How deeply should one understand other people's work before building on them? I have a few results during my math PhD that I used in a black-box or grey-box sense. I have a rough idea of how it is proved but the proof itself is extremely long and detailed. I cannot reproduce it off the top of my head without looking at the paper and going through it line by line again. Some of these proofs also themselves refer to other lemmas and proofs in different papers and it can become quite a deep rabbit hole.
I am at a stage now where I can either spend some weeks studying these proofs in detail or I can focus more new research and take these results as black box results. What is the right/expected thing to do from a scientific mindset? On the one hand, understanding everything from scratch would be nice but on the other hand, the reason we write lemmas is so that others can use them as a springboard to develop new ideas.
TL;DR how deeply do mathematicians understand other people's work before using their results? My practical concern would be my thesis defence but the broader scientific &quot;best practice&quot; would be good to know too.
This depends on the individual case and requires judgment.
On the one hand, understanding the past work completely gives you a better base of understanding going forward.
On the other hand, there are circumstances in which you understand the past work sufficiently to go forward, and you have enough trust in it to know that you won't be embarrassed in future for doing dead end work.
That complete understanding takes time, as you note. It is seldom necessary for a mathematician to recapitulate the entire history of mathematics since, say, Euclid, to do significant work.
For some problems in some (sub) fields, more understanding is needed. In others, less. But there is an old saying &quot;Don't let the perfect be the enemy of the good.&quot; (Usually attributed to Voltaire.)
If you are good enough to recognize problems with lines of thought and you have insight into the underlying structure of the field, then trust your judgement. If you smell a skunk, delve deeper, otherwise, move on. It isn't a perfect defense against error, but.... Voltaire, again.