Being affectionate with numbers, endlessly wondering about them, loving them, is, though impersonal and bloodless, no more strange perhaps than being possessed by the endless ramifications of cricket or trout fishing. Being consumed by numbers to the exclusion of all else, sounds deranged. The Hungarian mathematician, Paul Erdös, number theorist and combinatorialist extraordinary, eccentric, socially dysfunctional, obsessive, childishly egocentric, helplessly dependent on fellow number freaks to feed him, transport him, put him up and put up with him, was certainly outside the normal range, but not insanely so. Like most mathematicians, Erdös had a deep need to be ordered and structured, so requiring long immersion inside mathematical abstractions. He thought numbers more interesting and comforting than anything else in this world and was able to spend most of his waking life in contact with them. When he died in 1996, at the age of 83, he had worked on more problems, made more conjectures, proved more theorems, collaborated with more people, and written and co-written more mathematical papers (over 1500) than any mathematician in history.

His life, as told in Paul Hoffman’s breezy, informative and very readable biography, appears as one long anecdote. On the day he was born, his two young sisters died of scarlet fever and his mother, fearful for him, kept him at home and allowed him free rein. At three he could multiply three-digit numbers in his head, at four he showed an interest in prime numbers, at five he started thinking about squares and cubes of numbers; and so it went on. In his teens, he could boast to a younger mathematical prodigy of knowing 37 different proofs of Pythagoras’ theorem. By 21, he had obtained a PhD from the university in Budapest.

Hungary was by then in the grip of a rising tide of anti-semitism and Erdös decided that being a Jew would hinder his chances of doing mathematics; so he left and travelled the world as an itinerant mathematical wizard. He never had a permanent job, gave away money he made in excess of his needs, took to offering small money prizes for problems that caught his fancy, never learned to cook, drive a car, wash his clothes or become anything other than an appalling house guest, who thought little of waking his hosts in the middle of the night to shut his windows. He seems to have loved children, sought out mathematical prodigies to nurture, organised his moral sense round a personification of everything bad he called the Supreme Fascist, and died a virgin. He was devoted to his mother, travelling for many years with her in tow, from one university mathematics department to another, until her death. After that he travelled alone, discovered amphetamines, and for the last 17 years of his life criss-crossed a global net of collaborators at a frantic 15 cities a month. He’d turn up at fellow mathematicians’ doorsteps, declare ‘My brain is open,’ and wear them out with a gruelling 19 hours of speed-driven problem-solving punctuated by microsleeps.

I met Paul Erdös twice, and each time it was a non-event. First in Bristol in the Sixties. I was extending some results, I told him proudly, in the partition calculus he’d invented with his fellow Hungarian Richard Rado; I was also trying to settle a conjecture in infinite combinatorics – that there was no infinite descending sequence of countable order types. He ignored the first, questioned me about the second, thought for a few mom ents, then shrugged – which I read as ‘better you than me’ – and turned to a colleague of mine who had just rediscovered a combinatorial result known as Sperner’s lemma. Erdös was particularly fond of this little result and wanted to buy its rediscoverer a coffee.

The second time was in Memphis, a year or so before he died. His mathematician host asked if I’d keep the great man company before dinner. Somewhat reluctantly, I agreed. Sure enough, first question: what was I doing? I started to tell him about the work I’d done on the semiotics of mathematics, how mathematics was a giant network of waking dreams, journeys in ideal worlds, a fusion of imagination and writing, and so on. He looked at me as if I were an idiot and without a pause started into ‘Let *n* be a number, and suppose *n*-star is the number of ways you can . . .’ I should have known he’d try to avoid talking about mathematics as a subject; mathematicians get upset – G.H. Hardy, at the beginning of his famous ‘Apology’, found it a ‘melancholy experience’ – reflecting on their trade, they feel gloomy talking about it rather than doing it. And there he was, scribbling symbols, still doing it. Frail, alternately catnapping or relentlessly present, with the residual narcissism of the old, his vitality concentrated in the dark buttons of his pupils, looking ancient and Jewish, he reminded me of my father, who had died a few years earlier at 91, still proud of his mathematical acumen and challenging me to test his powers of mental arithmetic.

