What Is Mathematics, Really?

Since the time of Plato, a major area of discussion within mathematics has been over whether mathematicians create or discover new mathematics. Those who favor the discovery side believe that mathematical objects and concepts already exist in some abstract book of knowledge and the discoverer simply turned to the right page in the book. On the other side are people who believe that the mathematical objects and concepts have no independent pre-existence and appear for the first time in any form when a mathematician expresses them. Of course, with any two widely disparate positions, there are many people who take a middle ground. Some have invoked the "mind of God" as the location of the pre-existence, yet invocation of a deity is not necessary to argue the pre-existence position. Hersh puts forward complete descriptions of both these positions and then lists the mathematical principals who have put forward arguments on one side or the other. It is a list of most of the significant figures in the history of mathematics, which is an indication of how dynamic the field has been. Some of the major discoveries in mathematics have been counter-intuitive and led to alterations in the very foundations of mathematics. This is not a book that can easily be digested by the non-mathematician. To understand how significant a new discovery was, it is necessary to have a solid grasp of the mathematics. For example, the consequences and significance of Georg Cantor's discovery that there are different levels of infinity cannot be understood without knowing how mathematicians struggled with the concept for centuries. This book would make an excellent text for an upper-level course in the philosophy of mathematics and is engaging reading for all practicing mathematicians. It does all people working in any field good to take time out on occasion to study exactly what that field is and how it relates to the world.

First, I need to disclose that I'm not a mathematician or a philosopher. I'm a lawyer with an interest in jurisprudence (philosophy of law) and the nature of legal reasoning. "What Is Mathematics, Really?" is one of the most though-provoking books I've ever read. It has helped me to make progress on jurisprudential problems that I had formerly been attacking in largely fruitless ways. The book thus filled a particular need for me. But I think anyone interested in intellectual history or the placing of math in context with other fields will find this book fascinating.

It is a very good book.The scope of this book all inclusive and philosophical ideas are very well described and put in perspective especially on foundations of mathematics.Plus,a very clear exposition.Highly recomended.Dr.A.Gelman

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.

If you are very smart and enjoy thinking at the leading edge, this book can help you do that even better. Hersh, a mathematician and math teacher for 30 years, takes on the hard problems of what is knowledge and where does it come from, comparing the Platonist, formalist, structuralist, and humanist views. The work parallels what Kuhn has done for the philosophy of science. In particular he does a good job of showing why Platonism -- the commonly accepted view that math (read truth) exists a-priori and we merely "discover" it -- really does not hold up. Curiously, I found this deeply liberating, as it opens up much more breathing space for original thought, highlighting the role of the mathematician (thinker) as creator, as much as discoverer. "Must" reading for anyone who considers himself or herself at or near the genius bracket. I scaled my rating back to "9" because as the book acknowledges, it is the pioneer of a new genre (philosophy of mathematics) and hence it cannot deliver (on the first try) a complete answer. But what it does deliver is an extremely useful beginning.