Behind the blacked-out windows of a wartime British train, cold and uncomfortable, two men sit hunched over a pad of paper, scribbling furiously. One of them, plumply middle-aged with an indefinably foreign air, is doing most of the writing. His companion, a younger Englishman, contributes comments and thoughts. It is April 1943. On the coarse, government-issue paper, John von Neumann is writing his first computer program.

The historical significance of the event is limited: others have attempted such feats before. This program never runs. The device for which it is written is not a true computer - even the word "computer" is not yet in use.

But the 6.44 train from Bath to London is witness to a moment that encapsulates some of the central trends of 20th century intellectual history. In the middle of a war for the survival of European civilisation, one of the century's most powerful minds is applying the full force of mathematical theory to a crudely mundane task, the automated calculation of bomb blast patterns.

The themes from this moment resonate throughout von Neumann's life. The triumph of logic; the application of pure theory to savagely practical problems; the decisive shift in intellectual pre- eminence from Europe to the US - these themes, which he exemplified to perfection, will be seen by future generations as our century's legacy.

Von Neumann's contributions - to mathematical theory, to the atom bomb project, to the creation of the computer, to cold war missile strategy, and to economics - are significant in themselves. But they also reflect the tenor of the times, the brash mid-century belief in the power of reason to harness and tame the physical world. John von Neumann was not only one of the most striking figures of the century; he was also its pattern-card.

The young Janos Neumann was born in Budapest in 1903. His father was a lawyer and banker. The family was well-off and lived in a sort of dynastic apartment house, three generations together. Though Jewish, they were scarcely enthusiastic believers: treating the great Jewish holidays, in the words of his younger brother Nicholas Vonneuman, essentially as "non-denominational family reunions".

John and his two brothers went to the Lutheran Gymnasium (high school), because, says Nicholas, "it had the highest academic standards in Budapest". Though they had special lessons in Hebrew, they never really mastered the script.

If so, this was the only subject John von Neumann failed to master at school. Like his brothers, he was taught French and German before he was six. He started English later, so though he spoke it fluently he retained a warm Hungarian accent all his life.

Other subjects came easily: Nicholas remembers that his brother would merely spend a few minutes each evening glancing at his homework assignment for the next day, then toss the books aside. Despite his lack of preparation he was always able to participate in lessons - to his classmates' annoyance - as knowledgeably as if he had spent hours in study.

At the age of six, von Neumann could divide eight-figure numbers in his head. In adult life, competing against an early computer in speed of calculation, he won. Throughout his life, he possessed total recall, the ability to recite whole pages of text after reading them through once.

And he read prodigiously: his father bought a complete library in an estate sale and von Neumann read his way from one end to the other, enjoying particularly the 44-volume General History of the German historian Wilhelm Oncken. For the rest of his life he was able to hold his own with professional historians in subjects as diverse as the Byzantine Empire or the early history of Latin America.

His preference for reading sometimes led him into trouble. During the brothers' piano practice, Jancsi (as he was known) repeated scales over and over again, ignoring the music he was supposed to be learning. "It turned out," says Nicholas "that he had a book on the stand and was reading it - but for the sake of peace he kept playing a scale."

It was at the Gymnasium that von Neumann's mathematical brilliance was recognised. The school's maths teacher paid a call on von Neumann's father. "He warned him: don't let John deteriorate into a child prodigy," says Nicholas. Instead, a parallel mathematics course was devised with a tutor from Budapest university, and it was jointly with him that John published his first academic paper, at the age of 17.

The first world war - in which Hungary as part of the Austro-Hungarian empire had half-hearted involvement on Germany's side - did little to interrupt this harmonious way of life. But when the war ended and the Austro-Hungarian empire collapsed, the short-lived Hungarian Soviet Republic seized power. The von Neumanns left the country until the regime was overthrown, leaving John with a lifelong mistrust of communism.

It was clear that if John wanted a mathematical career, he would have to leave Hungary more permanently. The home of advanced mathematics was Germany, especially Berlin and the ancient university of Göttingen where David Hilbert held sway. So while simultaneously studying chemical engineering in Zurich (in deference to his father's wishes for a more practical education), John acquired a PhD in maths from the University of Budapest and moved to Berlin, from where he travelled frequently to Göttingen.

The central mathematical project of the period was the attempt to reconstruct the whole of the subject on an entirely logical and provable basis. The aim was to ensure that every mathematical theorem formed part of a single, coherent structure of unshakeable reasoning, based on a handful of self-evident axioms. This "formalist" project, best summarised by Hilbert in his list of unsolved problems drawn up in 1900, was doomed to failure: in 1931, Kurt Gödel proved that every mathematical system powerful enough to be interesting will always contain unsolvable paradoxes.

But the underlying approach remained extremely powerful. Von Neumann himself made several attempts to advance the Hilbert programme, and Alan Turing, the British mathematician, solved one of Hilbert's most important questions, the Entscheidungsproblem (or "decision problem") with a paper in 1936 that forms the theoretical basis for computer science.

More broadly, the formalist approach, with its emphasis on rigorous argument from basic truths, has come to pervade the physical and social sciences. Von Neumann started by applying this approach to the foundations of set theory. He went on, in 1927, to provide a formalisation of quantum mechanics, the theory of the structure of atoms and molecules, putting it on a rigorously mathematical basis and, incidentally, providing an accommodation between competing "wave" and "particle" theories.

The next year, 1928, he also provided his first contribution to "game theory", a branch of mathematics that he largely invented and later (with the economist Oskar Morgenstern) applied to economics. It is part of the underpinning of derivatives trading and "risk technology".

