2014;
291 pp;
Softcover

Print ISBN: 978-0-8218-9420-0

Product Code: MBK/83

List Price: $39.00

Individual Member Price: $31.20

**Electronic ISBN: 978-1-4704-1412-2
Product Code: MBK/83.E**

List Price: $39.00

Individual Member Price: $31.20

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#### Supplemental Materials

# Experiencing Mathematics: What do we do, when we do mathematics?

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*Reuben Hersh*

The question “What am I doing?” haunts many
creative people, researchers, and teachers. Mathematics, poetry, and
philosophy can look from the outside sometimes as ballet

—Yuri Manin, Max Planck Institute, Bonn, Germany

What happens when mid-career a mathematician
unexpectedly becomes philosophical? These lively and eloquent essays address
the questions that arise from a crisis of reflectiveness: What is a
mathematical proof and why does it come after, not before, mathematical
revelation? Can mathematics be both real and a human artifact? Do
mathematicians produce eternal truths, or are the judgments of the mathematical
community quasi-empirical and historically framed? How can we be sure that an
infinite series that seems to converge really does converge?

This collection of essays by Reuben Hersh makes an important
contribution. His lively and eloquent essays bring the reality of mathematical
research to the page. He argues that the search for foundations is misleading,
and that philosophers should shift from focusing narrowly on the deductive
structure of proof, to tracing the broader forms of quasi-empirical reasoning
that star the history of mathematics, as well as examining the nature of
mathematical communities and how and why their collective judgments evolve from
one generation to the next. If these questions keep you up at night, then you
should read this book. And if they don't, then you should read this book
anyway, because afterwards, they will!

—Emily Grosholz, Department of Philosophy, Penn
State, Pennsylvania, USA

Most mathematicians, when asked about the
nature and meaning of mathematics, vacillate between the two
unrealistic poles of Platonism and formalism. By looking carefully at
what mathematicians really do when they are doing mathematics, Reuben
Hersh offers an escape from this trap. This book of selected articles
and essays provides an honest, coherent, and clearly understandable
account of mathematicians' proof as it really is, and of the existence
and reality of mathematical entities. It follows in the footsteps of
Poincaré, Hadamard, and Polya. The pragmatism of John Dewey is a
better fit for mathematical practice than the dominant “analytic
philosophy”. Dialogue, satire, and fantasy enliven the philosophical
and methodological analysis.

Reuben Hersh has written extensively on mathematics, often from the
point of view of a philosopher of science. His book with Philip Davis,
The Mathematical Experience, won the National Book Award in
science. Hersh is emeritus professor of mathematics at the University
of New Mexico.

#### Table of Contents

# Table of Contents

## Experiencing Mathematics: What do we do, when we do mathematics?

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Permissions and acknowledgments xiii14 free
- Acknowledgments xvii18 free
- Overture 120 free
- The ideal mathematician (with Philip J. Davis) 524
- Manifesto 1332
- Self-introduction 1736
- Chronology 2140
- Mathematics has a front and a back 3554
- Part I. Mostly for the right hand 4160
- Introduction to part 1 4362
- True facts about imaginary objects 4766
- Mathematical intuition (Poincaré, Polya, Dewey) 5170
- To establish new mathematics, we use our mental models and build on established mathematics 7392
- How mathematicians convince each other or “The kingdom of math is within you” 89108
- On the interdisciplinary study of mathematical practice, with a real live case study 115134
- Wings, not foundations! 125144
- Inner vision, outer truth 131150
- Mathematical practice as a scientific problem 137156
- Proving is convincing and explaining 147166
- Fresh breezes in the philosophy of mathematics 157176
- Definition of mathematics 163182
- Introduction to "18 unconventional essays on the nature of mathematics" 167186
- Part II. Mostly for the left hand 173192
- Introduction to part 2 175194
- Rhetoric and mathematics (with Philip J. Davis) 177196
- Math lingo vs. plain English: Double entendre 191210
- Independent thinking 195214
- The “origin” of geometry 199218
- The wedding 205224
- Mathematics and ethics 207226
- Ethics for mathematicians 213232
- Under-represented, then over-represented: A memoir of Jews in American mathematics 217236
- Paul Cohen and forcing in 1963 227246
- Part III. Selected book reviews 233252
- Introduction to part 3 235254
- Review of Not exactly ... in praise of vagueness by Kees van Deemter 237256
- Review of How mathematicians think by William Byers 241260
- Review of The mathematician’s brain by David Ruelle 247266
- Review of Perfect rigor by Masha Gessen 251270
- Review of Letters to a young mathematician by Ian Stewart 255274
- Review of Number and numbers by Alain Badiou 257276
- Part IV. About the author 263282
- An amusing elementary example 265284
- Annotated research bibliography 267286
- Curriculum vitae 271290
- List of articles 273292
- Index 279298 free
- Back Cover Back Cover1311

#### Readership

The book is of interest to everyone who wonders what math really is, whether they are students, teachers, mathematicians, philosophers, or otherwise.

#### Reviews

I view Hersh as a hero: not a perfect idol, but more a real-life hero who through stubborn hard work and prolific writing has made a difference in the types of conversations we are having now about ourselves and the ways we relate to the world. ... [A]s I dove into the book, I found myself fascinated by its riches, and surprised that [it] worked so well. ... I think most readers would agree that the sequencing of the articles actually worked ... many will enjoy getting a complete overview of Hersh's argument. ... I remain enthralled by Hersh's ideas and impressed by his persistent defense of his controversial but significant perspective. I believe the American Mathematical Society has done a service to the mathematical community by putting together this collection. ... Reuben Hersh's collection is full of provocative ideas, offering perspectives on our profession that may help us understand better ourselves and our craft and even to teach our students better. This volume will remain on my easy-to-reach shelf for a long time to come.

-- MAA Reviews

... I found the author's arguments powerful and compelling, and conveyed with great clarity and concision. ... It is refreshing to occasionally step back and talk about mathematics rather than doing it, and this book provides solid rhetorical ammunition.

-- LMS Newsletter