The foundation myth of the internet invariably involves an iconoclastic and romantic technology entrepreneur, who, free from government restraint, enlists free-floating venture capitalists in building the Next Great Thing. It's a myth that borders on delusion, for some of the key technologies that led to the internet were underwritten by government subsidies and arose in the context of larger-than-life geopolitical battles.

Thus, cryptography, which powers much of today's electronic commerce, advanced in the background of the second world war, while packet switching  a cold war-era technology that made the internet possible  was to guarantee resilient communications in the event of a nuclear attack. More recently, 9/11 and the wars it unleashed have magically transported biometric technologies such as automated facial recognition from the battlefields of Afghanistan and Iraq into our offices and living rooms.

In Turing's Cathedral, George Dyson shows that the history of the modern computer belies the foundation myth as well. Dyson, who has previously written on the history of the Aleut kayak and a failed American attempt to send a mission to Mars, traces one particular effort to build and operate a computer  the unassumingly named Electronic Computer Project (EPC) based at the Institute for Advanced Study (IAS) at Princeton.

EPC was underwritten by various parts of the American government shortly after the second world war. The idea was to use computers to forecast the consequences of a thermonuclear explosion; eventually, the IAS computer was also put to more peaceful uses in biology and meteorology.

The project's godfather  the Hungarian émigré John von Neumann  was a polymath whose interest in computing had roots in both politics and academia. A superb mathematician who also made landmark contributions to economics and game theory, Von Neumann believed that computers might push mathematicians  who constituted the most powerful group at the institute  to appreciate the theoretical challenges posed by applied work. At the same time, his aversion to totalitarianism made him eager to help bolster the military might of his adopted homeland.

It took a genius of Von Neumann's scale to overcome the immense opposition to the project at the institute, which was a fascinating microcosm of intellectual life at the time (Dyson's book is worth reading for its treatment of the institute's early history alone). Building and operating a computer on the institute's premises meant opening its doors to engineers  a development that professional mathematicians, averse as they were to any work that didn't require chalk, blackboard, paper and pencils, didn't like at all. The institute's humanists hated mathematicians and engineers alike and, now that the war was over, didn't shy away from expressing their discontent.

It didn't help that Einstein, who was then at the institute, opposed the idea of "secret war work" and feared that "the emphasis on such projects will further ideas of 'preventive' wars." However, "preventive wars" were exactly what the hawkish Von Neumann wanted: in the immediate aftermath of the second world war, he briefly advocated the idea of a quick preventive war with the USSR to be followed by a Pax Americana. He also had no qualms about working with the government, eventually leaving the institute in 1953 to join the United States Atomic Commission  a government agency that would soon humiliate his friend and colleague Robert Oppenheimer by stripping him of his security clearance.

Strictly speaking, Von Neumann's was not the first computer. However, it played an extremely important role in getting the nascent computer industry off the ground. First of all, its origins in academia made it easier to get working scientists to pay close attention to what computing had to offer. Second, Von Neumann wanted to ensure that any work that the institute did on the EPC was put in the public domain and widely disseminated rather than patented by engineers (this noble effort was marred by Von Neumann's consulting gig with IBM  not well-publicised at the time  which required him to grant all of his own subsequent inventions to the company). Third  and most important  Von Neumann chose not to optimise his computer to do only pressing or lucrative tasks; he knew that its most useful applications had not been anticipated yet. By arguing that "the projected device is so radically new that many of its uses will become clear only after it has been put into operation", Von Neumann helped to usher in the era of general-purpose computing which, alas, may now be finally coming to a close, as consumers embrace single-purpose apps and tightly controlled computing devices.

While Dyson doesn't shy away from discussing obscure technical and theoretical aspects of Von Neumann's computer, he also provides ample social and cultural context. Gottfried Leibniz, Francis Bacon, and Bishop Berkeley appear next to more contemporary luminaries such as Norbert Wiener (the originator of cybernetics), Vladimir Zworykin (a pioneer of television) as well as numerous members of the Huxley family (Aldous, Julian and Thomas). Dyson, who grew up at the institute, where his father Freeman Dyson was a fellow, also brings a charming personal touch to the narrative.

Alas, the book is not perfect. Dyson, who spent a decade writing and researching it, bombards the reader with a mind-boggling stream of distracting information that adds little to his tale. We get to learn of the discrepancy between the British and Canadian war records of Jens Fredrick Larson, the architect of the institute's main hall; the price of oysters served at lunch meetings of its building committee; the price of nappies in Los Alamos hospitals in the 1950s.

Dyson's efforts to connect Von Neumann's cold war computing to today's Silicon Valley result in a slew of untenable generalisations. Is it really true that "Facebook defines who we are, Amazon defines what we want, and Google defines what we think"? Occasionally, Dyson makes mystical claims that no serious historian would endorse. What to make of his statement that "only the collective intelligence of computers could save us from the destructive powers of the weapons they had allowed us to invent"? This is a very odd way to tell the story of numerous disarmament campaigns, of fervent antiwar activism of the 1960s, of the emergence of groups like Computer Professionals for Social Responsibility that sought to draw clear ethical boundaries between academia and the defence industry. Surely, all of that mattered more than the "collective intelligence of computers"?

Despite these shortcomings, Turing's Cathedral is an engrossing and well-researched book that recounts an important chapter in the convoluted history of 20th-century computing. An equally rich history of Google and Amazon is long overdue.

