Philosophies of Mathematics Review

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Philosophies of Mathematics ReviewThis is a splendid book, teeming with virtues. But along with its many merits there is one drawback, which I'll mention later. First, the good stuff. Philosophies of Mathematics is primarily a textbook, designed to introduce logicism, intuitionism, and Hilbert's formalism to beginners at a moderately high level. It does this exceptionally well. The expositions are crisp and clear from start to finish. The authors say their book is accessible to anyone with a bit of elementary logic. Well, maybe. Those familiar with mathematics texts know the expressions `self-contained, but assumes a level of mathematical maturity.' I suspect that something like that goes for this book; philosophy undergraduates with no mathematics might be overwhelmed. But for the slightly more advanced reader (with, say, basic logic and one or two university mathematics courses), this book is the perfect introduction to the topics it covers.
What does it cover? The classic period in the philosophy of mathematics, namely, Logicism, Intuitionism, and Hilbert's formalism, here called Finitism. This, of course, is not particularly unusual. What is novel, and what makes the book such a valuable introduction, are three meaty, technical chapters on set theory, intuitionistic mathematics, and Gödel's theorem. Thus, after a standard discussion of the logicism of Frege and Russell showing how set theory develops out of logic, there is a very nice (though somewhat compact) presentation of basic set theory. The standard axioms of ZF are given, all the basic operations are defined, then it is shown how the real numbers (in the form of Cauchy sequences) can be derived. The discussion includes Cantor's theorem and the hierarchy of infinite numbers. This is done in forty-odd pages. There is enough material and in enough detail for a reader to get a feel for the plausibility of a reduction of all of mathematics to set theory. And if set theory can be reduced to logic, then one can get a feel for the scope of the classical logicist programme. I say `classical' because the authors do not take up the current "neo-logicist" programme, except to mention its existence.
The chapter on Intuitionism is devoted to Brouwer but also much influenced by Dummett. There is no mention of Bishop. The discussion even of Brouwer is perhaps a bit thin when it comes to what motivates him philosophically. But even the most ambitious would-be expositors find Brouwer's talk of the perception of time daunting. The emphasis is on provability and they draw the consequences for logic in considerable detail. Following this chapter is another devoted to constructive mathematics wherein George and Velleman prove a number of intuitionistic versions of classical theorems. This is extremely welcome, since most philosophical discussions merely wave their hands when it comes to the mathematical details.
The finitism chapter gives a standard account of Hilbert's formalism. The aims and achievements are outlined and so are the Gödel results that brought Hilbert's programme to a crashing halt. The incompleteness results are then given a chapter of their own. The exposition of Gödel is particularly good, and particularly thorough for a book on the philosophy of mathematics. The necessary technicalities are all here. Detailed proofs are given most of the time, and skipped in favour of a discussion of plausibility only when the technicalities require a great effort for a small payoff in understanding.
There is no other philosophy of mathematics book with extensive exercises. Philosophy of mathematics is a technical subject in itself and it relies on extensive knowledge of other technicalities. The exercises in this book will help to bring students up to speed.
I said there is one drawback. The book deals with the "classical" period in the philosophy of mathematics. There is not a word about current issues. There is no mention of naturalism (Maddy, Kitcher), or structuralism (Hellman, Resnik, Shapiro), or fictionalism (Field); there is no discussion of "experimental" mathematics, of the role of computers, of visualization, or of the interaction of mathematics and the sciences. Excellent though Philosophies of Mathematics is, one cannot get a proper sense from reading it of the field as it exists today. The authors are aware of this and in the preface express the view that `...contemporary work is best evaluated against a backdrop of understanding that encompasses these three great historical attempts to tame the phenomenon of mathematics.' There is no quarrelling with this. It's an excellent introduction to and guide through a golden age in the philosophy of mathematics.Philosophies of Mathematics Overview

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