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112 of 116 found the following review helpful:
a beautifully written introduction to (some of) what mathematics isApr 18, 2007
By Nim Sudo
Like many mathematicians, I often wish that I could give my non-mathematical acquaintances a better idea of what I actually do, and I was hoping that this book would serve that purpose. However, this book isn't so much about what mathematicians do and why, but rather about what mathematics is, i.e. what certain basic mathematical concepts mean. The first 7 chapters roughly cover the following topics:
1) What does it mean to use mathematics to model the real world?
2) What are numbers, and in what sense do they exist (especially "imaginary" numbers)?
3) What is a mathematical proof?
4) What do infinite decimals mean, and why is this subtle?
5) What does it mean to discuss high-dimensional (e.g. 26-dimensional) space?
6) What's the deal with non-Euclidean geometry?
7) How can mathematics address questions that cannot be answered exactly, but only approximately?
The eight and final chapter makes a few remarks about mathematicians.
The writing is spare and beautiful. For each topic, the book takes just enough space to give the reader some food for thought, then moves on. I especially liked the middle four chapters. I would definitely recommend this book to students in lower-division undergraduate math courses who are curious about or puzzled by the above questions.
The book touches on some philosophical questions. In doing so, the book flies close to some subtleties (such as Godel's theorem and the Banach-Tarski paradox) without acknowledging them (which is reasonable enough for a Very Short Introduction). Also, one can argue with some of the philosophical statements. For example, is mathematics discovered or invented? The author espouses the axiomatic approach (which is pretty much how mathematics is written), whereby mathematicians invent the rules and discover the consequences of the rules. I would want to emphasize that these rules are not completely arbitrary, but often there is some intuitive notion that one is trying to capture. In this regard, here are two specific statements in the book that I take issue with: 1) The author argues that i does not have a Platonic existence, on the grounds that one could replace i with -i in all mathematical statements without changing anything. OK, but if this is supposed to imply that complex numbers are invented rather than discovered, then I am not convinced. 2) The author suggests that in teaching students who make mistakes such as x^(a+b) = x^a + x^b, it might be good pedagogy to introduce exponentiation axiomatically and then deduce facts such as x^3=xxx from the axioms. However I think that if one does not already understand that x^3=xxx, then working with exponentiation axiomatically will just be meaningless symbol manipulation, of the kind that I encourage beginning calculus students to unlearn. I think that it makes more sense to build up a solid understanding of what x^n means when n is a whole number, and only then generalize.
Anyway, the nitpicking in the previous paragraph should probably just be regarded as evidence of the thought-provoking nature of the book.
The author has posted a number of additional essays on related topics on his webpage. These tend to be a bit more mathematically advanced than the book, but not too much, and are also good reading.
99 of 105 found the following review helpful:
Pragmatic MathematicsSep 26, 2005
By Peter Reeve
An introduction to mathematics could be just that; elementary arithmetic and geometry, or it could be an outline history, or finally, it could introduce the philosophical aspects of the subject. Gowers does none of those, although he does touch on the history and philosophy of mathematics. This is really an introduction to higher mathematics, for readers who have reached what in Britain is GCSE standard, roughly eleventh grade in the US.
Philosophically, Gowers is a pragmatist. To him, problematic concepts like infinity and irrational numbers have meaning in as much as they are useful, and are true in as much as they give true results. As a European, Gowers credits Wittgenstein with these ideas. An American author would have credited William James. Gowers sidesteps rather than resolves philosophical problems, thus giving reassurance to mathematicians and irritation to philosophers.
The book is a random selection of topics rather than a continuous narrative, but succeeds because each topic is fascinating and the writing is clear throughout.
Under "Further Reading", Gowers includes his own website address, where you can find sections that did not make it into the book. What a good idea! The site is as full of good stuff as the book, and gives links to further sites that will give you as much mathematics as you will ever want.
140 of 162 found the following review helpful:
A Very Good IntroductionSep 20, 2003
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.
11 of 11 found the following review helpful:
Very good overview of mathematical thought.Nov 02, 2008
By Theodore Rice
Gowers is a very good mathematician. As a Field's Medalist (the equivalent of a Nobel Prize) he knows his stuff. This book is about how mathematicians approach problems and think about their subject. There is little "higher mathematics" in this book. If you want to know how mathematicians think about problems, this book is for you.
Too often mathematics is seen as "formulas and rules." This book dispels these myths by showing mathematics is about ideas and problems.
32 of 39 found the following review helpful:
Mathematics: A Very Short IntroductionFeb 19, 2009
By Sam Adams
Gowers, in The Princeton Companion to Mathematics, makes the rough but useful distinction of 3 kinds of mathematics: the algebraic, the geometric, and the analytic (where 'analytic' is not used in the philosophical sense but in a sense derived from the mathematical branch called 'analysis'). In the book under review, Gowers favors the geometric and the analytic, giving the algebraic only perfunctory attention.
I read the book because I like Gowers' approach in the Princeton Companion and I was curious what he had written here. The book has a very limited point of view and is as likely to disappoint as delight, depending on the type of mathematics the reader is inclined to enjoy the most.
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