## Synopsis

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science.

## Excerpt

Forty years ago, as a machinist's helper, with no thought that mathematics could become my life's work, I discovered the classic, *What Is Mathematics?* by Richard Courant and Herbert Robbins. They never answered their question; or rather, they answered it by *showing* what mathematics is, not by telling what it is. After devouring the book with wonder and delight, I was still left asking, "But what is mathematics, really?"

This book offers a radically different, unconventional answer to that question. Repudiating Platonism and formalism, while recognizing the reasons that make them (alternately) seem plausible, I show that *from the viewpoint of philosophy* mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. I call this viewpoint "humanist."

I use "humanism" to include all philosophies that see mathematics as a human activity, a product, and a characteristic of human culture and society. I use "social conceptualism" or "social-cultural-historic" or just "social-historic philosophy" for my specific views, as explained in this book.

This book is a subversive attack on traditional philosophies of mathematics. Its radicalism applies to philosophy of mathematics, not to mathematics itself. Mathematics comes first, then philosophizing about it, not the other way around. In attacking Platonism and formalism and neo-Fregeanism, I'm defending our right to do mathematics as we do. To be frank, this book is written out of love for mathematics and gratitude to its creators.

Of course it's obvious common knowledge that mathematics is a human activity carried out in society and developing historically. These simple observations are usually considered irrelevant to the philosophical question, what is mathematics? But without the social historical context, the problems of the philosophy of . . .