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Mathematics: A Very Short Introduction [Paperback]

Timothy Gowers
4.6 out of 5 stars  See all reviews (30 customer reviews)

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Book Description

October 2002 0192853619 978-0192853615 1st
The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as "Is it true that mathematicians burn out at the age of 25?") It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.

About the Series: Combining authority with wit, accessibility, and style, Very Short Introductions offer an introduction to some of life's most interesting topics. Written by experts for the newcomer, they demonstrate the finest contemporary thinking about the central problems and issues in hundreds of key topics, from philosophy to Freud, quantum theory to Islam.

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Editorial Reviews


`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'.

Product Details

  • Series: Very Short Introductions
  • Paperback: 160 pages
  • Publisher: Oxford University Press; 1st edition (October 2002)
  • Language: English
  • ISBN-10: 0192853619
  • ISBN-13: 978-0192853615
  • Product Dimensions: 9.1 x 5.9 x 0.4 inches
  • Shipping Weight: 2.1 ounces (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (30 customer reviews)
  • Amazon Best Sellers Rank: #34,678 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
101 of 103 people found the following review helpful
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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96 of 102 people found the following review helpful
5.0 out of 5 stars Pragmatic Mathematics 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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138 of 160 people found the following review helpful
5.0 out of 5 stars A Very Good Introduction September 19, 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.
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11 of 11 people found the following review helpful
5.0 out of 5 stars Very good overview of mathematical thought. November 2, 2008
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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Most Recent Customer Reviews
5.0 out of 5 stars Provides an excellent introduction to real (and imaginary) mathematics
If you even are remotely interested in increasing your likeness for higher level math, then get this book. Prof. Read more
Published 1 month ago by Rafid Hoda
5.0 out of 5 stars An enjoyable and enlightening book
The author covers a wide range of topics in a small volume, from complex numbers to limits to hyperbolic geometry. Read more
Published 2 months ago by Michael Massimilla
5.0 out of 5 stars Unique Introduction to Mathematics
Gowers' unique book focuses on the subject matter and epistemology of mathematics: what mathematicians think about and how they establish results. Read more
Published 2 months ago by James W. Lamb
5.0 out of 5 stars Very happy with the book
There was always a sort of discontent in my mind that I couldn't grasp why it is not possible to divide by 0. Read more
Published 3 months ago by Neori
4.0 out of 5 stars abstraction
Important perspective for the reason to teach an abstract math early on.
Very well written - provides great starting points for further exploration in the subject.
Published 6 months ago by pistol Pete
3.0 out of 5 stars Gives information about math for a general reader.
It tells about math much like other books. There are no special insights from Gowers. I had hoped he would discuss his concept of the "width of a proof".
Published 7 months ago by DesertRat
5.0 out of 5 stars Informative, Well-written
Covers the topics that I was interested in reading about.
This book appears to be an original work by the author, although there may be
other similar titles.
Published 10 months ago by Lloyd Rice
3.0 out of 5 stars Mathematics by T Gowers
The book is a short and very personal kind of tutorial. Its reasonably explanatory in a textual rather than a mathematical way. Read more
Published 16 months ago by qualquan
5.0 out of 5 stars This book is great
I've enjoyed reading it. It seems to me that Timothy as a broad view and a deep insight in the matter and beyond as well. Read more
Published 17 months ago by Pascal Doret
5.0 out of 5 stars modern mathematics explained
In a short book the principles of modern mathematics are explained in a clear way. Very inspiring and very stimulating.
Published 17 months ago by J
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