Mathematics: A Very Short Introduction | 
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Product Description
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.
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Customer Reviews:
• A Beautifully Written Introduction To (some Of) What Mathematics Is 18 April, 2007
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.
- Reviewed by customer ID: A56TXS76PETEV
• Great Overview 12 November, 2007
This book is a great overview of higher level mathematics. It uses examples that anyone with basic high school math skills can understand. The language is not overly simplistic, nor is it hard to follow. A great value and a great book.
- Reviewed by customer ID: A28L0OY5CY1V2P
• A Neat Book 08 February, 2007
Just the kind of book that a highschool level student or an underclassman college kid might enjoy. Full of good fundamental ideas and very interesting mathematical concepts. Written at a level which is very approachable and readable, even for that majority of the population which didn't become math majors.
- Reviewed by customer ID: A33RAR1QOKG6EK
• Pragmatic Mathematics 26 September, 2005
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.
- Reviewed by customer ID: ABTUNH7645QJL
• Informative, Yet Not Introductory 06 September, 2008
An informative and interesting book at times, although readers seeking a genuine introduction to the subject of mathematics should look elsewhere. This book should more properly be titled "A Very Short Introduction to Advanced Mathematics," or similar.
Gowers is obviously comfortable writing about mathematical concepts--he fills his pages with references to complex numbers, fractional dimensions, the atlas of a three-dimensional manifold, and so on--than he is writing about histories or people. If you're curious to know why the parallel postulate does not hold within spherical or hyperbolic geometries, you will find Gowers helpful. But if you seek to understand what mathematics is, where it comes from, who its major practitioners are and have been, or what the current ideas in the field are--in other words, if you want a more human, less concept-driven overview of the field-- you will likely be disappointed. These things are mentioned, but only in a hurried, fragmented way.
As I read through the book I grew to dislike the author's smug tone, which essentially says, "Seventy-five percent of my argument may be going over your head--but I'm a mathematics professor and you're not." I'm surprised that Gowers' editor didn't do a better job of reminding him for whom he was writing!
In all, though, this is an interesting little book that may serve to stimulate more reading in higher-level math, if you're already somewhat oriented in that direction. If you just want to understand more about the basic ideas of math, its history, and so on, you will probably find Gowers' book difficult and offputting.
- Reviewed by customer ID: A1WSGPIQXV2S66
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