The methods of mathematics are the main topic of the course, not a long list of finished mathematical results with such highly polished proofs that t… - Richard Hamming

Assuming you rise to the top, please remember: what made you great may not be appropriate for the next generation.

I must address the topic of whether the effort required for excellence worth it. I believe it is — the chief gain is in the effort to change yourself, in the struggle with yourself, and it is less in the winning than you might expect. Yes, it is nice to end up where you wanted to be, but the person you are when you get there is far more important. I believe a life in which you do not try to extend yourself regularly is not worth living — but it is up to you to pick the goals you believe are worth striving for.

In the face of almost infinite useful knowledge, we have adopted the strategy of "information regeneration rather than information retrieval." ...most importantly, you should be able to generate the result you need even if no one has ever done it before you—you will not be dependent on the past to have done everything you will ever need in mathematics.