comment by Douglas_Knight ·
2012-12-12T03:46:42.979Z · LW(p) · GW(p)
By the standards of other fields, mathematics doesn't have "substantial" controversies, but here are a few examples, including the two mentioned by Morendil and Jonathan L.
1. A century ago there was controversy about constructive mathematics and about axioms like choice. I think these controversies were resolved by driving out the people with high standards into logic and theoretical CS (Informatics). The higher standards seem more useful in those places, but to call the current standards "uncontroversial" is, I think, misleading. Mathematicians claim to accept that constructive proofs or proofs in weaker systems are in some sense better, but in practice they never think it worth the loss of elegance. In recent decades, the desire to computerize calculation has increased the popularity of constructive proof some. For example, the original controversial non-constructive proof by Hilbert of his Basis Theorem (1890) is usually taught following Noether's very short proof (1921) rather than using Buchberger's constructive Gröbner bases (1965).
1b. Bourbaki and especially Grothendieck actually work in stronger systems than ZFC. It is not clear to me how controversial this is, but there have been awkward conversations (eg, on the FOM mailing list) about the status of Grothendieck's results.
2. Mathematicians have a strong consensus about the ideal of proof to which they aspire: formal ZFC, but they don't actually produce formal proofs. The real standard is informal proofs and a belief that they can be debugged to a formal proof. This is not a precise standard. People are generally nervous about very long proofs. They are not very nervous about the thousands of pages of SGA that go into Deligne's proof of the Ramanujan conjecture because most of that is modular. It breaks into small pieces which are easy to understand and which are reused in many places, thus extensively tested. But the proof of the classification of finite simple groups (famously condemned by Serre around 1980) is made up huge modules that are used for nothing else. In fact, it contained an error at a high level, that the organizers failed to observe that one of the assigned modules had not been delivered. There will be more confidence in the second generation proof because (1) it is projected to be only 5000 pages and (2) the same people will prove the modules as will use them.
2b. When someone rewrites a proof in his own words, it is much more reassuring than if he simply claims to have read and understood it. Modifying the proof, either for simplification or to prove something else is also helpful. But it is generally rude to express such doubts publicly. Here is someone getting in trouble for voicing such concerns. There are lots of decades old results that are controversial in some circles.
2c. It is quite common for top mathematicians to provide fewer details of proofs. Often other people write longer papers filling in the details. The status of the result before this is sometimes controversial, but again often not publicly.
3. The computer-generated proofs of the four-color theorem are controversial for two different reasons. One reason is that since they don't exist in a human mind, they don't provide insight. The other was that since they were of a different form than usual proofs, they were not trustworthy. In fact, there have been many generations of these proofs and I am told that each new one claims that the previous one missed some cases. The latest one generated a formal proof, so it is computer-checked and as defensible as anything. But the first controversy remains.
4. Physicists often make claims about mathematics. For example, that the scaling limit of the self-avoiding walk on the square grid exists; and that the limit has a particular fractal dimension. Is this a controversy that the mathematicians do not accept the physicists' "proof'? Or do they just mean different things by that word? Anyhow, the physicists go on to say that the limit is a conformal field theory, and the mathematicians reject this as a meaningless statement, though they try to salvage it, eg, through SLE.
5. People make calculation errors all the time. Long calculations can be controversial, but they probably should be trusted less than they are. It is generally better to compute many things in parallel, so that an error in the calculation infects all the answers and is immediately apparent.