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Philosophies of Mathematics

Philosophies of Mathematics

Alexander L. George, Daniel Velleman

ISBN: 978-0-631-19543-6 December 2001 Wiley-Blackwell 240 Pages


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This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

1. Introduction.

2. Logicism.

3. Set Theory.

4. Intuitionism.

5. Intuitionistic Mathematics.

6. Finitism.

7. The Incompleteness Theorems.

8. Coda.



"George and Velleman manage to accomplish a difficult feat: on the one hand, they explain, clearly and rigorously, a number of highly technical accomplishments of twentieth-century mathematical logic, making plain the relevance of the mathematical work for philosophy; yet, on the other, they presuppose little more from their readers than a first course in basic logic. The examples they choose to explicate their points are carefully selected and illuminating. This is a splendid book." William Ewald, University of Pennsylvania <!--end-->

"This book includes just the right mix of helpful historical exposition and clear, tight philosophical argument. It is extremely well written and does an excellent job of making difficult material accessible. There is nothing else currently available that discusses in a single volume such a wide range of important material. The authors are to be commended for a job well done." Andrew Irvine, University of British Columbia

"This is a well-written, informative and innovative introduction to philosophies of mathematics. It is a very valuable addition to the existing literature." Wilfried Sieg, Carnegie Mellon University

  • Systematically presents both philosophical and mathematical aspects of the three central approaches to mathematics: logicism, intuitionism, and formalism.

  • Requires only that students have a basic familiarity with quantificational logic and very fundamental parts of set theory and analysis.

  • Comprehensive and accessible.