TITLE: How much mathematicians rely on other results?
QUESTION [7 upvotes]: When mathematicians do research, do they think every detail so that they know exactly every step what it takes to achieve the result from axioms? Or is it possible to work as a researcher without knowing every detail? For example, is it enough to know that for example Wiles proved the FLT and Helfgott proved the weak Goldbach conjecture or should I work through the details if I will apply those results in some publications?

REPLY [4 votes]: This all depends a bit on what you mean by "knowing all the details".
From my experience an knowledge about this, most researchers depend heavily on other people's results. And one can't know every single detail of the results that one use. We have to rely on the fact (hope) that when something is published, then other people have checked the details. That is not to say that articles don't contain typos and even greater inaccuracies. It is very common to find smaller mistakes in papers.
I think that one thing that makes a researcher good is his or her ability to use results without knowing even the background of those results. You are somehow able to distill what "you need" from a paper without having to know a lot about it. You of course have to be convinced that the results used actually apply to your situation. That can't be any uncertainty about the statements that you are using. And the more your know, the better. You might then, of course, run your ideas by someone who knows more about that area to check that you haven't misunderstood anything.
This is the same "problem" that a Ph.D. student can "run into" when starting to do research. There is this whole world out there, and you are somehow expected to absorb what you need without having to spend years learning all the details. It can be challenging to go from a graduate level class where all details are presented, to having to just "accept stuff". This is, in my opinion, where a good advisor is essential. 
All that said, you, of course, want to know the details of your own research. That is, you need (in my opinion) to know the details of how these results are used.