18 Unconventional Essays on the Nature of Mathematics

1. Front Matter

   Pages i-xxi

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2. A Socratic Dialogue on Mathematics

   • Alfréd Rényi

   Pages 1-16

3. “Introduction” to Filosofia e matematica

   • Carlo Cellucci

   Pages 17-36

4. On Proof and Progress in Mathematics

   • William P. Thurston

   Pages 37-55

5. The Informal Logic of Mathematical Proof

   • Andrew Aberdein

   Pages 56-70

6. Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology

   • Yehuda Rav

   Pages 71-96

7. Towards a Semiotics of Mathematics

   • Brian Rotman

   Pages 97-127

8. Computers and the Sociology of Mathematical Proof

   • Donald MacKenzie

   Pages 128-146

9. From G.H.H. and Littlewood to XML and Maple: Changing Needs and Expectations in Mathematical Knowledge Management

   • Terry Stanway

   Pages 147-159

10. Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics

    • Rafael Núñez

    Pages 160-181

11. Does Mathematics Need a Philosophy?

    • William Timothy Gowers

    Pages 182-200

12. How and Why Mathematics Is Unique as a Social Practice

    • Jody Azzouni

    Pages 201-219

13. The Pernicious Influence of Mathematics upon Philosophy

    • Gian-Carlo Rota

    Pages 220-230

14. The Pernicious Influence of Mathematics on Science

    • Jack Schwartz

    Pages 231-235

15. What Is Philosophy of Mathematics Looking for?

    • Alfonso C. del Ávila Palacio

    Pages 236-249

16. Concepts and the Mangle of Practice Constructing Quaternions

    • Andrew Pickering

    Pages 250-288

17. Mathematics as Objective Knowledge and as Human Practice

    • Eduard Glas

    Pages 289-303

18. The Locus of Mathematical Reality: An Anthropological Footnote

    • Leslie A. White

    Pages 304-319

19. Inner Vision, Outer Truth

    • Reuben Hersh

    Pages 320-326

This book comes from the Internet. Browsing the Web, I stumbled on philosophers, cognitive scientists, sociologists, computer scientists, even mathematicians!—saying original, provocative things about mathematics. And many of these people had probably never heard of each other! So I have collected them here. This way, they can read each other’s work. I also bring back a few provocative oldies that deserve publicity. The authors are philosophers, mathematicians, a cognitive scientist, an anthropologist, a computer scientist, and a couple of sociologists. (Among the mathematicians are two Fields Prize winners and two Steele Prize w- ners. ) None are historians, I regret to say, but there are two historically o- ented articles. These essays don’t share any common program or ideology. The standard for admission was: Nothing boring! Nothing trite, nothing tr- ial! Every essay is challenging, thought-provoking, and original. Back in the 1970s when I started writing about mathematics (instead of just doing mathematics), I had to complain about the literature. Philosophy of science was already well into its modern revival (largely stimulated by the book of Thomas Kuhn). But philosophy of mathematics still seemed to be mostly foundationist ping-pong, in the ancient style of Rudolf Carnap or Willard Van Ormond Quine. The great exception was Proofs and Refutations by Imre Lakatos. But that exciting book was still virtually unknown and unread, by either mathematicians or philosophers. (I wrote an article en- tled “Introducing Imre Lakatos” in the Mathematical Intelligencer in 1978.

• Counting
• mathematics
• philosophy of mathematics
• physics
• proof

From the reviews:

"This is not a book about foundations or formal logic. It is … about the philosophy of mathematics, but perhaps more about the human practice of mathematics. … This is a terrific collection of essays. … all of the essays give us new insight into that curious thing we do when we do mathematics." (William J. Satzer, MathDL, January, 2006)

"The current book is destined to become a classic as well for numerous reasons which will soon become transparent. … Reuben Hersh is to be commended for orchestrating this masterful collection of essays from distinguished individuals … . I predict that this book will become a classic for the coming generations of mathematics philosophers as well as for mathematics educators interested in changing dominant conceptions of what is mathematics, finally!" (Bharath Sriraman, Zentralblatt für Didaktik der Mathematik, Vol. 38 (4), 2006)

"Editor Reuben Hersh has orchestrated a highly provocative book, particularly for mathematicians, but also for anyone interested in the philosophy and practice of mathematics. … This is a very appealing collection of essays. … I highly recommend this book to those wondering how math is carried out, both mathematicians and laymen. I also recommend it to educators interested in changing the dominant view of math and how to do math." (Hector Zenil, Computing Reviews, January, 2008)

• Department of Mathematics and Statistics, University of New Mexico, Albuquerque, USA

  Reuben Hersh

REUBEN HERSH is professor emeritus at the University of New Mexico, Albuquerque. He is the recipient (with Martin Davis) of the Chauvenet Prize and (with Edgar Lorch) the Ford Prize. Hersh is the author (with Philip J. Davis) of The Mathematical Experience and Descartes' Dream, which won the National Book Award in l983, and What is Mathematics, Really?