The Princeton Companion to Mathematics edited by Timothy Gowers et al
Princeton, £60; 1,008pp Buy the book

The Numerati: How They'll Get My Number and Yours by Stephen Baker
Jonathan Cape, £17.99; 272pp Buy the book

WITHOUT mathematics our world would be very different - no internet, no mobile phones, no satellite navigation, no passenger aircraft, no CDs, no digital cameras, no MRI scanners. Yet this almost total reliance of our society on mathematics is largely unknown and unappreciated. There are many ways to drag mathematics out of the wings and into the limelight, and these two books follow very different strategies.

The Princeton Companion to Mathematics contains contributions by about 100 of the world's leading mathematicians, and its editor, Timothy Gowers, is a winner of the Fields Medal, the highest honour in mathematics. It concentrates on pure mathematics, done for its own sake without any specific application in mind. Its coverage is broad, but it is not an encyclopaedia.

The core is a section on the different branches of mathematics. At school we encounter a few basic concepts in arithmetic, algebra, geometry, calculus, probability and not much else. The professional community recognises about 100 major branches of mathematics, with names such as “algebraic topology” and “stochastic processes”. They represent the hidden world of frontier research, and they are as unlike school mathematics as composing a symphony differs from playing scales on a piano.

It is almost impossible to convey much about these frontier activities without employing technical terms, formulas and abstract concepts. The Companion makes a heroic attempt to keep these to a minimum, but the minimum is fairly big. A mathematics undergraduate would feel comfortable with this part of the book, but a school student would struggle.

Most of the rest of the book is accessible to anyone with an interest in mathematics and some basic high school background. There are biographies of mathematicians and a discussion of the origins of modern mathematics. There are essays on applications of mathematics to areas such as chemistry, biology, traffic flow, transmission of digital information, money, medicine, music and art.

The Companion is an ambitious attempt to do something valuable, and it is not the fault of contributors or editor that it is impossible to realise this ambition completely. The book conveys the breadth, depth and diversity of mathematics. It is impressive and well written, and it's good value for money.

The Numerati is short, has no formulas, and no overt mathematical concepts beyond ordinary numbers. It's written in a breezy journalistic style and it avoids sensationalism even when this must have been tempting. It is about the numerical data that supermarkets, banks and internet service providers collect and how they use it - or hope to use it. The numerati have got your number. They've been collecting data for years but until recently had no idea what to do with it.

Your supermarket loyalty card is not free. You pay for it by allowing the supermarket to collect information about your purchases. They know that you always buy a particular brand of razor. Occasionally you pick up a chocolate bar at the checkout. They know how frequently you do this, what your total spend is. So far they've used this data in limited ways. But soon the numerati will be making more effective use of your personal data to persuade you to buy stuff that you wouldn't otherwise have bought. “You like Cherry Coke. How much would Pepsi have to slash the price of its Wild Cherry Cola to entice you to switch?” The hold over you will be even greater when they can track your shopping cart as you pass through the store, link that to your loy-alty card and flash messages on a screen pointing out bargains that they think will appeal to you. And this kind of thing will also happen in your relations with your employer, your vote, your doctor, even your love life.

Is this mathematics? Or is it just numbers? The data may be numbers, but what you do with it involves high-powered maths. Vast amounts of mathematics lurk just under the surface of our lives, making everything possible.

Some of these developments are good, some bad - loss of privacy, even a police state. Another key question, strangely missing from the book, is: will we let these things happen? As the numerati build ever more effective weapons to control our lives, we may decide not to play their games. For every advertisement on the web there is a free add-on to block it. And ultimately, we can spend only what we earn. The numerati may encourage us to spend it on them, rather than on the opposition, but this arms race has a cost. Like air miles, it could become self-defeating - expensive to operate and of little value once everyone is doing it.

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