An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Experiencing Mathematics: What do we do, when we do mathematics?

Reuben Hersh University of New Mexico, Albuquerque, NM
Available Formats:
Softcover ISBN: 978-0-8218-9420-0
Product Code: MBK/83
List Price: $42.00 MAA Member Price:$37.80
AMS Member Price: $33.60 Electronic ISBN: 978-1-4704-1412-2 Product Code: MBK/83.E List Price:$39.00
MAA Member Price: $35.10 AMS Member Price:$31.20
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $63.00 MAA Member Price:$56.70
AMS Member Price: $50.40 Click above image for expanded view Experiencing Mathematics: What do we do, when we do mathematics? Reuben Hersh University of New Mexico, Albuquerque, NM Available Formats:  Softcover ISBN: 978-0-8218-9420-0 Product Code: MBK/83  List Price:$42.00 MAA Member Price: $37.80 AMS Member Price:$33.60
 Electronic ISBN: 978-1-4704-1412-2 Product Code: MBK/83.E
 List Price: $39.00 MAA Member Price:$35.10 AMS Member Price: $31.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$63.00 MAA Member Price: $56.70 AMS Member Price:$50.40
• Book Details

2014; 291 pp
MSC: Primary 00;

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 en pointe, and at other times as the flight of the bumblebee. Reuben Hersh looks at mathematics from the inside; he collects his papers written over several decades, their edited versions, and new chapters in his book Experiencing Mathematics, which is practical, philosophical, and in some places as intensely personal as Swann's madeleine.

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.

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

• Chapters
• 1. Overture
• 2. The ideal mathematician (with Philip J. Davis)
• 3. Manifesto
• 4. Self-introduction
• 5. Chronology
• 6. Mathematics has a front and a back
• Part 1. Mostly for the right hand
• 7. Introduction to part 1
• 8. True facts about imaginary objects
• 9. Mathematical intuition (Poincaré, Polya, Dewey)
• 10. To establish new mathematics, we use our mental models and build on established mathematics
• 11. How mathematicians convince each other or “The kingdom of math is within you”
• 12. On the interdisciplinary study of mathematical practice, with a real live case study
• 14. Inner vision, outer truth
• 15. Mathematical practice as a scientific problem
• 16. Proving is convincing and explaining
• 17. Fresh breezes in the philosophy of mathematics
• 18. Definition of mathematics
• 19. Introduction to "18 unconventional essays on the nature of mathematics"
• Part 2. Mostly for the left hand
• 20. Introduction to part 2
• 21. Rhetoric and mathematics (with Philip J. Davis)
• 22. Math lingo vs. plain English: Double entendre
• 23. Independent thinking
• 24. The “origin” of geometry
• 25. The wedding
• 26. Mathematics and ethics
• 27. Ethics for mathematicians
• 28. Under-represented, then over-represented: A memoir of Jews in American mathematics
• 29. Paul Cohen and forcing in 1963
• Part 3. Selected book reviews
• 30. Introduction to part 3
• 31. Review of Not exactly ... in praise of vagueness by Kees van Deemter
• 32. Review of How mathematicians think by William Byers
• 33. Review of The mathematician’s brain by David Ruelle
• 34. Review of Perfect rigor by Masha Gessen
• 35. Review of Letters to a young mathematician by Ian Stewart
• 36. Review of Number and numbers by Alain Badiou
• Part 4. About the author
• 37. An amusing elementary example

• 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.

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
2014; 291 pp
MSC: Primary 00;

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 en pointe, and at other times as the flight of the bumblebee. Reuben Hersh looks at mathematics from the inside; he collects his papers written over several decades, their edited versions, and new chapters in his book Experiencing Mathematics, which is practical, philosophical, and in some places as intensely personal as Swann's madeleine.

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.

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

• Chapters
• 1. Overture
• 2. The ideal mathematician (with Philip J. Davis)
• 3. Manifesto
• 4. Self-introduction
• 5. Chronology
• 6. Mathematics has a front and a back
• Part 1. Mostly for the right hand
• 7. Introduction to part 1
• 8. True facts about imaginary objects
• 9. Mathematical intuition (Poincaré, Polya, Dewey)
• 10. To establish new mathematics, we use our mental models and build on established mathematics
• 11. How mathematicians convince each other or “The kingdom of math is within you”
• 12. On the interdisciplinary study of mathematical practice, with a real live case study
• 14. Inner vision, outer truth
• 15. Mathematical practice as a scientific problem
• 16. Proving is convincing and explaining
• 17. Fresh breezes in the philosophy of mathematics
• 18. Definition of mathematics
• 19. Introduction to "18 unconventional essays on the nature of mathematics"
• Part 2. Mostly for the left hand
• 20. Introduction to part 2
• 21. Rhetoric and mathematics (with Philip J. Davis)
• 22. Math lingo vs. plain English: Double entendre
• 23. Independent thinking
• 24. The “origin” of geometry
• 25. The wedding
• 26. Mathematics and ethics
• 27. Ethics for mathematicians
• 28. Under-represented, then over-represented: A memoir of Jews in American mathematics
• 29. Paul Cohen and forcing in 1963
• Part 3. Selected book reviews
• 30. Introduction to part 3
• 31. Review of Not exactly ... in praise of vagueness by Kees van Deemter
• 32. Review of How mathematicians think by William Byers
• 33. Review of The mathematician’s brain by David Ruelle
• 34. Review of Perfect rigor by Masha Gessen
• 35. Review of Letters to a young mathematician by Ian Stewart
• 36. Review of Number and numbers by Alain Badiou
• Part 4. About the author
• 37. An amusing elementary example
• 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.