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Introducing Philosophy of Mathematics

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The philosophy majors may find the level of the book elementary for their taste and sophistication. It also may be the case that this feeling is somewhat provoked further by the author's very honest struggle to make the exposition as much clear as possible. After all the title of the book claims only to "introduce" the philosophy of mathematics and does it neatly and cleanly. It puts very judicious pro and counter argumentations for each theory of mathematics (and it covers almost all of the standard theories) which is first presented cleanly and seriously enough. In this way you can genuinely understand, for example, how and why so much fatal was Russel's blow to Frege's so laboriously established logical foundations. Is this that much elementary?

Being understandable should be rather a merit of a book as this, and as such it deserves all commends and praising.

In her Preface, the author says: "This book is intended as an upper-level undergraduate text or a lower-level graduate text for students of the philosophy of mathematics." (ix) She says in her Guide to Further Reading that books to "consult" as a next step after reading hers are Stephen Körner's 1962 The Philosophy of Mathematics, Stewart Shapiro's 2000 Thinking About Mathematics, and James Brown's 1999 Philosophy of Mathematics. (191)

The author is a clear and careful writer, but the elementary level of exposition limits the satisfaction of reading. It is a common failing. As Saunders Mac Lane wrote in his 1986 Mathematics: Form and Function: "The various schools on the foundations [of mathematics] have correspondingly various attempts to answer these questions, none of them generally convincing . Often - especially in the work of philosophers - they are anchored almost exclusively in the most elementary parts of Mathematics - numbers and continuity. Much more substantive material is at hand." (4) Speaking of Mac Lane, there is no discussion of Category Theory in Introducing Philosophy of Mathematics. The book is, however, a clear introduction to its subject, albeit elementary. Readers with little mathematical sophistication may feel they're involved in some significant way with mathematics itself. It is a peril in philosophy, that students may presume to be equipped preternaturally to render analysis and clarification upon subjects they know little about. Grade school arithmetic and acquaintance with Zeno's paradoxes along with having overheard that non-euclidean geometry is useful in physics is not preparation enough to form a philosophy of mathematics. It is more than enough, however, to read this book.

Brief Contents:
Acknowledgements
Preface
1. Infinity
2. Mathematical Platonism and realism
3. Logicism
4. Structuralism
5. Constructivism
6. A pot-pourri of philosophies of mathematics
Appendix: Proof ex falso quod libet
Glossary
Notes
Guide to further reading
Bibliography
Index