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

Reuben Hersh
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
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Reuben Hersh is a mathematician who trained with Peter Lax (and wrote about him). He has published several research papers in a range of topics, but to most of us he is known as the author (or co-author) of one of several fascinating and/or inspiring and/or intriguing and/or exasperating books: The Mathematical Experience, What is Mathematics, Really?, Loving and Hating Mathematics: Challenging the Myths of Mathematical Life, Descartes’ Dream: The World According to Mathematics. All of them have been read by mathematicians and other mathematical enthusiasts with much interest, earnestness, skepticism, disappointment, and other strong emotions.

Reviewing the reviews of his works, it is clear that Reuben Hersh is not always found to be philosophically sophisticated or comprehensive and conclusive in his coverage of the many points he raises, but one thing is undeniably certain: He raises significant questions about the nature of our profession and the social and cultural dimensions of our mathematical existence. Philosophers may have tried to dismiss his ventures into their territory, but North American philosophy of mathematics is now forever changed. One may even call the new interest in mathematical practice a revolution inspired by the persistence of Hersh and his collaborators. While most mathematicians kept silent for decades about how what philosophers were interested in about mathematics did not always resonate with the sentiments of the actual practitioners of the field, I think I would not be totally incorrect in asserting that it was at least partially Hersh and his collaborators who managed to eventually awaken enough interest in how we, the real living warm-blooded mathematicians, do this thing that we do.

As a mathematician interested in the humanistic dimensions of our discipline, 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. Compare today with fifty years ago, if you were alive then and doing math:

  • the AMS hosts several blogs!
  • the joint math meetings these days regularly feature art exhibits, poetry readings, and standup comedy!
  • we are talking about how mathematicians live!
  • we worry about teaching and learning!
  • the Ideal Mathematician of Hersh and Davis from the 1980s is a joke, and certainly not the norm, for most new members of our professional community!
  • we care about what proof is and how to teach it to our students, we care about the subjectivities in our research communities, and we care about these in public!

Hersh and company (this review is meant to be about Hersh so I will not attempt to make a list of names, but some will know to whom I might be referring to here) managed to make this shift.

In 2014, The American Mathematical Society published a book that presents to the world Hersh’s oeuvre, a collection of most of his articles, essays, and book reviews that reflect his philosophy of mathematics. Hersh once called this a humanistic philosophy, in some newer articles he prefers to call it a pragmatist philosophy. But the main idea remains the same: that mathematics is ours, it is we human mathematicians who create mathematics in the social and cultural and historical contexts we live in. As he himself put it in What is Mathematics Really? in 1997, “mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context.” Mathematics lives somewhere in the realm shared with other social-cultural-historical constructs, such as war, peace, religion; it is part of our common human heritage.

I was delighted to have the chance to review this book for the MAA, which I thought would give me a more complete view on the man and his works. But I was also a bit concerned. A collection of papers written at different times stapled together without much stitching would not automatically make a good book. The unavoidable repetition, especially given that the author is making one fundamental argument in all his work, could get tedious. Sections could feel uncomfortable, disconnected, random.

However as I dove into the book, I found myself fascinated by its riches, and surprised that this all worked so well. Granted, I was already a convert before starting this volume, and I indeed agree with most of what Hersh is saying, but I think most readers would agree that the sequencing of the articles actually worked. Kudos to the author and the editors who must have thought carefully about how to do this! I also did not get annoyed by the repetitions, which were surprisingly fewer than I had expected,. When they occurred, they usually added unexpected nuances while reinforcing a point. And of course, the foundational argument is not as simplistic as one may first think, so the many dimensions of the issue come up in different ways and are treated differently in each essay. You might even observe the author solidifying and clarifying his own stance as years go by, if you are into that kind of historical analysis. But in any case, many will enjoy getting a complete overview of Hersh’s argument.

The book is organized into four parts, not including the preamble material. The preamble material itself is nontrivial, it includes a reprint of “The Ideal Mathematician” and an article published earlier in Synthèse, “Mathematics has a Front and Back,” to set the stage for what is to come in the first part. Hersh offers a self-introduction and includes here a manifesto to summarize his philosophy for the innocent. He points out which of the following articles will make which parts of the argument and the interested reader can refer back to the manifesto along the way, though this is definitely not a requirement to follow the thread.

