on 29 April 2003
Professor Gowers explains the key concepts of mathematics in this accessible and well-constructed book. Written for a general audience (although some basic scientific knowledge is an advantage), this book avoids the popularisation of mathematics, but instead focuses instead on the central ideas of abstraction and axiomisation which underpin all of modern mathematics.
After an introductory chapter on mathematical models in the sciences, Professor Gowers covers topics including numbers, limits, dimension and approximation in six short chapters. A final chapter gives thought-provoking answers to questions such as what are the connections between mathematics and music, and what is beauty in mathematics. The format inevitably means that some topics are omitted due to lack of space - there is little background on the history of mathematics for example - but that does not detract from the central theme of the book.
Professor Gowers' enthusiasm for his subject comes across on every page.
on 23 December 2003
I have bought several books in the "Very Short Introduction..." series all of which have been excellent and this one in particular is extremely clear and interesting. The book is aimed at the intelligent layperson and gives a general introduction to what maths really is and how professional mathematicians think about it. This approach is in contrast to most people's experience in school where tedious and repetitive calculation is the norm (which is a real shame). Various areas in maths are looked at and the ideas behind them are explained rather than the reader being hit with big formulae and funny looking symbols.
on 30 September 2002
It is amazing how Tim Gowers, one of the greatest mathematicians
of our time, managed to convey the spirit and content of math to the general reader, and not only. Even professional mathematicians will learn a lot from his insightful remarks.
This ``little'' book is destined to become a classic of popular
The world of mathematics elicits many different emotions, from fear to reverence, from apprehension to fascination. On a surface this may seem surprising, since mathematics is supposed to deal rational thought, and should be as removed from emotional considerations as it gets. And yet, it is precisely this dispassionate rationality that makes those unaccustomed to mathematical thinking recoil, and those with a mathematical bend of mind rub their hands in glee. For the truth is, mathematical thought relies heavily on emotions, and mathematicians are fascinated with beauty and elegance of mathematical creations.
In the book "Mathematics: A Very Short Introduction" the author Timothy Gowers tries to convey some of this beauty. This is a very readable and intelligent short introduction, and probably the best short introduction to mathematics out there. It takes reader through some basic mathematical problems, and showcases the methods and procedures that mathematicians use in their work. If you are math-phobic, you will not have to deal with any complicated mathematical equations, and all of the problems and proofs that are offered in the book are straightforward and intuitive, and require a very minimal level of mathematical knowledge. The fact that the book attempts to "do" mathematics, as opposed to just tell about it, is one of its more rewarding aspects. It makes this an intelligent read, and rewarding no matter whether you are a complete mathematical "outsider" or someone with an advanced degree in a math-intensive field.
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.
on 12 February 2013
I have already a growing number of "very short introduction.." on my bookshelf and quite enjoy reading them.
I think its a very good series except for the text font and layout which I am not a big fan. I find them hard to read sometimes but they do have the advantage of being pocket size so handy.
This one on Mathematics brought be back to my young age at school. I would have been so pleased to have a math teacher such as Mr Gower.
He has a talent to explain complex things simply. Some worked examples may be a bit derouting for some of us who dont use maths every day but you honnestly do not have to read everything. I read this book with a relaxed attitude, trying to enjoy more than to learn. The book is also loaded with diagrams which helps you further to understand some key concepts.
What I found fascinating was that some maths conjectures are still not resolved to this date. Finally, the last sections on "FAQ" is very useful and instructive.
on 28 December 2004
This book demystifies puzzling concepts such as infinity, curved space, n-dimensional space and fractional dimensions. His emphasis on the abstract method - the focus on what mathetical objects DO rather than what they really ARE - as the key to understanding all these concepts is amazingly powerful, truly an eye-opener.
A basic knowledge of mathematics is an advantage
on 3 March 2014
Having read a few of the Very Short Introduction (VSI) series, I wanted to revisit some of the joy of university life by returning to mathematics, the subject which I studied as an undergraduate. With the title as it is, one might wonder what sort of level as it’s pitched at. Here, one could be lulled into a false sense of security by mistaking it for “Arithmetic: A Very Short Introduction”. Do not expect this to be “a very simple introduction”. To anyone who has studied maths at university, this will be a very simple book. To anyone studying maths at A-level, they should find it a little challenging in places, but it should provide good food for thought, building on some familiar principles. The author says that it should be OK for anyone with a good GCSE grade, though I would express a little scepticism at that sentiment.
That said, I do think it’s a marvellous little book. One of the first things that Gowers discusses is the cumulative nature of maths. i.e. some things can be very simply stated, but only in terms of other things which need to be well-defined and understood. The danger then is how far back do you regress to be able to find ground which is widely understood?
Gowers deals with this brilliantly by having his opening chapter on mathematical modelling. In so doing, he grounds mathematics in the real, the physical, the tangible, instead of diving off into the realms of pure mathematics straight away. Though I must admit, the appeal to me comes about primarily because he enunciates the way I have thought of maths for most of the last 3 decades.
