Gian-Carlo Rota composed these thoughts, in 1996:
Ten Lessons I wish I had been Taught
…I have been collecting some random bits of advice that I keep repeating to myself, do’s and don’ts of which I have been and will always be guilty. Some of you have been exposed to one or more of these tidbits. Collecting these items and presenting them in one speech may be one of the less obnoxious among options of equal presumptuousness. The advice we give others is the advice that we ourselves need. Since it is too late for me to learn these lessons, I will discharge my unfulfilled duty by dishing them out to you. They will be stated in order of increasing controversiality.
- Lecturing
- Blackboard Technique
- Publish the same results several times. … The mathematical community is split into small groups, each one with its own customs, notation and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation, and who will rightly claim it as his own.
- You are more likely to be remembered by your expository work.
- Every mathematician has only a few tricks. A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdos‘ work. You admire contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos’ work could be reduced to a few tricks which Erdos repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. 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!
- Do not worry about your mistakes.
- Use the Feynmann method.
- Give lavish acknowledgments. I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else. One day, I tried an experiment. After writing a rather long paper, I began to draft a thorough bibliography. On the spur of the moment, I decided to cite a few papers which had nothing whatsoever to do with the content of my paper, to see what might happen. Somewhat to my surprise, I received letters from two of the authors whose papers I believed were irrelevant to my article. Both letters were written in an emotionally charged tone. Each of the authors warmly congratulated me for being the first to acknowledge their contribution to the field.
- Write informative introductions
- Be prepared for old age.