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Publisher:

Oxford University Press

Publication Date:

2014

Number of Pages:

618

Format:

Paperback

Price:

65.00

ISBN:

9780198701514

Category:

Monograph

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Nick Scoville

09/29/2014

This is a collection of over a dozen essays written by the author on the historical development of both mathematical logic and the philosophy of mathematics, covering the beginning of the 20^{th} century to the 1940s, the “golden age” of mathematical logic. The author is clear about his goals; he writes on page vi of the Preface that “The essays … were written with a single major aim, namely that of reaching a deeper understanding of the interaction between developments in mathematical logic and the foundations of mathematics and logic from 1900 to 1940.” The author succeeds not only in achieving his stated goal, but also in achieving the more modest goal of giving a general and understandable overview of the history of mathematical logic and philosophy that can be appreciated and interiorized by the non-specialist. In addition to utilizing some of the more well-known sources, the author has integrated information that was not widely available from various archives around the world. The text is divided into five parts.

Part I is a single essay giving a history of mathematical logic. Here the aim is historical rather than philosophical. Some topics that are covered include the history of the controversy over the axiom of choice, Zermelo’s proof that every set can be well ordered, and the work of Peano, Padoa, and Pieri paving the way for the work of Hilbert.

Part II focuses on mathematical foundations. We see much of the evolution of the positions of Hilbert as well as the influence that Russell had on Hilbert’s work. There is an interesting essay on Wittgenstein and his contribution to the foundations of mathematics. In particular, this essay investigates the claim that Wittgenstein did not know much mathematics. Part II concludes with two essays on Gödel’s contribution to the foundation of mathematics.

Phenomenology is the focus of Part III. The first essay in this section discusses Weyl’s intuitionism, developing the position originally proposed by Brouwer, which resulted from his rejection of the law of the excluded middle. The third essay summarizes a correspondence between Weyl and Becker discussing fundamental questions as to the nature of physics and geometry. Tarski, Quine, and nominalism are the subjects of Part IV, while Tarski’s theory of truth is the topic of Part V.

While the text is primarily historical and philosophical, it does not shy away from explaining the details of the mathematics and the more technical parts of philosophy at times. One familiar with the basics of first order logic, propositional logic, model theory, etc. will most fully appreciate the book, but such knowledge is not necessary as these more technical parts can easily be brushed over without compromising understanding of the relevant points. This book will appeal to specialists as well as people curious about the history of mathematical logic and philosophy. Since the essays are self-contained, one can read any essay without knowledge of other essays, making the book an excellent and readable resource for any mathematician.

(nscoville@ursinus.edu)is an assistant professor of mathematics at Ursinus College. His areas of interest are homotopy theory, discrete topology, and the history of topology. He considers himself an amateur scholastic. His website is at http://webpages.ursinus.edu/nscoville/.

Preface

**Part 1: History of Logic **

1. The Development of Mathematical Logic from Russell to Tarski, 1900-1935

**Oart 2: Foundations of Mathematics **

2. Hilbert and Bernays on Metamathematics

3. Between Russell and Hilbert: Behmann on the foundations of mathematics

4. The Russellian influence on Hilbert and his school

5. On the constructivity of proofs

6. Wittgenstein's constructivization of Euler's proof of the infinitude of primes

7. Between Vienna and Berlin: The immediate reception of Godel's incompleteness theorems

8. Essay Review of Godel's Collected Works (volumes IV and V)

**Part 3: Phenomenology and Mathematics **

9. Hermann Weyl: Predicativity and an intuitionistic excursion

10. Mathematics and Phenomenology: the correspondence between O. Becker and H. Weyl

11. Geometry, Physics and Phenomenology: Four letters of O. Becker to H. Weyl

12. Das Abenteuer der Vernunft: O. Becker and D. Mahnke on the phenomenological foundation of the exact sciences

**Part 4: Nominalism **

13. Harvard 1940-1941: Tarski, Carnap and Quine on a finitist language of mathematics for science

14. Quine and Tarski on nominalism

**Part 5: The emergence of semantics: truth and logical consequence **

15. Neurath and Kokoszynska on the semantic conception of truth

16. Tarski on models and logical consequence

17. Tarski on Categoricity and Completeness: an unpublished lecture from 1940

18. Archival Appendix. "On the completeness and categoricity of deductive theories" (1940), By Alfred Tarski.

Bibliography

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