Part I, “Mostly for the right hand,” is the main chunk of the book, where Hersh argues explicitly and coherently about his philosophy of mathematics. His disregard for foundational questions and his lack of enthusiasm for formal proof may turn off some readers, but the overall argument is clear. I think if it is read without prejudice will allow one to get what Hersh has been trying to say all these years, though possibly not to the complete satisfaction of everyone. This first part may indeed be viewed as an excellent compilation for anyone interested in learning more about the philosophy of mathematical practice. Philosophers should take notes. Sure, read philosophers writing about this too, but here is the mathematician telling you of his craft. Perhaps his words, his arguments, when taken as a whole (though presented in bits and pieces in bite-size articles and essays), are warranted.

I know not all mathematicians agree with the way Hersh represents our discipline. Many thoughtful mathematicians uphold platonic views (and mostly a lot more nuanced than the version sometimes caricatured by Hersh and company). But Hersh’s perspective is quite illuminating, and can be so even for the Platonist. He tries sincerely to reconcile what we know of mathematics and what we know of ourselves with what is out there, and I bet that at least parts of his argument will ring true, even to the Platonist. In particular his discussions of where mathematics lives, of how humans get to interact with this thing outside of their individual existence, are quite good and would be relevant to any Platonist.

Part 2, “mostly for the left hand,” is a lot more eclectic in the flavors it offers its reader. To give you a sense of the diversity of material here, I can tell you that in this section, you will learn about Paul Cohen’s forcing argument to prove the independence of the Continuum Hypothesis from the Zermelo-Frenkel axioms (“Paul Cohen and Forcing in 1963”) right after having read about the original underrepresentation and the following overrepresentation of Jews in US mathematics (“Under-represented Then Over-represented: A Memoir of Jews in American Mathematics”). So if you only care to read a coherent collection of essays, you should stop at this point. But if you do stop, you will miss out on some interesting essays; besides the ones mentioned above, I think most readers would in particular enjoy reading and thinking about Hersh’s two essays on ethics.

Part 3, Selected Book Reviews, mostly complements the first part, and most of the reviews reprinted here add to our understanding of Hersh’s overall philosophy of mathematics. As someone who enjoys reading and writing book reviews, I was thrilled to read each of these; Reuben Hersh is nothing if not opinionated.

The last part, Part 4 is About the Author, and ogives us an annotated research bibliography of Hersh, as well as a complete curriculum vitae.

After reading all this, 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. Throughout my adult years, I have only read one other book I took more notes on, and that was a three-volume encyclopedia on mathematics and society which I reviewed for the College Mathematics Journal: Encyclopedia of Mathematics and Society, by Sarah J. Greenwald and Jill E. Thomley (College Mathematics Journal, 44  (September 2013) 332–335). 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.

Gizem Karaali is associate professor of mathematics at Pomona College and a founding editor of the Journal of Humanistic Mathematics, which has so far published two essays by Reuben Hersh. Since May 2013, she is also the associate editor of the Mathematical Intelligencer.

  • Overture
  • The ideal mathematician (with Philip J. Davis)
  • Manifesto
  • Self-introduction
  • Mathematics has a front and a back
  • Chronology
  • References

Mostly for the right hand

  • Introduction to part 1
  • True facts about imaginary objects
  • Mathematical intuition (Poincaré, Polya, Dewey)
  • To establish new mathematics, we use our mental models and build on established mathematics
  • How mathematicians convince each other or "The kingdom of math is within you"
  • On the interdisciplinary study of mathematical practice, with a real live case study
  • Wings, not foundations!
  • Inner vision, outer truth
  • Mathematical practice as a scientific problem
  • Proving is convincing and explaining
  • Fresh breezes in the philosophy of mathematics
  • Definition of mathematics
  • Introduction to "18 unconventional essays on the nature of mathematics"

Mostly for the left hand

  • Introduction to part 2
  • Rhetoric and mathematics (with Philip J. Davis)
  • Math lingo vs. plain English: Double entendre
  • Independent thinking
  • The "origin" of geometry
  • The wedding
  • Mathematics and ethics
  • Ethics for mathematicians
  • Under-represented, then over-represented: A memoir of Jews in American mathematics
  • Paul Cohen and forcing in 1963

Selected book reviews

  • Introduction to part 3
  • Review of Not exactly ... in praise of vagueness by Kees van Deemter
  • Review of How mathematicians think by William Byers
  • Review of The mathematician's brain by David Ruelle
  • Review of Perfect rigor by Masha Gessen
  • Review of Letters to a young mathematician by Ian Stewart
  • Review of Number and numbers by Alain Badiou
  • An amusing elementary example
  • Annotated research bibliography
  • Curriculum vitae
  • List of articles