In the Light of Logic

by Solomon Feferman Oxford University Press, 1998, 352 pp. $ 60.00 US, ISBN 0195080300 REVIEWED BY ANDREW ARANA 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. They are not. Mathematical logic is a mature mathematical subdiscipline, with its own problems generated by reflecting on what is known from other logic problems and solution attempts. Like any mature mathematical subdiscipline, what counts as a good problem is largely determined by factors ‘internal’ to the subdiscipline, such as how the problem contributes to other work in progress and to what is already known. Foundations of mathematics, on the other hand, has a different standard. It raises questions about the objects and structures of mathematics: what are they, and how do we know anything about them? It raises questions about mathematical statements: how should we go about discovering and justifying them? It raises questions about mathematical proofs: what is a proof, what kinds of proofs do we prefer, and for what reasons? Foundations of mathematics is therefore not a mathematical subdiscipline at all, but rather a body of reflections on mathematics itself. A striking insight reached by David Hilbert and others in the early twentieth century was that the foundations of mathematics could be studied by the application of mathematical logic. By taking mathematical objects and structures to be described by axioms in formal languages, these axioms and