Turns out I'm writing diary entries less often than graydon these days... I can only half-blame not having time due to the baby, since I am finding time to spend on Orkut (ncm introduced me to it almost four months ago). I just posted a reply to a query for advice on starting study of mathematics from the Orkut Mathematics forum; maybe it has wider appeal:
- Recognise that cultivating an intuition of mathematics and knowing how to give rigorous proofs of mathematical theorems, while related, are quite different kinds of knowledge, and mathematical intuition is more fundamental;
- Learn to judge whether mathematics is good, whether it is powerful or beautiful, and not just whether it is correct;
- The best way you can spend time is find someone with similar tastes in mathematics and spend time regularly with them discussing problems, papers you have read, crazy ideas you have had, and so on. I didn't do this until I started my doctorate; now I regret....
- Leverage what you know: because you are confident with programming, try coding up programs that find witnesses to existential theorems you prove. There's a close analogy between proving theorems and writing programs.
- Don't make the mistake of just choosing computer-relevant mathematics: the core subjects of mathematics complement each other and were you to avoid analysis, say, you would finish your degree with only a skewed and incomplete grounding in mathematics.