A colleague raised the above question with me; more precisely he said:
Suppose that a mathematician were resolved not to publish any theorems
unless they had checked the proof of every theorem that they cite (and
recursively the proofs of all the theorems that those rely on etc.).
Can they have a career in pure mathematics?
With the obvious proviso:
Of course, there are a few well-known theorems, like the classification
of finite simple groups, whose proofs are virtually impossible for any one
person to check at all. But one can have a perfectly good mathematical career
without ever citing any of those.
It seems to me that complete checking might be possible, though perhaps
only in narrow fields of mathematics, but does anyone know of mathematicians
who actually do it? (or try to)
Best Answer
Possible or not, this should be a goal:-) Let me put it slightly differently: you should understand every result that you use. First of all, a theorem that you use can be wrong. So whenever you rely your proof on a theorem that you did not check, you take a risk. There are many known cases when a result was "accepted" by a mathematical community, and then turned to be either wrong or unproved. If your proof relies on a theorem that you do not understand this really means that you don't fully understand your own proof.
In the cases like finite simple group classification, you should clearly state in your publication that your proof depends on it. And in general, if you write a proof which relies on the theorem that you do not fully understand, you should make as clear as possible, where exactly and how you use this theorem.
EDIT. When you cite a result you endorse it. You are essentially saying that on your opinion it is correct. Now suppose you are simply asked to endorse some result: just to tell your opinion, whether it is correct or not. Would you endorse it publicly in print, without checking the proof ? On my opinion, citing a result in your paper without any comment is the same.