem
It would be common to argue that a counterexample to a theorem would demonstrate that the formal theory is inconsistent. But Asvin’s claim essentially says that the formal theory might not be inconsistent – that is, one has a counterexample that is unable to be integrated into the formal system. But it would be quite surprising – arguably inconceivable – if we were able to demonstrate a pair of numbers and show that they could multiply to give a counterexample to Fermat’s last theorem in a way that couldn’t be integrated into ZFC.
The broader points about the Italian school, the foundational crisis of set theory, etc. are particularly correct. It seems to me much more believable that the large cardinal hypotheses needed to sustain proper category theory turn out to be inconsistent; and new foundations of reasoning become necessary to support the large collections it calls for.
But even these seem to be missing the point Asvin is trying to make. It is still more likely that we simply discover that the informal reduction to formal set theory we have been doing is in fact impossible, a convenient lie.
Continued Ray 2.2.