Paul Erdös was not a grand creative master like Poincaré or Hilbert or Cantor or von Neumann, whose work changed the face of 20th-century mathematics. He was a mathematician’s mathematician whose extraordinary gift for collaboration sprang from a genius for problems: formulating tantalising cognitive nuggets that you could play with in your head and (essential requirement) get somewhere with; and for proofs: inventing ingenious, logically tight arguments that solved them. His aesthetic insistence that proofs not be ugly, opaque or contrived was an integral part of how he thought: his highest praise was to say a proof came straight from ‘the Book’, an imagined compilation of all the most perfect (surprising, elegant, economic, natural, unimprovable) proofs. Inspired by him, two German mathematicians have just published *Proofs from the Book* (dedicated to Erdös), a book of the Book which gathers 30 such perfect demonstrations, including some of Erdös’s favourites, from the combinatorial fields he so loved.

If one had to pick a mathematical genius at the furthest pole from Erdös, it might well be John Forbes Nash, 1994 Nobel Laureate in Economics, the subject of Sylvia Nasar’s biography. Where Erdös was impish, kind, open to all and monkishly pure, Nash was overbearing, secretive and abrasive, with a stormy marriage, an illegitimate son and several complicated liaisons with men. Erdös was a problem-solver delighted with progress, given to clever formalisms, inching towards the truth in many ways at once: Nash, a self-proclaimed conqueror, iconoclast and revolutionary who wrestled with ultra-difficult problems in the depths. Erdös all speed, lightness, constant movement: Nash heaviness, depth and endurance – carrying the same problem around with him in his head, for months on end.

Nash shot to fame from a legendary obscurity when he was given the Nobel Prize (for work on game theory done four decades earlier), after recovering spontaneously in the late Eighties from thirty years of disabling paranoid schizophrenia. With a nice sense of the dramatic, Nasar starts her story on a May afternoon in 1959, with Nash, 30 years old, and with ‘a vast, distorted universe whispering in his head’, flopped in an armchair in the secure lounge of Maclean Hospital, an asylum near Boston where he had been committed for observation by MIT psychiatrists and by his 26-year-old wife, whom he was threatening to divorce. His visitor, the Harvard mathematician George Mackey, pained by Nash’s pronouncements about alien beings, asks how he could believe that extraterrestrials were sending him messages to recruit him to save the world. ‘Because,’ Mackey remembers the reply, ‘the ideas I have about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.’ Odd question, asking the mad why they aren’t rational, but an uncannily lucid and prescient answer from a man at the beginning of a thirty-year nightmare of mathematical and supernatural wandering, in which his formidable mind would be transformed into a numerological machine generating elaborate connections between the everyday details of his world and the cosmos. Nash renounced his American citizenship, thought himself to be emperor of Antarctica as well as the left foot of God, and was possessed by paranoid formulas and fearful ravings about punishment, humiliation and triumph.

*A Beautiful Mind* is an astonishing achievement that opens a vivid, poignant window onto Nash’s disintegration into delusional turmoil. Careful, sympathetic, minutely researched where there are facts and convincingly extrapolated where there are none, it takes us from Nash’s birth in 1928 in a small town in West Virginia to a picture of the dissension at the Nobel economics committee in Stockholm before its members finally agreed on Nash as its 1994 recipient. The story begins in earnest with the 20-year-old Nash arriving at Princeton, the mecca of American mathematics, to start a graduate degree. Brash, brilliant, dismissive of anybody that couldn’t follow him, seeking interviews with Princeton legends such as Einstein and von Neumann, pitting his wits against all and sundry, he made an immediate and never to be forgotten impact. Within little more than a year, he’d proved some difficult but minor mathematical results, invented a two-person Go-like board game which captured the interest of his fellow students, and written the 26-page thesis on the theory of non-co-operative games whose ramifications, not recognised or anticipated at the time, were to have such an impact on economics.

