Written: July 31, 2008.
I read with great interest, in the Aug. 2008 issue of the Notices of the Amer. Math. Soc., Mel Nathanson's fascinating and provocative opinion about "mathematical truth", in which he correctly laments about the unfortunate unreliability of the mathematical literature, and justifiably claims that most referees do not do a thorough job, and hence that there must be lots of undetected errors.
Mel also mentions the two (relatively) recent "biggies", Andrew Wiles' proof of FLT and Grisha Perelman's proof of Poincare. He claims that very few people checked the proofs completely, and the proofs are considered correct by political intimidation of the big shots. While I am sure that mathematics is (almost) as political as any other human endeavor, after all, mathematicians are mere people, and the big shots decide what is important (what they call "mainstream") and what is not (what they call "marginal"), it is too pessimistic to have serious doubts about FLT or Poincare. Mathematicians are, after all, human beings, ambitious and spiteful. I am sure that Gerd Faltings tried very very hard to find a gap in the Wiles-Taylor proof, and was very disappointed that he couldn't. He was hoping to prove it himself, and if Wiles' proof has a mistake, he would still have a chance. Ditto for the army of (mostly Chinese) young mathematicians who scanned Perelman's "sketch" and filled-in the details. They would have loved to gain fame by spotting an error or gap, so the "free enterprise" mathematical culture is a good guarantee against major errors in big stuff.
As for the small stuff, which is 99.99 percent, it is true that it probably has lots of errors, mostly minor, and easily corrected, but probably also some major ones. But who cares? Most mathematical papers are leaves in the web of knowledge, that no one reads, or will ever use to prove something else. The results that are used again and again are mostly lemmas, that while a priori non-trivial, once known, their proof is transparent. Of course, a notable exception is the Classification "Theorem", mentioned by Nathanson, that while essentially correct, still awaits a full watertight proof (eventually it should be at least computer-checkable, if not computer-generated).
There are two ways to improve the reliability of mathematical truth. First computerize! Computers are much more reliable than humans, and as more and more mathematics is becoming amenable to computer checking, this is the way to go. One can also learn from computer programmers efficient ways of handling large data with modularity and other tricks of the trade, and having cross-references that avoid unnecessary repetition.
The other remedy is to abandon that stupid habit of anonymous refereeing. Why should a referee toil so hard, if he or she remains anonymous, and he or she does not get paid a penny for their effort? A much more reasonable system would be that every submitted paper would have, in addition to author(s), also checkers, whose name will appear along with the authors. The duty of the checkers would be to check the paper line-by-line. They should get paid by the authors, either with cash, or by bartering. "If you will check my paper, I will check yours". If a checker messes up and approves an erroneous result, he or she will have bad rep, and it should be even possible to do "rate the checker", and checkers with a high rating could charge more for their trouble. An even more efficient reward system is to pay the checker a flat fee plus a (large!) bonus for each mistake that they spot.
Once a paper comes checked, the editor can make a "political" decision, with or without the help of referee henchmen, whether the paper is "good enough for the Annals" or should be recommended to be submitted to the Fibonacci Quarterly. This is an entirely subjective and political decision. But at least the formal correctness will be (almost) ensured.
Mathematics is arguably the most certain body of knowledge, but of course, nothing is certain in this world, and it is a distinct possibility that the Pythagorean Theorem, and even 2+2=4, are wrong, and it just so happened that Nature and/or God programmed the human mind so that it will overlook the gaps in their alleged proofs. Complete certainty (even of death and taxes, and certainly of mathematical facts) is an unreachable ideal, but one can at least try to improve the reliability along the lines that I suggested.