The 1920s and early 1930s represented the high water mark of European dominance of advanced scientific theory. Political tensions were rising; anti-Semitism was exiling central European Jewish academics; and the US was increasingly using its economic might to establish intellectual and cultural predominance. A symbol of this was the establishment of the Institute of Advanced Study at Princeton in 1930, with Albert Einstein as a founding member. Von Neumann joined in 1933. By the end of the decade Princeton, not Göttingen, was the world centre of mathematics and theoretical physics.

By now, newly married and still in his early 30s, von Neumann's breadth of interests and intellectual fertility had made him world famous within his chosen discipline. "Mathematicians prove what they can; von Neumann proves what he wants," went the joke.

His early slimness had given way to a comfortable plumpness. His hair was receding, but his dark brown eyes retained their expressive, engaging character. He was the opposite of an ivory-tower academic: universally known as "Johnny" to his American friends, he was convivial, fond of unprintable jokes, with a reputation for ogling pretty women and driving erratically. His comfortable home in Princeton became a part of the hard-drinking academic social scene; his first marriage broke up, and he married again in 1938.

He worked relentlessly, tossing off another economics paper in 1937 which is in many ways the bedrock of modern growth theory. When the war came, he was enlisted to apply his mind to the mathematical problems of industrialised warfare, starting with guns and bombs, and moving on to help with the Manhattan Project to build an atomic bomb. His "implosion lens" was adopted for the trigger of the second bomb, Fat Boy, which destroyed Nagasaki in 1945.

By then, his restless travels between top-secret military establishments had led him into a chance meeting with the designers of ENIAC, a fast electronic calculating machine. Von Neumann was already fascinated with the potential of such machines - his trip to Bath in 1943 had been to investigate a mechanical calculator. Though this experience led nowhere in the short term, it kindled a spark. As von Neumann wrote to John Todd, his British travelling companion, it was "a decisive impulse which determined my interest in computing machines".

That interest led him to a leading role in the conceptual design of early computers.

The second world war over, von Neumann accepted a string of commercial consultancies. But he was increasingly drawn into the defence establishment. An earnest advocate of the development of the H-Bomb, he later described himself as "violently anti-communist, and a good deal more militaristic than most". After his death, Life magazine reported that he had said in 1950: "If you say why not bomb them [the Russians] tomorrow, I say why not bomb them today? If you say today at five o'clock, I say why not one o'clock?"

Though frequently quoted as evidence that von Neumann favoured a first strike "preventive war" against the Soviet Union, this is a weak reed on which to build such a claim. It seems more likely it was characteristic - though tasteless and bloodthirsty - banter designed to emphasise his hawkishness.

There is no doubt, however, of his commitment to a strong US nuclear force: he chaired the ICBM committee, which persuaded President Eisenhower to launch the missile race, and was a full-time member of the Atomic Energy Commission, which was responsible for the nuclear weapons that would sit on the rockets.

Von Neumann was not an unthinking militarist: he believed that the world's only chance of avoiding destructive conflict was world government, but - since he could not see any practical way of achieving this - he was determined to defend his adopted country with as much armed might as possible.

On a wider canvas, his thinking represented the century's characteristic belief in the ability of the force of intellect to shape the physical world. In his later years, as well as working for the government, he explored the underlying theory behind self-replicating automata; and how to forecast and - ultimately - control the weather. Although he was an early convert to the theory that carbon dioxide emissions were causing global warming, he believed - at the same time - that scientists could darken the ice-caps to adjust the earth's weather patterns. He was worried that this might become a weapon, threatening mass destruction just as deadly as nuclear war.

At the peak of his influence, in 1955, he fell ill with cancer. His friend, the physicist Eugene Wigner, later wrote: "When von Neumann realised he was incurably ill, his logic forced him to realise that he would cease to exist, and hence cease to have thoughts. . . It was heartbreaking to watch the frustration of his mind, when all hope was gone, in its struggle with the fate which appeared to him unavoidable but unacceptable." He died in 1957.

Four decades after von Neumann's death, surrounded as we are by the science and technology to which he gave birth, his character still presents a puzzle. This is surprising, because he was such a publicly accessible man, who wrote so much science and so many letters, and who knew so many people. Several interesting books have been published about him, but they are mostly either the tributes of old friends or they treat him as a stereotype. One such effort sees him as the "bad scientist" who cheerfully worked on weapons of destruction, compared with "good scientists" who had qualms; another as the cold- hearted rationalist who believed that game theory led inexorably towards a nuclear first strike.

It is hard to find justification of these views in the facts of von Neumann's life. His attitude seems to have been more heavily influenced by traditional - and understandable - central European pessimism than by any such caricatures. But he is a difficult man to understand. Alan Turing, a more private figure, is curiously more accessible.

What makes von Neumann so inscrutable is that he was not a tortured genius - to whom everything may be forgiven - but in many ways a normal person who devoted himself with ferocious energy and mental clarity to some of the central issues of the age. As with any such effort, there were mistakes, exaggerations and misjudgments. But in those, he was just as much a representative figure of our era as in his scientific discoveries or his military advice.

Paul Halmos, von Neumann's assistant, summed up his career: "The heroes of humanity are of two kinds: the ones who are just like all of us, but very much more so, and the ones who, apparently, have an extra-human spark. We can all run, and some of us can run the mile in less than four minutes; but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue. Von Neumann's greatness was of the human kind. We can all think clearly, more or less, some of the time, but von Neumann's clarity of thought was orders of magnitude greater than that of most of us, all the time."

I am grateful to Nicholas Vonneuman for talking to me about his brother's early years. His memoir, John von Neumann As Seen By His Brother, is privately published at $20 by Nicholas A. Vonneuman, 1396 Lindsay Lane, Meadowbrook, Pennsylvania 19046 USA.