[FOM] The certainty of mathematical proof

[Staffan Angere]

Dear FOMathematicians,

in relation to the storm of protests that seems to have started due to Voevodsky's position here, I would like to offer my perspective, which I guess lies somewhere between the average FOM member and "radicals" such as Voevodsky. As Tim Chow pointed out, what counts as a "proof" is somewhat context-relative. This, I would say, is probably because certainty is context-relative. In some contexts an error has to be less likely than in others, in order for us to count it as certain. It is, in some sense, clearly possible that PA is inconsistent, even if we, by pure chance, haven't run into any inconsistency yet. It is, however, extremely unlikely that it is.

What may be going on is that in at least parts of FOM, greater certainty is required than in other parts of mathematics for something to count as a proof, just as greater certainty is required in mathematics than in mathematical physics. Perhaps proving PA to be consistent requires greater certainty than proving, say, ZF to be so, since PA may be seen as more "fundamental" (at least by the persons in question). In that case, one could give a perfectly valid proof of the consistency of ZF, while suspending judgement on PA.

I think this resembles Lewis's analysis of knowledge in "Elusive knowledge" somewhat. As Lewis says, a belief that p is knowledge only if it holds in every possible situation, but what is counted as a possible situation is context-dependent. Mentioning a possibility makes us have to take it into account, and we can thus no longer exclude as a possibility. In our case, Voevodsky and many members of FOM would be talking past one another, either because they are in different contexts, or because they consider different possibilities to be relevant in the contexts in question. The second of these differences, however, concern a normative question rather than a factual one. How certain must a proof be for it to be eligible to be called a "true" proof?

I tend to agree with Voevodsky that the question of whether PA is inconsistent is still open, even if I would not say that I doubt its consistency. One of the things Gödel showed was that it always will be open, unless we find an actual inconsistency in it - at least of the standards of proof required are such as to include taking the falsity of PA as a live possibility.

Best regards,
Staffan Angere
Department of Philosophy
University of Lund