Customer Reviews

What Is Mathematics, Really?

5 star:		(5)
4 star:		(2)
3 star:		(1)
2 star:		(1)
1 star:		(1)

 

 

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.

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.

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.

Brint Montgomery is spot-on calling this "a choppy rough draft," but I'd say s/he gives it one star too many.

Hersh wanders through the landscape of ideas about math without much of a plan: he visits some regions so often the reader loses patience, while wholly ignoring others. He harps to the point of total tedium on his not-very-exciting notion of mathematics as a social undertaking, without really examining what that implies. Are we to think the idea "two" for * * is some sort of social construct? I'll bet anything Australopithecus was hard-wired to know the difference between one lion and two. Anyway, does counting up to five really belong in the same category with group theory? Hersh just assumes so, leaving a whole slew of fascinating questions out of his purview.

But Hersh writes clearly, if sometimes a bit too cutely, covers a lot of ground, and has some fascinating proofs and explanations in the back of the book. I'd be happy to have bought it just for the explanation and short proof of Gödel's Incompleteness Theorem. Bottom line: glad I bought it, hope something a lot better comes along soon.

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.

Hersh starts the book by considering the four dimensional equivalent of a cube, or a 4 cube, deducing that it would have 81 parts and then asking in what sense it exists. Perhaps the obvious answer is it is a mathematical object though you could also say it is something imaginary thought up by humans or that it exists in some mysterious way of the type proposed by Plato. So far so good.

He then spends most of the book trying to argue that mathematics is simply a human construction and philosophers who think like Plato must be deluded religious right wingers. Which is fair enough as a debating point but a bit silly. For example he mentions those arguing maths is more than a human pastime sometimes point out that if we met intelligent aliens, they would have discovered the same maths so it's more than a human invention. He then counters this, believe it or not, by saying we wouldn't be able to understand what the aliens were saying and so the argument is not valid. But come on .. of course they would have discovered the same maths and given a few days to learn their language we could discuss it. In fact when searching for alien intelligence one of the things SETI look for are stings of prime numbers in the data and of course this is more than just a human invention. Still the book is entertaining anyway.

This book has four parts: In the first the author discusses his ideas about his philosophy of mathematics. The second and longest part is historical, divided into mainstream philosophies of mathematics and "humanists and mavericks". There follows a short summary and some interesting and more technical notes.

There are basically three philosophies of mathematics: Platonism, Formalism and Constructivism. Reuben Hersh proposes an alternative: Humanism. The three basic philosophies deal mainly with the problem of foundations and view mathematics as a source of indubitable truth. The problems with foundations (the paradoxes), the failure of Hilbert's program (Gödel's theorem) and recent controversial proofs, such as the Four Colour Theorem, breathe air to this new kind of philosophy, perhaps not so new, since we can find its origins already in Aristotle. The humanist philosophy looks at what mathematicians do. It is no so different from what other scientists do. Mathematics is fallible and corrigible and mathematical rigour varies with the ages(remember A. Wiles first proof of Fermat's conjecture, classical calculus infinitesimals or Pasch missing gap in Euclid's axioms). Mathematics is not so different from music. Music exists by some biological or physical manifestation, but it makes sense only as a mental and cultural entity. RH defines mathematics as "the study of the lawful, predictable, parts of the socio-conceptual world". Mathematics is part of our culture and history and mathematical ideas match our world for the same reason that our lungs match earth's atmosphere.

Solving problems and making up new ones is the essence of mathematics. It is the questions that drive mathematics. It is a pity that math teachers forget about this when they teach and professional mathematicians often forget it when they write their papers.

Is mathematics invented or discovered? It has been a long standing controversy subject of discussions such as Alain Connes and a French neurologist. Hersh thinks both. After you invent a new theory (example group theory) you must discover its properties (find, for example, how many simple finite groups exist). And you may have to invent a trick to discover the solution of a problem.

To sum up: this book a "dimythization" of mathematics. Mathematics is just a human endeavour,but a highly beautiful, interesting, sophisticated and applicable human endeavour.

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.

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.

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