TITLE: Is it possible to have a research career while checking the proof of every theorem that you cite?
QUESTION [57 upvotes]: 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)

REPLY [10 votes]: It is worth quoting what Hirsch says about J. H. C. Whitehead (footnote 23 on page 95 of Hirsch's contribution to the Smalefest volume.)
"Whitehead was very good about what he called "doing his homework," that is, reading other people's papers. "I would no more use someone's theorem without reading the proof," he once remarked, "than I would use his wallet without permission." He once published a proof relying on an announcement by Pontryagin, without proof, of the formula $\pi_4(S^2)=0$, which was later shown (also by Pontryagin) to have order 2. Whitehead was quite proud of his footnote stating he had not seen the proof. Smale, on the other hand, told me that if he respected the author, he would take a theorem on trust."