The best book I have read of it's type. Seemlessly incorporating Mathematics, Philosophy and History. Makes one want to really read everything in the Bibliography.

It was about five years ago. Physics suddenly seemed fascinating but I was struggling with math. My tutor suggested two books for me. One of them was this book. I cannot say this book was particularly helpful but it gave me a good sense of what mathematics is: its people, culture, history, and philosophy. Quite unlike E.T. Bell's Men of Mathematics, this book does not contain romantically presented stories of some math heros. And unlike some popular math books by Ian Stewart, it does not attempt to explain (rather unsuccessfully) some esoteric theories. It is just as the title suggest--what a mathematical experience can be. A book of this kind is rare.

P.S. Now, some five years later, I am not sure if mathematical knowledge maintains a separte existence as Plato had thought, and as the authors believe. (Ref. Plato, Phaedo)

This book was a sheer joy to read and digest. The authors skilfully comingle history, mathematics, philosophy, and biography. The result is a truly fantastic voyage into the meaning and gist of discovery and conjecture. In chapter after chapter important ideas like Fourier analysis, Non-Cantorian Set Theory, and Objects and Structures are scrutinized in a very interesting manner.

The deeper you go into the book the more will you revel in the sheer majesty and scope of the topics. I had to read the chapter on Inner Issues twice to really get everything out of the text. Topics such as Teaching and Learning are very insightful and full of little hidden gems.

If you are prepared to expend some effort and if you wish to know what mathematics "really is like", grab this book. I am sure this will become a permanent treasure in your library and you will peruse it often long into the night.

Sea Shapes allows children to learn about shapes in their environment, creatures of the sea, and how to draw them. Bright colors and simple shapes help to make learning fun.

This book is another in the "Teachers First, Technically Proficient Last" genre. Hersh has written a Political Manifesto of the radical socialistic, liberal, "modern" teacher's misconception of what it means to teach. He is more interested in "cute" partitionings of (genuinely brilliant) philosophers of mathematics and mathematicians into "right" and "left" (the word "wing" left for the reader to insert), than he is in explaining mathematics (Really). Any discussions of mathematics are so trivial that he only makes a couple of basic errors (the most gross being employment of Platonic concepts); the rest is simply an expression of his apparent resentment at things being what they are rather than what he happens to want them to be. This whole book could have been written with one sentence; Mathematics is a social enterprise,and that's not even correct. Mathematics is a human enterprise (usually best done individually). The book will NOT explain what mathematics is really or otherwise. In its alleged field it is akin to Wilson's Consilience and Shapiro's Philosophy of Mathematics and even more unevenly written.

This book comes across as some kind of extended constructivist/pragmatist complaint. Disjointed in its execution, it gives the appearance of a bunch of lectures too-quickly thrown together. Some weak arguments appear here and there, a few even coming across as downright silly. Perhaps its because Hersh has a simplistic, even at times sophomoric understanding of philosophy. He also has the lazy-man's habit of quoting huge tracts of other peoples writings without giving any sort of application or interpretation. On the up side, the book does have an encyclopedic breadth, so it's not a complete waste of time, even given its weaknesses. I took down several references. Did I like the book? Yes. Hersh should have retained an editor, or perhaps spent another year tidying it up. One more thing: Hersh is very anti-theistic. He downgrades Platonism on the basis that nobody believes in God anymore. He really should get out more, or at least read some sociology. The vast majority of the human race and even westerners believe in God. Hence, maybe Platonism in mathematics isn't so crazy after all.

The book offers the best kind of live, seriously thought out, philosophy of mathematics--in real contact with mathematical practice and teaching. Hersh writes from a deep love of mathematics and a deep concern to make it accessible to others, and for him both of those motivate philosophic reflection on the nature of mathematics.

Hersh notes that mathematics is a social enterprise. People may pursue it alone in their rooms, and even do the greatest thinking that way (as Andrew Wiles did some great thinking in near secrecy on the way to proving the Fermat theorem). But what they think about is not their sole creation (witness the many enthusiastic citations Wiles gives to what he owes others). What we call "proofs" in actual practice are not complete deductions in formal logic, nor simply "whatever persuades you". They are reasonings that live up to a socially recognized standard.

Hersh believes, and argues, that students who understand the social nature of mathematics will approach it with more interest and less fear than those who think it is inhuman perfection. Actually, I think he is wrong about that. Students today generally believe literature is a social product, but they still too often think that "getting it" is an arcane and uninteresting skill of English teachers. But Hersh's view deserves careful consideration and you can learn from him whether you agree in the end or not.

I will also say that Hersh's descriptions of earlier philosophies of mathematics are not always historically very accurate. And though he has genuine concern to give sympathetic accounts of them (before giving his own refutation) he does not always succeed. But neither are his versions notably worse than the versions in other similar books. For accurate accounts of Plato or the 20th century giants Poincare, Hilbert, Brouwer, and so on, you have just got to read the originals.

Anyone interested in philosophic thought about math, and not just solutions to one or another specific technical problem in the philosophy of math, should read this book. But don't only read this one.