Ten Lessons for Mathematicians, according to Gian-Carlo Rota

Ten Lessons I wish I had been Taught (1994), by Gian-Carlo Rota.

7. Use the Feynman method

You have to keep a dozen of your favorite problems... Every time you hear or read a new trick or a new result, test it... to see whether it helps. Every once in a while there will be a hit, and people will say: "How did he do it? He must be a genius!"

This is the best advice. I must immediately think up at least one problem to get my chance of being a GENIUS!

1. TODO

1. Lecturing

An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field.

YES!

try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person's work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.

Good way to actually use lectures for they live-ness. After all, if this live-ness is not used, lectures are worse than recordings.

See also "8. Give lavish acknowledgments"

3. Publish the same result several times, 4. You are more likely to be remembered by your expository work, 9. Write informative introductions

Publish several times to scoop up ALL the credits, and add more examples to make it easy to read, and make sure people read YOUR papers, not the original discoverers'.

5. Every mathematician has only a few tricks

But on reading the proofs of Hilbert's striking and deep theorems in invariant theory, it was surprising to verify that Hilbert's proofs relied on the same few tricks. Even Hilbert had only a few tricks!

6. Do not worry about your mistakes

This is known by all mathematicians, I think. A good mathematician must be able to think robustly, so that mistakes remain local and the global flow of ideas remains on track. A big error should "smell" bad even if you can't see the precise logical problem.

A short-sighted mathematician will never achieve anything more than an automatic proof assistant!