Written: Dec. 7, 2005.
What is Mathematics? Most mathematicians don't know and don't care. Mathematics is what mathematicians do. This brilliant insight gives rise to Naturalism (e.g. Penelope Maddy) and Humanism (or Social Constructivism(?)) preached by Philip Davis and Reuben Hersh.
Well, both Naturalism and Humanism seem like good ideas, but they pertain to present day mathematics that is still largely done by humans. In fifty years (at most) human mathematicians will be like lamp-lighters and ice-delivery men. All serious math will be done by computers. Let's hope that human philosophy will still survive, but we need to adjust naturalism to the practice of math in the future and to the way it will be done by machines. Of course, we don't know exactly how, so let's put this project of Naturalist mathematical philosophy on hold and wait to see how things turn out in fifty years.
Tim Gowers said that we are all formalists, but most of us don't know it (and if we knew, we wouldn't care). I kind of agree, but this is only a corollary of a more profound truth: EVERYTHING IS COMBINATORICS. Classify Lie Algebras? It is just root systems and Dynkin Diagrams. Finite Groups? The Monster is a Combinatorial Design. Even when it is not obviously combinatorics, it could be made so. If it is too hard for us, then we need a computer! But computer science is all Discrete Math, alias combinatorics. In a way Logic is too. But Logic is so low-level, like machine language. It is much more fun and gratifying to work in Maple, and do higher-level combinatorics.
I am also a trivialist. Of course, I am only following in the footsteps of King Solomon who said
("hevel havalim hakol hevel"), meaning that everything is nonsense, as well as other wise people like Greg Chaitin and Stephen Wolfram, who realized that we humans, and even our computers, can only prove trivial results. Since all knowable math is ipso facto trivial, why bother? So only do fun problems, that you really enjoy doing. It would be a shame to waste our short lives doing "important" math, since whatever you can do, would be done, very soon (if not already) faster and better (and more elegantly!) by computers. So we may just as well enjoy our humble trivial work.
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