What Is Mathematics, Really?

Rating:
Summary: Really philosophy of mathematics
Review: 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.

Rating:
Summary: A choppy rough draft in philosophy of mathematics
Review: 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.

Rating:
Summary: At times full of empty rhetoric
Review: This book is interesting, and has made me think hard about my own views. Mostly that is because I disagree with a lot of the beliefs touted by Hersh.

Hersh starts out with an approach to the hypercube, always a fascinating topic for me. Pages later he is trying, very unsuccessfully though he doesn't realize, to decimate the "old fashioned" views about numbers, physical objects, social conventions, and basically everything Intelligible. His arguments are so terrible they basically ruin the book. He uses every form of slimy rhetoric to be convincing. Read these pages with an open eye and you'll see how often he plays on the reader's biases and preconceptions to make his point seem clear.

For the rest, the book is quite inclusive. A lot of this is interesting. But all of these ideas have existed without the help of Hersh, who would seem to have problem accepting this fact unless we reduced the timespan of their existence to include Hersh's closest biological relatives, the rest of us humans.