- "Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation" because "many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community."
- Recognizing mathematical truth is harder than one might think at first. "If a theorem

has a short complete proof, we can check it. But if the proof is deep, difficult, and already fills 100 journal pages, if no one has the time and energy to fill in the details, if a “complete” proof would be 100,000 pages long, then we rely on the judgments of the bosses in the field. In mathematics, a theorem is true, or it’s not a theorem. But even in mathematics, truth can be political."

Yes, proofs of several published theorems contain mistakes. However, more often than not, the mistakes can be fixed and the results turn out to live to see the day after all.

Of course, I expect that mathematicians will want to improve upon the social notion of proof that underlies their profession. Quoting again from the opinion piece by Nathanson:

How can the reliability of mathematical proofs be improved? When should a proof be deemed to be "complete"? As a computer scientist, I'd say that a proof is really "complete" when it can be verified by a computer-based proof checker and it has been independently verified using a few such software tools. I know that this is a very stringent requirement, which will most likely never be implemented, but either we are ready to accept that mathematical truth is an unreliable but remarkably robust notion or we enlist the help of our computers to try and make it more reliable.

We mathematicians like to talk about the “reliability” of our literature, but it is, in fact, unreliable.

Part of the problem is refereeing. Many (I think most) papers in most refereed journals are not refereed. There is a presumptive referee who looks at the paper, reads the introduction and the statements of the results, glances at the proofs, and, if everything seems okay, recommends publication. Some referees do check proofs line-by-line, but many do not. When I read a journal article, I often find mistakes. Whether I can fix them is irrelevant. The literature is unreliable.

Do any of you think that proof-checking of research-level mathematical proofs will become commonplace in our lifetime?

Addendum dated 5 August 2008: I should have known that Doron Zeilberger would have commented on Nathanson's opinion piece. Of course, Dr. Z's opinion is thought provoking as usual. You can read it here. In summary, Dr. Z suggests two ways for improving the reliability of mathematical knowledge.

- In his words, "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."
- Abandon anonymous refereeing.

Regarding the second suggestion, I am not so sure that abandoning anonymity in refereeing would have such a great effect on the reliability of mathematical literature. The bottom line is that one should exercise the greatest amount of professionalism in all aspects of the academic profession, regardless of whether one does something anonymously or not. Excellent referees are a great resource to have, and editors and PC members soon build a trusted core of referees that can be relied upon to provide high-quality feedback to authors and editors alike. Referees who do not do a good job tend to be ignored after a while, as do editors who act as black holes for the papers that are submitted to them.