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4.6 out of 5 stars
Mathematics: A Very Short Introduction (Very Short Introductions)
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7 of 7 people found the following review helpful
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.
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39 of 41 people found the following review helpful
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.
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21 of 22 people found the following review helpful
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.
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33 of 36 people found the following review helpful
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
science writing.
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2 of 2 people found the following review helpful
on 5 January 2011
It's well known that some concepts in mathematics can be utterly mind-bending - concepts such as curved space, higher dimensional geometry, infinity, and more besides. To be able to work with any of these, you often need to let go of the urge to visualise something and instead settle for merely conceptualising it. You need, in other words, to feel comfortable thinking in a thoroughly abstract way. This book is an introduction to that way of thinking. It's not one of those books that joyously recaps all the rules and syntax you'd forgotten since leaving school. Rather, its aim is simply to give you the means to grapple with core aspects of advanced mathematics.

While this book doesn't overwhelm with technicalities (a British GCSE or equivalent in the subject should be sufficient prior knowledge for most readers), the rules and processes are not ignored, and there are brief introductions/reminders of topics such as Euclidean geometry, irrational numbers and so on. However the author's aims and emphasis are decidedly more psychological and philosophical. Why should we accept the axioms proposed by mathematicians? What exactly is a number? What does mathematical proof actually mean? These questions first need to be explored before embarking on the path to more esoteric concepts such as multi-dimensional geometry and manifolds (incidentally, useful for anyone interested in string theory).

The chapters are sensibly structured and Gowers's enthusiasm for the subject radiates from each page without ever sinking into that patronising "Hey kids, maths is fun!" style of writing. The last chapter contains a selection of genuinely interesting FAQs, including some good suggestions as to why so many people seem to strongly dislike mathematics, compared with say biology or English literature (about which the non-enthusiast may just feel indifferent). Overall I found this more interesting that expected; a useful take on the subject that lays the groundwork for some of the more advanced areas of mathematics.
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12 of 13 people found the following review helpful
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
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2 of 2 people found the following review helpful
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.

7/10
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22 of 25 people found the following review helpful
VINE VOICEon 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.
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1 of 1 people found the following review helpful
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!
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8 of 9 people found the following review helpful
I've recently gotten interested in Mathematics again (after finding it very difficult and boring at school) and was looking for a good general book to serve as an introduction. I really don't think I could have done any better than this. If you have even the slightest interest/curiosity about maths - you should buy this book.
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