“A quaint and ceremonious village of puny demigods on stilts,” wrote Albert Einstein to the queen of Belgium. It was November 1933 and Einstein had been in Princeton, New Jersey, for just a month having left Europe after the Nazis’ rise to power. The celebrated physicist wasn’t referring to the ordinary citizens of the famous university town but to some of his new colleagues.

Einstein’s recruitment by the Institute for Advanced Study was like a modern-day star signing for a football club backed by a Russian oligarch. His presence there soon attracted a stellar collection of mathematicians. The brightest of these was a 30-year-old Hungarian.

John von Neumann was a mathematical prodigy who published his first paper at 18. As far removed from the caricature of the bookish, socially awkward mathematician as possible, he was a charming, party-loving, hard-drinking raconteur who mastered ancient Greek, Latin, French, German, English and Hungarian as a child growing up in Budapest. He was also capable of chillingly cold calculation, and an attitude to human life in the context of the cold war that can seem breathtaking in its ruthlessness. Nevertheless, despite being the real-life model for Stanley Kubrick’s Dr Strangelove, he helped usher in the computer age.

There was certainly no doubt about his intellect. To entertain his friends, he would recite verbatim passages from books read years before. An unnamed historian, so the story goes, would only attend one of Von Neumann’s legendary parties if they didn’t discuss Byzantine history, since he wanted to maintain his reputation as the world’s foremost expert.

Non-Euclidean geometry, set theory, the prisoner’s dilemma, Gödel’s incompleteness theorems, self-replicating machines, game theory and nonlocality are among the astonishing range of topics that science journalist Ananyo Bhattacharya covers as he takes us on a whistle-stop tour through Von Neumann’s restless mind. It was after tackling some of the most vexing questions in mathematics that Von Neumann turned his attention to physics and did something quite remarkable.

With his extraordinary capacity to see beyond the superficial complexities of a problem and break it down to its most fundamental form, Von Neumann reduced quantum mechanics to its essentials, and in the process showed that two competing formulations of the theory – Erwin Schrödinger’s wave mechanics and Werner Heisenberg’s matrix mechanics – were, in fact, one and the same. It was a result that inspired him to build the first rigorous framework for the new science of the atomic realm.

During the second world war, Von Neumann quickly became an expert in the science of ballistics and explosions. Brought in as a consultant to the Manhattan Project, the top-secret effort to make an atomic bomb, he worked out the best way to detonate it. Solving the equations required extensive number crunching with the aid of mechanical desk calculators. Looking at ways to speed up the time-consuming work, Von Neumann came across the Electronic Numerical Integrator and Computer, a machine being developed for the army to compute firing tables for artillery. You could almost live inside the ENIAC, as it occupied a room 30 feet wide and 56 feet long. It consisted of some 18,000 vacuum tubes and banks of wiring and switches arranged in panels 8 feet tall. Von Neumann helped turn it into the world’s first programmable digital computer.

In an effort to understand conflict and cooperation, Von Neumann co-wrote a 640-page book on game theory, which had a huge impact on economics, psychology and military strategy. He applied game theory to nuclear war, openly advocating a pre-emptive strike on the Soviet Union. “With the Russians it is not a question of whether but when,” he told Life magazine. “If you say: ‘Why not bomb them tomorrow,’ I say: ‘Why not today?’ If you say: ‘Today at 5 o’clock,’ I say: ‘Why not one o’clock?’” He may have been prepared to send humanity back to the stone age but, as revealed in this splendid new biography, Von Neumann did much to create the world we live in now.

• The Man from the Future: The Visionary Life of John von Neumann is published by Allen Lane (£20). To support the Guardian and Observer order your copy at guardianbookshop.com. Delivery charges may apply.