A book written by Paul Halmos, I’ve very much enjoyed it and have found the following quotes particularly inspiring.
Learning by examples
By now, having been a student for over 60 years, registered in schools of various sorts (from grade school through Ph.D.) for 16 of them, I think I know how to study. After prelims I didn’t, not yet, but I began to grope toward finding a method that would be right for me. If I had to describe my conclusion in one word, I’d say examples. They are, to me, of paramount importance. Every time I learn a new concept (free group, or pseudodifferential operator, or paracompact space), I look for examples -and, of course, non-examples. The examples should include, whenever possible, the typical ones and the extreme degenerate ones.
Inductive vs Deductive
Mathematics is not a deductive science-that’s a cliche. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does, but it is different in its degree of precision and information. Possibly philosophers would look on us mathematicians the same way as we look on the technicians, if they dared.
My greatest strength as a mathematician is the ability to see when two things are the “same”. When, for instance, I kept puzzling over David Berg’s theorem (normal equals diagonal plus compact), the insight came when I noticed that his mess resembled the proof that every compactum is a continuous image of the Cantor set. From then on it did not need great inspiration to use the classical statement, rather than its proof, and the outcome was a new and perspicuous way of getting Berg’s result.
I could cite many instances of the same kind of thing. Some of the most striking ones occur in duality theories. Examples: the study of compact abelian groups is the same as the study of Fourier series, and the study of Boolean algebras is the same as the study of totally disconnected compact Hausdorff spaces. Other examples, not of the duality kind: the classical method of successive approximations is the same as Banach’s fixed point theorem, and probability is the same as measure theory.
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That kind of insight makes mathematics clean; it trims off the fat and shows what’s really going on. Does it advance mathematics? Are the great new ideas really just recognitions that two things are the same? I often think so-but I am not always sure.
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