TITLE: Request for examples: verifying vs understanding proofs
QUESTION [54 upvotes]: My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an important difference between merely verifying that a proof is correct and really understanding it. Bourbaki put it as follows:

[E]very mathematician knows that a proof has not really been “understood” if one has done nothing more than verifying step by step the correctness of the deductions of which it is composed, and has not tried to gain a clear insight into the ideas which have led to the construction of this particular chain of deductions in preference to every other one.
[Bourbaki, ‘The Architecture of Mathematics’, 1950, p.223]

We are interested in examples which, from the perspective of a professional mathematician, illustrate this phenomenon.  If you have ever experienced this difference between simply verifying a proof and understanding it, we would be interested to know which proof(s) and why you did not understand it (them) in the first place.  We are especially interested in proofs that are no longer than a couple of pages in length.  We would also be very grateful if you could provide some references to the proof(s) in question.
We are sorry if this isn’t the appropriate place to post this, but we were hoping that professional mathematicians on MathOverflow could provide some examples that would help with our research.

REPLY [4 votes]: A good proof is a proof that makes us wiser. If the heart of the proof is a voluminious search or a long string of identities, it is probably a bad proof. If something is so isolated that it is sufficient to get the result popped up on the screen or a computer, then it is probably not worth doing. Wisdom lives in connections. If I have to calculate the first 20 digits of π by hand I certainly become wiser afterwards because I see that these formulas for π that I knew take too much time to produce 20 digits. I will probably devise some algorithms which minimize my effort. But when I get two millions of digits of π from the computer using somebody else's library program I remain as stupid as I was before.

Yuri I. MANIN, Good Proofs are Proofs that Make us Wiser