@MISC{Feferman_inthe, author = {Solomon Feferman}, title = {In the Light of Logic}, year = {} }

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Abstract

Poincaré famously compared the logician’s understanding of mathematics to the understanding we would have of chess if we were only to know its rules. ”To understand the game, ” Poincaré wrote, ”is wholly another matter; it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of moves a sort of organized whole. ” [P, pp. 217-218] The Dutch mathematician L.E.J. Brouwer took a position similar to Poincaré’s: genuinely mathematical reasoning is not simply a matter of logical inference. It is, as Poincaré put it, a matter of mathematical insight. Despite those views concerning logic, Poincaré and Brouwer believed that the foundations of mathematics ought to be studied, and indeed carried out fundamental work in this area. This might strike contemporary ears as a bit odd, but it is a consistent view. Mathematical logic and the foundations of mathematics are frequently lumped together, as though they are the same.