My Brain is Open: The Mathematical Journeys of Paul Erdos.

By Bruce Schechter.

*Oxford University Press; 224 pages; $30 and £22.50*

PAUL ERDOS was both the most prolific and best travelled of 20th-century mathematicians. Between his first paper as an undergraduate of 18 and his death in 1996 at the age of 83, he produced almost 1,500 papers with nearly 500 different collaborators in all parts of the world. As a Jewish student in Budapest, he knew that he would have to leave Hungary to pursue his career: he left for England in 1934, went to America in 1938 for a year at the Institute for Advanced Study at Princeton University, and continued to travel for the rest of his life.

Bruce Schechter's elegantly written book gives a picture of a mathematician at the height of his powers. Arriving with little warning, Erdos would spend up to 19 hours a day in feverish collaboration, often working with several groups of mathematicians at the same time. A week or two later he would move on to the next group of collaborators, leaving his exhausted hosts to write up the details. Mr Schechter conveys both the intense sociability and the essential isolation of a man whose life was devoted to the pursuit of mathematics: despite his hundreds of friends, he had no family after the death of his mother in 1971, and travelled alone for most of his life.

The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.

By Paul Hoffman.

*Hyperion; 320 pages; $22.95. Fourth Estate; £12.99*

THIS book concentrates more on Erdos in his declining years. Despite his brilliance, Erdos was unworldly and eccentric. He kept few possessions and had no permanent job or bank account; he gave away what money he earned or won. Paul Hoffman's rich store of anecdotes that tell us not only about Erdos but about the mathematical community he worked in. Erdos will be remembered both for his fundamental research in pure mathematics, and for the hundreds of conjectures he leaves behind. Both Mr Hoffman and Mr Schechter give fascinating accounts of a man whose extraordinary talent was not just knowing how to find the right answers, but knowing how to ask the right questions.

The Language of Mathematics: Making the Invisible Visible.

By Keith Devlin.

*W.H. Freeman; 300 pages; $24.95 and £14.95*

ALMOST 50 years ago, G.H. Hardy wrote in “A Mathematician's Apology” that “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” Keith Devlin presents mathematics as a science of patterns. He argues that mathematics makes visible the invisible structures that surround us, a formulation that includes the worlds of physicists and astronomers as well as the domains of computers, financial markets and statistical analysis.

His book is devoted to a collection of fascinating case studies, ranging from Sumerian accounting tokens and the problem of the best way to stack oranges, to Fermat's last theorem and modern cryptography. Mathematics can be notoriously inaccessible, and Mr Devlin succeeds both in giving us a glimpse of the internal beauty of the subject and in demonstrating its usefulness in the external world. “The Language of Mathematics” is lucidly written and richly illustrated, and remains accessible and enthusiastic throughout. Occasionally, Mr Devlin's way with words runs away with him. In an intriguing contrast, he suggests that whereas fossil fuels powered the industrial age, “in today's information age, the principal fuel we burn is mathematics.” Hhm. You use up fuel; you do not use up numbers. And though you can't run your car on the petrol in mine, we can both divide two into ten simultaneously. Still, this is an excellent starting place for someone who wants to know what mathematics is about.

The Moment of Proof: Mathematical Epiphanies.

By Donald C.Benson.

*Oxford University Press; 342 pages; $30 and £24.95*

THE classical case of mathematical epiphany is Archimedes. The king of Syracuse ordered him to find out if baser metals had adulterated the gold in a crown. In the bath, Archimedes hit upon the principle of buoyancy and was so excited by his discovery that he ran naked down the street shouting “Eureka” (“I have found it”). With similar enthusiasm, Donald Benson hopes to communicate “the experience and joy of mathematical discovery”. A professor emeritus at the University of California, Mr Benson clearly loves his subject. He presents a wide variety of mathematical results in a book that is a cross between popularisation and mathematical text, complete with propositions, theorems, proofs and marginal boxes with notation and definitions. Some may find Mr Benson's book hard going in places and occasionally unclear, as in his discussion of the nature of proof. For the enthusiastic reader, however, there is a plentiful supply of elegant proofs and engaging problems.

The Mathematical Brain.

By Brian Butterworth.

*Macmillan; 446 pages; £20*

A PROFESSOR of neuropsychology at University College, London, Brian Butterworth is interested in the building blocks of mathematics: numbers. How do we come to “describe and represent our world in terms of numbers”? Mr Butterworth thinks we are born with counting ability because the brain contains a “number” module that can distinguish “numerosities” of up to four or five. Mathematical thought derives both from this natural ability and from concepts learnt culturally. How much maths you have depends in the end, he thinks, on psychological factors and on the time you spend at it. Evidence for innate abilities is drawn from many sources, including prehistoric cultures, the etymology of number words and the way that children and chimpanzees learn. This is an interesting and controversial book, with implications not only for philosophy and psychology, but for the way we teach mathematics to children.

This article appeared in the Review section of the print edition under the headline "Mathematics for all"