First, some quick background: I'm a recent computer science grad planning on grad school in the near future, probably in artificial intelligence, and since I feel my mathematical foundations are lacking, I've decided to embark on a self-study project to fill in my (substantial) gaps. I mention this to illustrate three things: First, I have enough mathematical maturity to understand what proofs are, how to build them, and their importance from a theoretical standpoint. Second, for me, math is just a tool. Third, time is of the essence.
With this in mind, on to the question. In the course of studying the various texts I plan to study, it seems I have a few options when encountering a theorem with a proof:
- Learn the theorem statement and skip the proof entirely.
- Study the proof only until I accept it as valid.
- Study the proof until I understand it intuitively, and can "see" why it works.
- Do option #3, but revisit the proof over time until the intuitive understanding is cemented in memory.
Going down the list, each path grants more understanding but at the cost of more time, so given my constraints, it seems #1 would be best. And in any case, as I understand it, the whole purpose of a theorem is to be an intellectual shortcut, so that when a certain set of conditions hold you can draw a conclusion without worrying about all the details in between. However, I can't shake the feeling that if I go this route, I'll regret it down the road.
So, finally, my question is, which of the four paths should I take, assuming any of them? Moreover, how should the approach change depending on how "central" the theorem is or the nature of the proof?