For reasons that I cannot now reconstruct, a few moments ago the phrase "Sperner's Lemma" popped into my head. Thirty-nine years ago, while working on the lectures I gave in a graduate course called "The Use and Abuse of Formal Methods in Political Philosophy" [out of which came my book on Rawls and my article on Nozick, as well as my tutorial on a blog two years ago devoted to the subject], I undertook to master and then to teach a formal proof of the Fundamental Theorem of Game Theory, due to John van Neumann, which states that every two person zero sum game with mixed strategies has a solution. The key move in that proof, at least in the form in which it is given in Luce and Raiffa's Games and Decisions, is an appeal to the famous Fixed Point Theorem of L. E. J. Brouwer. [The proof given by Luce and Raiffa may actually be due to Nash. I am not sure now.] Back then, I located and mastered a proof of the fixed point theorem in a math book [there are many such proofs] which used a theorem due to Kakutani, in the course of which there is an appeal to Sperner's Lemma. I actually expounded the entire proof of von Neumann's theorem, with the proof of the Fixed Point Theorem, in my course. Lord knows what the students made of it all.
My curiosity piqued by the idle thought, I did what any normal red-blooded American boy would do: I looked on Wikipedia. There, sure enough, was a lovely article about the Fixed Point Theorem and another even lovelier article about Sperner's Lemma. Fully understanding what I found in those two articles is, alas, beyond me. Which got me thinking, as I often have, that one of the many things I regret is that I did not study more math. That and my embarrassing inability to master foreign languages are my two intellectual deficits, I feel [others, of course, may have a longer list of my failings.]
As I noted on this blog some long while ago, My grandniece Emily is now making a serious study of Mathematics, a fact that gives me enormous vicarious pleasure. Go Emily!
Do you feel knowledge of more math might have increased your contributions to philosophy?
ReplyDeleteThat is difficult to say, but I suspect the answer is yes -- assuming that I have made any contributions to philosophy at all! Certainly what math I did learn was indispensable in my study both of rational choice theory, etc. and in my engagement with the economic theories of Marx. But aside from all that, it is so beautiful! Not having a grasp of mathematics is like having a tin ear for music.
ReplyDeleteSo was Marx brilliant at math like everything else?
ReplyDeleteNo, inm fact, but he had astonishingly good intuitions about formal strustures, even though he appears to have known very little formal math.
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