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Philosophies of Mathematics

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This 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.

Eschewing interesting anecdotal tidbits, this short book aims for the heart of the principal controversies over the foundations of mathematics. The reader is given the basic logic and mathematics needed to understand the main points of logicism, Zermelo-Fraenkel set theory, intuitionism, finitism and Godel's incompleteness theorems. The chapters are well-written and lucid. You will doubtless pick up something of value if you merely read the book. You will gain more if you study it and do some of the exercises (which do not come with an answer section).

Most philosophy of mathematics survey books give a general account of various theories and programs without actually getting into technical details or doing some math. What distinguishes this book is it actually gets into the technical details. Instead of just saying "logicists derived math from logical principles" it shows how this was attempted. The same goes for deriving mathematics from set theory, and deriving it from intutuionistic logic, etc. This is not a breezy read and needs to be studied closely, but for someone who wants more substance after reading some non-detailed introductory accounts of philosophy of mathematics, it's an excellent book.

If you want to become or are a practicing Mathematician and want to get a acquainted with what is really going on underneath the naive and mechanical side of Mathematics, do yourself a favor an read/study this book. In some 200 pages the authors give you so much content!!! I know this will be one of the books I keep reading and which content I will keep thinking about, so I heavily earmarked and annotated it.
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The only two issues I have with this book after my first thorough read are:
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1) what I believe to be a mistake on page 136 while presenting -choice sequences- it is stated on the 3rd paragraph : "It might help to think of choice sequences as a sequence of rational numbers that is generated by someone else"; now look closely at the definition of Cauchy sequences (32) on page 72. Is it safe to assume, that the totality of real numbers can be covered by an infinite, converging sequence of -rational- numbers?
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2) on page 148 in which the authors mentioned David Hilbert's somewhat semiotic view of the foundations of Mathematics "In the beginning was the sign" ... they say that "Much scholarship has been devoted to fine questions regarding this intuition", but then they end their reference to it by saying "we will not delve to deeply into these matters here". Why not giving the interested readers some more information about where all the scholarship devoted to these fine questions can be found?
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Please, if you know better about any of these questions , I am asking for clarification/help!
Thank you