Evgeny Morozov is the author of The Net Delusion (Penguin)

At first sight  and it's a long first sight, lasting a good 200 of the book's 340 brilliant and frustrating pages of text  Turing's Cathedral appears to be a project for which George Dyson has failed to find a form. Ostensibly the story of the building of one of the earliest computers at Princeton in the late 1940s and early 50s, it keeps digressing wildly. The Institute for Advanced Study's MANIAC gets under construction over and over, in chapter after chapter, only for Dyson to veer off again into the biographical backstories of the constructors, and a myriad of alternative intellectual hinterlands, from hydrogen bomb design to game theory to weather prediction, by way of the café society of interwar Budapest. It's not that these aren't relevant. They are; but they aren't introduced in the cumulative, surreptitiously spoon-feeding way in which good pop-sci writing usually coaxes a linear narrative out of complex material.

If this is a cathedral, it doesn't have anything as geometrical as a nave. It's a mass of separate structures joined by spiders' webs of coloured string. But it isn't a failure. It isn't one thing at all. It's three successes: three separate and different and differently impressive books Dyson might have written, all bizarrely shredded and mixed into a heap whose sorting is left as an exercise for the reader. Some of it is a painstaking oral history of MANIAC, built on an archivist's certainty that everything is worth rescuing from entropy that can possibly be known about the dawn of the digital computer. Truly everything, from interviews with as many of the surviving engineers as possible in the 1990s, to the institute's cafeteria manager's unexpected history testing Blériot monoplanes in 1912, and the director's complaint in 1946 that the engineers were putting too much sugar in their tea. This part of the book is a monument (or rather a bit-stream of a monument).

Some of it is an intellectual biography of MANIAC's chief architect John Von Neumann and the circle around him, determined to do justice to the polymathic range of his genius, and therefore dipping into everything he contributed to, from bomb design to game theory to robotics. Alan Turing, after whom the book is misnamed  it should really be called "Johnny's Web"  only comes into the picture seriously on page 242. He is merely the collaborator of Von Neumann who happened to stand along the particular out-raying string of his interest that happened to lead to the intellectual foundations of the digital age. But since Dyson himself is passionately interested in those, in comes the third separate thing the book is, a speculative, even visionary account of the philosophy of programming.

This last, marvellous element dominates the end of the book; and having reached it, and begun to be able to make sense in retrospect of the digressive tangle that came before, you ask yourself whether its design might possibly have been consciously, artfully non-linear. A kind of literary equivalent to the whole-genome shotgun method, maybe, with the shredding of multiple projects handing over to us the job of sequencing and unification. But it feels less willed than that, more the interference pattern of three different ambitions, none of which the author was ready to relinquish. And it does, no denying, take persistence. Is it worth persisting? Absolutely. Let me give you, appropriately enough, three reasons why.

One: no other book about the beginnings of the digital age brings to life anything like so vividly or appreciatively the immense engineering difficulty of creating electronic logic for the first time; of creating originally, and without a template, the pattern of organisation which has since become absolutely routine, and been etched on silicon at ever smaller micron-distances in chip foundries. The very word "foundry" insists that logic is a commodity, a material, the steel of the information age. But it didn't start like that. It started as an elaborate, just-possible accomplishment, requiring both conceptual brilliance and ingenious hands-on tinkering. It had to be built from scratch at the macro level, as an assemblage of valves and hand-wired circuits and cathode-ray tubes, fed by power at many different voltages, and protected from hazards ranging from roofing-tar to thunderstorms to the magnetic fields of passing trams. When Dyson describes the MANIAC being designed into its casing "like the folding of a cerebral cortex", you know he means  specifically that, like a brain. He has read the error logs in which the baffled pioneers tried to work out which of a hundred causes produced each failure, from a simple error in coding logic to the finicky failure of adjacent phosphor spots to stay distinctly charged. "I know when I'm licked." "This now is the 3rd different output." "To hell with it!"

Two: no other book has engaged so intelligently and disconcertingly with the digital age's relationship to nuclear weapons research, not just as a moral quandary to do with funding, but as an indispensable developmental influence, producing the conceptual tools that would unlock the intellectual power of the computer. The "Monte Carlo" method (Von Neumann and Stanislaw Ulam) was born as a means to track the probability of a thermonuclear reaction staying supercritical in a hydrogen bomb. If there had been no branching paths of scattering, splitting, absorbing or escaping neutrons to be modelled, there might well have been no algorithms to simulate the probabilistic paths of evolution, finance, climate. Conversely, if there had been no Monte Carlo algorithm running at electronic speed on Maniac itself, there would have been no American H-bomb in 1952, vaporising 80m tons of Enewetak Atoll in a red cloud boiling half the sky.

Three: no other book  this is where we get visionary  makes the connections this one does between the lessons of the computer's origin and the possible paths of its future. Dyson takes his cue from Turing and Von Neumann's ability to see all the way to the limits of the digital architecture they were themselves proposing and struggling to substantiate for the first time.

In the late 1940s they were already thinking about the essential rigidity and (from one point of view) logical inefficiency of machines which, unlike living information processors, can only do one thing at a time, leaving the whole elaborate structure of the rest idle. As Dyson puts it: "There is a thin veneer of instructions, and then there is a dark empty 99.9%." Yet the "Von Neumann architecture" of a memory passing individual bits to a processor, each with its own unique memory address, is not the only possible one, and not the only one considered by Von Neumann, for that matter. Dyson believes that the birth of other architectures atop the reliable substrate of the digital-as-we-know-it is now imminent. Some of his suggestions may be, let's say, in advance of the evidence, like the idea that Google represents a first sketch of what Turing called "an oracle machine", supplementing its own deterministic states with the non-deterministic input of human queries. But then so were many of Turing's and Von Neumann's ideas a little previous, to say the least.

Most of us should persist in reading this for the scrambled richness of its history. But I suspect that one of its afterlives is going to be as a source of koans for coders, troublingly simple questions to be copied out, and sellotaped to workstations, and stared at until  eureka!  something new happens in a human mind, and shortly thereafter in one of its electric surrogates.

 Francis Spufford's Red Plenty is published by Faber.