The originality and fecundity of Nash’s thesis lay not in its mathematical ideas, which were, by general consent, little more than an application of well-known fixed-point theorems, but in the concept of a game he formulated. The mathematical theory of games, as invented by von Neumann, was essentially a theory of zero-sum, two-person games (e.g. noughts and crosses or most card games) in which one player’s gain is another’s loss and in which the idea was to determine the existence of winning strategies or to prove there weren’t any. Nash’s idea of a game was much wider and allowed for situations involving conflict and mutual gain, and he demonstrated that, under appropriate assumptions, such games could end in positions of equilibrium – that is, stable standoffs – if each player independently chose his best response to the other player’s best responses. This seemingly simple idea would, in the hands of a generation of social scientists who developed it, ultimately allow a great number of situations in the social sciences, from business competition to peace negotiations, to global economics, to be analysed in mathematical terms.

Over the next few years, Nash abandoned game theory and moved on to mainstream research. From Princeton he went to the Rand Center, a government-sponsored think-tank whose Cold War protocols required that he be instantly dismissed after being caught in a local police sting operation aimed at homosexuals, and then the mathematics department of MIT, from where, within a few years, he was committed to Maclean Hospital. Before this he had had a cruelly dominating and unsatisfactory affair with a young Boston nurse with whom he had a son and entered on a stormy marriage with a beautiful and aristocratic young physics student who’d courted him and who, despite divorcing him, never abandoned him – taking him in and looking after him when his plight became too much to bear.

Bouts of involuntary incarceration, with years of drug treatment and a brutal insulin shock treatment in the Sixties failed to improve his condition. He became a hollowed out ghost who for two decades wandered the halls of Princeton’s mathematics department, muttering, cadging the occasional cigarette and leaving cryptic messages on blackboards. Then, in the late Eighties, something lifted. He started to come to – to know people and himself, spontaneously and mysteriously to re-enter the world. Had this not happened, it’s possible, as Nasar’s convincing re-creation of their machinations reveals, that the Nobel committee would not have risked the awkwardness and potential ridicule of awarding their prize to a madman; Nasar would then not have written her book, and Nash would only have been a name in social science textbooks.

Mathematics has more than its share of obsessionals and single-minded fanatics. The many recent commentaries on the Unabomber, Theodore Kaczynski, evince little surprise that he was a brilliant mathematician gone bad; as if, in the popular mind, serving the god of number and being an isolated revolutionary sending letter-bombs shared the same weird, potentially unstable source. Undeniably, a large number of mathematicians, I’d say most, find in mathematics a haven from the mess and conflicts of their feelings, from the pain and treachery of social life and the turbid confusion of everyday thought; they gladly escape imprecision and uncertainty in fortress mathematica, where all is crystal line structure ruled by everlasting truth. The retreat from the world into mathematical certainty, well attested to by Hardy and Bertrand Russell, and discernible in the pixie-like detachment of Erdös and the mad withdrawal of Nash, is there in Kaczynski, but transposed: the intellectual certainty of mathematics has become unchallengeable moral truth, a system of ethical axioms whose theorems Kaczynski murderously forged into a holy war against technology.

But fortress mathematica is also a temple in which infinity and the unfathomable complexities of number induce a secular divinity, a god of the atheists whose ancient connection to the secrets of the Universe continues to be worshipped. The Pythagorean fantasy that the Universe is constructed out of numbers is alive and well within popular, as well as high scientific culture. The low form, numerology, seems mad from the start, as in the cabbalistic pronouncement, one of many chalked on blackboards by Nash, that ‘Mao Tse-Tung’s Barmitzvah was 13 years, 13 months and 13 days after Brezhnev’s circumcision’; or the seductively hard-edged, computer-aided version in the just released film *p*, directed by Darren Aronofsky, whose protagonist, a suitably deranged mathematician searches the digits of the decimal expansion of pi (the ratio of the circumference of a circle to its diameter) for the hidden 216-digit sequence which he thinks will reveal the pattern behind the stock market but which the New York Hasidic Jews tell him will be the secret name of God.

In its high, and highly respectable form, the Pythagorean fantasy is endemic to Western rational thought, insofar as the mission and methodology of science, and particularly physics, is taken as central. Certainly mathematicians, when they ponder the utility of mathematics, and physicists in book after book about their subject, seem unwilling or incapable of renouncing the conviction of their patron saint Galileo that the book of the Universe was written (by God) in the language of mathematics.

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