From here, he starts to ask some more fundamental questions about the nature of numbers, including complex numbers (but not quaternions) and some “proper” algebra though he cunningly avoids the use of terms such group, ring or field whilst ensuring the reader is familiar with their rules by means of definitions followed-up with examples. He also touches on some rules regarding logarithms which perplex some people, but are dealt with very well.
He then goes on to probably the most important idea in maths: proof. Though touching on a little philosophy, he tries to skirt around it and give a robust exposition of what a mathematician means when (s)he talks about proof and how it differs from the more lackadaisical use of the word in everyday (and even some other areas of scientific) usage.
Though any book on serious mathematics probably ought to contain a good amount on calculus, Gowers avoids this quite ostentatiously. Rather, he lays the groundwork for an understanding of it with a chapter on limits and infinity. In so doing, one might think he’s dodged a potential bullet of losing the interest of readers, though I think that anyone who hasn’t done calculus but who has understood this chapter will be well-placed to start studying calculus.
Moving on, we start to get a bit more geometrical. The first of these two chapters looks solely at the idea of dimensionality. One might instinctively have an idea of what a dimension is or how to count them. However, maths is rather more refined than such instinctive generalities and Gowers gives us some examples that any student would find in a 1st year linear algebra course. If anything, this is the acid test for those considering doing maths at university; if this is incomprehensible to you, then it’s best to turn away. But if, on the other hand, you can see there’s something there that can be grasped, if you don’t quite get it exactly at this stage, then you’re standing in good stead. The second of the geometrical chapters looks at much more fundamental geometry, looking at the classic issue of Euclid’s 5th postulate and the consequences of abandoning it.
The last main chapter is on one of my least favourite topics: estimates and approximations. This was a topic I did to death in my 2nd year at uni, not getting on with the lecturer (an angry Scot called Alan) and having a nightmare on the exam, barely scraping a 2:1 in it. Gowers doesn’t hide from the fact that this is not neat maths, as most of the rest of the book is, though he spares the reader from an insight into the gory detail of numerical analysis (or num anal, as we disparagingly called it). If there’s any downside to this wonderful introduction, it is this chapter. Not because it is badly written or poorly explained, it isn’t. But merely because it lacks the sexy panache that the rest of the topics have.
This is redeemed somewhat at the very end when he puts in some frequently asked questions, along with his answers. If anything, these give a far better insight into the working life of the mathematician than anything found in the rest of the book. He addresses some myths and common (mis)conceptions, giving an honest assessment on some issues where he needn’t have done so.
Overall, it’s a great book. If you hated maths at school, then this isn’t the book to get you back into it and loving it. But for anyone with a keen interest in science and how it’s done, then this is a gem. If you know any A-level students considering doing maths at uni, give them this book. Hey, they are students, so £8 for a little book is a lot of money. Go on!
on 27 July 2013
I had expected this to be a book outlining some of the advanced theories and formulae of mathematics. On the contrary, it was largely concerned with rather fundamental aspects of maths, such as what is meant by numbers, the concept of infinity, how to handle multiple dimensions, geometry, etc. I found this unexpected approach to be quite refreshing and informative. It gave me a feel for how mathematicians think, and how they approach problems. Of particular interest were explanations on how mathematicians set about proving a theory, such as a confirming that the square root of two is an irrational number. Mathematicians take nothing for granted, and everything has to be proved, step by step, all the way back to first principles. Some insight was also provided on solutions that mathematicians regard as being elegant, or even beautiful, these being seen as a cut above more "ugly" proofs.
Admittedly, significant chunks of the book were beyond me and I was unable to follow the reasoning used, such as was the case with much of the chapter on geometry. Nevertheless, it has to be said that no advanced maths were needed to understand these sections but what would have come in handy was to have read the book with a pen and pad of paper at my side so that I could have jotted some points down to help reinforce the arguments being presented. In mitigation, I should point out that I read the Kindle version and flipping from one page to another, which was essential for following some points, is not so easy on a Kindle as with a conventional book. All the same, it is certainly a book that requires effort on the part of the reader to fully digest its contents.
Other than school text books, this is the first book I've read on maths. Whilst I found it interesting, I don't feel driven to read any further books on this subject. So it could be said that the book has failed spark a desire to find out more which should be the hallmark of an outstanding book.
on 20 October 2015
I am reading it in small instalments when I have time. I have enjoyed what I have read so far and gained some very valuable insights. Timothy Gowers writes in a very clear style, and (of course) is a very knowledgeable mathematician. I will give an update when I have read the whole book (it's 140 pages so even at my slow pace, it should not take too long to finish reading it). The book introduces some traditional topics, and also some "current" / "advanced" mathematical topics, in a clear and accessible way, which I am finding very interesting and useful (such as: modelling brains and computers, higher dimensional space, hyperbolic geometry and curved space).