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IT INSPIRES more questions about its philosophical foundations and its role in the world than probably any other discipline. What is mathematics for? Why is it difficult? Why do I need it?

These are some of the questions asked by Christina, a friend of Philip Davis whom he invokes in his delightful and informative book Mathematics and Common Sense. In a question-and-answer style narrative he addresses many others. How has mathematics changed in the last 100 years? Where would a fan of the subject go to find the latest hot material? What is the greatest challenge to modern mathematics? How much can an interested amateur do?

Let me add another question, asked by my teenage daughter: why do mathematicians bother to find out things that no one really needs to know? This reflects the principal theme of the book: the creative tension that Davis sees between the things in mathematics that are obvious and intuitive 鈥� common sense 鈥� and the things that are counter-intuitive and esoteric.

Of course, mathematicians do not own mathematics. Almost everyone uses elementary mathematics daily, for example, in counting money, time or animals. Almost everyone takes advantage of technologies whose design involves mathematics in an essential way, such as cars, aeroplanes, bridges, television, washing machines and computers. There is hardly an aspect of human existence to which mathematics cannot contribute understanding 鈥� though that is not to say that the value of human life can be reduced to a set of equations.

Mathematics is integral to many disciplines: as subjects become better understood, they become more mathematical. A good example is the life sciences, where there has been a major increase in the application of mathematics through the use of models related to blood flow, the heart, nerve fibres, genomics, the structure of viruses and so on. In asking a doctor to prescribe a treatment that will reduce our symptoms to an acceptable level, we are posing a classical problem of control theory, a highly mathematical subject, though one that up to now has not been applied frequently in medicine. It is not fanciful to think of a time when visits to the doctor will involve personalised mathematical simulations, for example to determine optimal treatment regimes based on genetic data and medical test results.

This leads to another fundamental question: how does mathematics evolve? Most areas of mathematics stem from an attempt to understand or predict something about the world. Mathematicians develop models that are used to make predictions, which are then compared with what actually happens, and this way the model is improved. During this process mathematical ideas used to analyse the model may themselves end up being transformed through the application of notions such as beauty, simplicity and the power of generalisation, which have little to do with the original problem. Later, the metamorphosed mathematics might be applied to similar, or perhaps completely different aspects of the world. A good example of this is the Radon transform, which evolved decades later into the lifesaving X-ray tomography machine.

The development of mathematical models has been helped by modern computers, but has also been complicated by them. Solutions to equations that our ancestors were helpless to solve can now be computed routinely. Or can they? In complicated problems, it is far from obvious whether the solutions produced by a computer are correct. When, for example, a partial differential equation is solved numerically, this is done using an algorithm intended to produce an approximation to the true solution. Plausible but faulty algorithms can easily lead to wrong answers, so validation of algorithmic codes, ideally using error estimates, is crucial. What鈥檚 more, the solutions to certain important equations, such as those that model weather and climate, may become so complicated that no computer in existence could compute them accurately. Thus modern scientific computation requires its own new mathematics to establish its validity.

This brings me to another important issue that Davis covers in his book: whether it is fruitful to divide mathematics into pure and applied. Traditionally, pure mathematics refers to the development of mathematics according to its internal logical structure and applied mathematics to the development of models that correspond to reality. Davis describes it more simply as the difference between being inward-looking and outward-looking. However, the two cannot be so cleanly partitioned. For example, you鈥檒l find linear algebra, probability theory and the study of partial differential equations on either side of the traditional pure/applied divide. Many of the great mathematicians of the past, such as Archimedes, Newton, Leonhard Euler, Karl Gauss, Augustin Cauchy, Bernhard Riemann and Jules-Henri Poincar茅, saw mathematics as a seamless whole. The polarisation of the subject arose in the 20th century, and nowadays this disharmony seems increasingly recognised as unhealthy and artificial.

How does all this resonate with the public? In one of the many intelligent and approachable essays that comprise his book, Davis discusses how mathematical developments are rarely covered by the media. A local television reporter visiting the offices of the American Mathematical Society in Providence, Rhode Island, is quoted as saying he had passed the building many times, and often wondered what on earth went on inside. 鈥淏ut I really don鈥檛 want to know,鈥� he said.

鈥淣otions such as beauty and simplicity can transform mathematics鈥�

Since Davis completed his book, one remarkable story from the world of mathematics has featured in newspapers, magazines, TV and radio around the world. In August Grigori Perelman, until recently a leading researcher at the Steklov Mathematical Institute in St Petersburg, Russia, declined to accept the award of a Fields medal, regarded as the equivalent of a Nobel prize, at the International Congress of Mathematicians in Madrid, Spain.

In 2002 and 2003, Perelman had posted three papers on the arXiv preprint server proposing a proof of the Poincar茅 Conjecture, one of the most famous problems of mathematics, unsolved for over a century. The importance of the problem, the brilliance of Perelman鈥檚 ideas, his failure to submit his work to a mathematical journal, his apparent indifference to fame and fortune, the possibility that he may also turn down the $1 million Clay prize should this be awarded to him, the efforts to fill in the details of his proofs by three pairs of mathematicians 鈥� all these combine to make this a tale of extraordinary mathematical and human interest.

I hope the Perelman story will change media attitudes towards mathematics, but I am pessimistic. Few non-specialist journalists have a good understanding of mathematics or science, which seldom feature in media studies courses. Often those journalists who do write about mathematics can find it difficult to persuade editors to give space to their stories. Good media coverage inspires students, and inspiring the young to study the mathematical sciences is vital for the future of the planet. In bringing some of these crucial issues to a wider audience, Davis has made a valuable contribution to the public understanding of the importance of the subject.