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Mathematics: A Very Short Introduction 1st Edition

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ISBN-13: 978-0192853615
ISBN-10: 0192853619
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Editorial Reviews

Review

`a marvellously lucid guide to the beauty and mystery of numbers' Gilbert Adair

About the Author


Timothy Gowers is Rouse Ball Professor of Mathematics at Cambridge University and was a recipient of the Fields Medal for Mathematics, awarded for 'the most daring, profound and stimulating research done by young mathhematicians'.
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Product Details

  • Series: Very Short Introductions
  • Paperback: 160 pages
  • Publisher: Oxford Paperbacks; 1st edition (October 2002)
  • Language: English
  • ISBN-10: 0192853619
  • ISBN-13: 978-0192853615
  • Product Dimensions: 7 x 0.4 x 4.2 inches
  • Shipping Weight: 2.1 ounces (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (38 customer reviews)
  • Amazon Best Sellers Rank: #78,326 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

110 of 114 people found the following review helpful By Nim Sudo on April 17, 2007
Format: Paperback
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?
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99 of 105 people found the following review helpful By Peter Reeve on September 25, 2005
Format: Paperback Verified Purchase
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.
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139 of 161 people found the following review helpful By Bibliophile on September 19, 2003
Format: Paperback
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.
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11 of 11 people found the following review helpful By Theodore Rice on November 2, 2008
Format: Paperback
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.
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