Philosophy of math under 200 pages!

If one expects a thorough course in basic math, this book may not be it - "Mathematics for the Million" by Lancelot Thomas Hogben should be your first choice. Nor does this book have much to say about the historical development of mathematics - for this there is no substitute for Morris Kline's "Mathematics for the Non-Mathematician" (which teaches the basic concepts of math simultaneously, aided by exercises).

This book aims to convey, I think, a sense of what mathematical reasoning is like. "If this book can be said to have a message, it is that one should learn to think abstractly, because by doing so many philosophical difficulties simply disappear," writes Gowers in the Preface. And at times it does feel as though you're reading a book written by a philosopher. For instance, p. 80-81 discusses "What is the point of higher-dimensional geometry?" (Of course Gowers is not a philosopher but a VERY distinguished mathematician.)

Incidentally, here's something that stumps me. Gowers says "[t]here may not be any high-dimensional [i.e., more than three] space lurking in the universe, but...." But I thought higher-dimensional space is what superstring theory is all about. And besides, Martin Rees, Andrei Linde and Alan Guth are now telling us there is an infinite number of universes outside our own, each taking a different number of dimensions - some fewer than three, others many more! Higher-dimensional space may not be as abstract as Gowers thinks.

Gowers's main point, however, is that higher dimensions have meaning and validity in mathematics quite independent of whether they are grounded in objective physical reality, or whether physicists use them or not.

This once again illustrates what Eugene Wigner called "the unreasonable effectiveness of mathematics." Mathematicians often develop concepts, like Riemannian geometry, n-dimension geometry (where n is over 20), etc., which are way ahead of developments in the empirical sciences, often without any idea whether they will become applicable to, say, physics. Steven Weinberg puts it this way: It's as though Neil Armstrong when visiting the Moon found the footsteps left behind by Jules Verne.

Rare indeed is the distinguished physicist who does not hold mathematics and mathematicians in high regard.

I find this book very stimulating to read (though not always easy to understand - my fault no doubt). It won't help you with school problems. Nor will it help with daily life. But it is deep and thought-provoking, explaining "just what IS mathematics?"

I have a minor point of disagreement over this sentence on p. 127: "Here is a rough and ready definition of a genius: somebody who can do easily, and at a young age, something that almost nobody else can do except after years of practice, if at all." This definition would seem to exclude some of the greatest scientists of all time: Einstein, Max Planck (who was already middle-aged when he discovered the quantum), not to mention Darwin, Benjamin Franklin, Niels Bohr, even possibly Newton. (It would exclude many non-scientific geniuses also, like Marlborough, who won the Battle of Blenheim at the ripe old age of 54.)

I pointed out to the author that his definition is actually appropriate for "prodigy" (and he seemed to agree). Indeed his statement is a very succinct definition of "prodigy."

Is this point worth discussing? It wouldn't have been, were the concept "genius" not so often used among mathematicians - to describe one another (with good reasons). I might add by the way that Gowers, a Fields Medallist, is a certified genius himself. (Gowers told me he disagreed on both charges.)

On reflection, Gowers's definition is not so much wrong as too exclusive. There are of course no simple ways to define "genius." Like "beauty," "genius" may be in the eyes of the beholder only - we think we recognize it when we see it. My feeling is that most prodigies are indeed geniuses - how else would you describe a six-year-old who understands trigonometry, or the 16-year-old who is a world champion in chess? - but many true geniuses are not and have not been prodigies when young. Einstein is one such example. And Darwin another (even more so). Perhaps Newton also.

I suspect that Gowers's error comes from his experience as a mathematician: many great mathematicians are indeed mathematical prodigies as children. (Think of John Von Neumann.) This rule is less true outside mathematics - and the further away from mathematics, the less true it becomes. Music is close to math for some reason - Mozart is an outstanding example - but war-making is obviously not. (Napoleon, who was good at math, and rose from nothing to Emperor of France at age 34, might disagree on the latter point. But then he later lost the war.)

Anyway, Gowers does say his definition is only "rough and ready," not complete in itself. This leaves room for other "definitions" of genius, as there indeed must be. Surely prodigy is that special kind of genius which catches people's attention instantly, and has some mysterious "magic" to it, which Gowers rightl stresses is not a necessary quality for success in mathematics. Von Neumann (always capitalized "V"), who mastered calculus by age 8, and went on to contribute to quantum mechanics, the Manhattan Project, the first mainframe computers (the "Von Neumann machines"), set theory, cybernetics, meteorology, the hydrogen bomb, and Game Theory, was a child prodigy with a photographic memory who fits Gowers's restrictive definition of genius - and indeed he was a genius by any definition. But as Gowers emphasizes, you don't have to be a Von Neumann to be a productive mathematician.

The following are Contents: Preface, List of Diagrams, 1 Models, 2 Numbers and abstraction, 3 Proofs, 4 Limits and infinity, 5 Dimension, 6 Geometry, 7 Estimates and approximations, 8 Some FAQs, Further reading, Index.

I'm surprised that calculus is nowhere to be found in the Index (as is Newton). If Gowers has discussed calculus in this book, I may have missed it. (But then I am no genius.) In any case a fuller discussion of calculus (and of Newtown) would seem desirable to me.

I can't think of a better book to carry around in your pocket than this. This book is outstanding.