When doing mathematics one constantly looks at the literature to find theorems on which to build some further results. So a crucial issue is: how do we know when a published paper in fact contains an error ? If we are interested in understanding the method used by the author to prove a given statement then of course one can check directly for oneself.
But in doing so one could overlook an error in the proof. Another situation is when one wants to use a theorem proved by somebody else as a black box, something one takes for granted without having the need to actually understand the proof first hand.
Well, when an error has been noticed, either the journal subsequently publishes an errata (this is very common in the case of typos which alter an otherwise valid result) or even a retraction when the paper cannot be salvaged. Or the author takes the responsability to warn potential readers of the flaw in one way or another (as for example the renowned Michael Harris duly does on his webpage).
But in both cases this is hardly satisfying. Indeed, what about people who do not notice the errata or do not find any warning on the web ? I think it is high time for the mathematical community, perhaps through the IMU, to set up an official website which would contain a list of all published maths papers. All mathematicians of the world would be invited to register on this website. Then those who have read carefully a given paper from the first to the last page coulg log in and either declare the paper to be correct to the best of their abilities or to point out an error.
At this point I’m not sure what would be best: that all users be provided with a pseudonym like user_6892345, or on the contrary that only real names be accepted. Because what would be very important is that all mathematicians would record, with complete honesty, all the things they have understood — and only them. Honesty is indeed the key here: we should avoid dishonnest claims that some result is incorrect just for the sake of harming its author’s reputation. Another thing to avoid would be arrogant people who claim to have understood some very complicated proof when they haven’t.
Provided these two issues can be satisfactorily resolved, for example by the use of a neutral third party, what in fact this website could be is a way of replacing altogether the refereing process. Indeed the latter is very questionable in many circumstances, from referees carelessly accepting a paper they haven’t read properly or haven’t been able to understand but are too ashamed to acknowledge, to outright colusions (the referee writing on his/her report that he/she would accept the paper under review provided its author cites some paper he/she overlooked — and which turns out to be a paper of the referee…). With this website we would en up with a system with a much improved visibility.
I’m not even evoking the issue about whether or not one should ultimately resort to an automated theorem prover, since programs capable of dealing with anything from extremely easy to extremely elaborated concepts are still some decades off anyway.
Update 29-march-2007: corrected typos and repaired a broken link.