Turing's Cathedral: The Origins of the Digital Universe (Penguin Press Science)

by George Dyson

Asked why he always purchased Cadillacs, he answered, “Because no one would sell me a tank.”

Von Neumann could deliver publishable text, and even mathematical proofs, on the first draft. “I write rather freely and fast if a subject is ‘mature’ in my mind,” he explained in 1945, apologizing for an undelivered manuscript, “but develop the worst traits of pedantism and inefficiency if I attempt to give a preliminary account of a subject which I do not have yet in what I can believe to be in its final form.”

“Von Neumann was one of the greatest of all mathematical artists,” says Goldstine. “It was never enough for him merely to establish a result; he had to do it with elegance and grace.”

According to Rear Admiral Lewis Strauss, von Neumann was “able to take the most difficult problem, separate it into its components, whereupon everything looked brilliantly simple, and all of us wondered why we had not been able to see through to the answer as clearly as it was possible for him to do.”

While World War I had been a battle for bigger guns, World War II (and the cold war that followed) became a battle for bigger bombs.

“So far I have chiefly worried about spherical and Gaussian measures of various functions.”53 This was shorthand for calculating the behavior of high explosives—the surprising thing about large explosions being not how much energy was released, but how unpredictable was the damage produced as a result.

“Physicists—particularly experimental physicists—are more in demand for defense work,”

In the interim, Stanley Frankel, a Berkeley graduate student of Oppenheimer’s who had been put in charge of the hand computing group, and Richard Feynman, a graduate student (and amateur safecracker) from Princeton who was game for any unauthorized challenge, managed to uncrate the machines and get them to work.

Feynman and Frankel were hooked. “Mr. Frankel, who started this program, began to suffer from the computer disease that anybody who works with computers now knows about,” Feynman later explained. “The trouble with computers is you play with them.” Feynman and Frankel, joined by Nicholas Metropolis, adapted the IBM machines to accelerate the work of the hand computing group. “If we got enough of these machines in a room, we could take the cards and put them through a cycle,” explained Feynman. “Everybody who does numerical calculations now knows exactly what I’m talking about, but this was kind of a new thing then—mass production with machines.”

Feynman secured permission from Oppenheimer to give a lecture to the recruits. “They were all excited: ‘We’re fighting a war! We see what it is!’ They knew what the numbers meant. If the pressure came out higher, that meant there was more energy released. Complete transformation! They began to invent ways of doing it better. They improved the scheme. They worked at night.”65 Productivity went up by a factor of ten.

“We used to go for walks on Sunday,” remembers Feynman. “We’d walk in the canyons, and we’d often walk with Bethe, and Von Neumann, and Bacher. It was a great pleasure. And Von Neumann gave me an interesting idea; that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of Von Neumann’s advice. It’s made me a very happy man ever since.”

Let the whole outside world consist of a long paper tape. —John von Neumann, 1948

“One cannot stumble on an idea unless one is running,” Zworykin advised those who joined his lab.

“A sufficiently approximate solution of many differential equations can be had simply by solving an associated difference equation,” Mauchly had written in August of 1942.

Virtually all the algorithms that humans had devised for carrying out calculations needed reexamination.

A fast internal memory, coupled to a larger secondary memory, and linked in turn to an unlimited supply of punched cards or paper tape, gave the unbounded storage that Turing had prescribed.

Mathematicians who had worked on applications during the war were expected to leave them behind. Von Neumann, however, was hooked.

“There he clearly stunned, or even horrified, some of his mathematical colleagues of the most erudite abstraction, by openly professing his great interest in other mathematical tools than the blackboard and chalk or pencil and paper. His proposal to build an electronic computing machine under the sacred dome of the Institute, was not received with applause to say the least.”

It wasn’t just the pure mathematicians who were disturbed by the prospect of the computer. The humanists had been holding their ground against the mathematicians as best they could, and von Neumann’s project, set to triple the budget of the School of Mathematics, was suspect on that count alone. “Mathematicians in our wing? Over my dead body! and yours?” Aydelotte was cabled by paleographer Elias Lowe.

“The device we are all dreaming about is something very much more than a computing device…. It is a device wherewith one carries out, accurately and rapidly, certain electrical and mechanical processes which are isomorphic with certain important mathematical processes.”

“These uses which are not, or not easily, predictable now, are likely to be the most important ones. Indeed they are by definition those which we do not recognize at present because they are farthest removed from … our present sphere.”

“Von Neumann was well aware of the fundamental importance of Turing’s paper of 1936 ‘On computable numbers …’ which describes in principle the ‘Universal Computer’ of which every modern computer (perhaps not ENIAC as first completed but certainly all later ones) is a realization,” Stanley Frankel explains.

“Johnny had by then a very definite idea of how and why he wanted this machine to function with the emphasis on the why,” remembers Klári. “He wanted to build a fast, electronic, completely automatic all purpose computing machine which could answer as many questions as there were people who could think of asking them.”

We have been trying to see how far it is possible to eliminate intuition, and leave only ingenuity. We do not mind how much ingenuity is required, and therefore assume it to be available in unlimited supply. —Alan Turing, 1939

In September of 1930, at the Königsberg conference on the epistemology of the exact sciences, Gödel made the first, tentative announcement of his incompleteness results. Von Neumann immediately saw the implications, and, as he wrote to Gödel on November 30, 1930, “using the methods you employed so successfully … I achieved a result that seems to me to be remarkable, namely, I was able to show that the consistency of mathematics is unprovable,” only to find out, by return mail, that Gödel had got there first.

“He was disappointed that he had not first discovered Gödel’s undecidability theorems,” explains Ulam. “He was more than capable of this, had he admitted to himself the possibility that Hilbert was wrong in his program. But it would have meant going against the prevailing thinking of the time.”

Gödel set the stage for the digital revolution, not only by redefining the powers of formal systems—and lining things up for their physical embodiment by Alan Turing—but by steering von Neumann’s interests from pure logic to applied.

Gottfried Wilhelm Leibniz, born in Leipzig in 1646, enrolled in the University of Leipzig as a law student at age fifteen. Our universe, Leibniz theorized, was selected from an infinity of possible universes, optimized so that a minimum of laws would lead to a maximum diversity of results.

Leibniz believed, following Hobbes and in advance of Hilbert, that a consistent system of logic, language, and mathematics could be formalized by means of an alphabet of unambiguous symbols manipulated according to mechanical rules.

Anticipating Gödel and Turing, Leibniz promised that through digital computing “the human race will have a new kind of instrument which will increase the power of the mind much more than optical lenses strengthen the eyes…. Reason will be right beyond all doubt only when it is everywhere as clear and certain as only arithmetic has been until now.”

Leibniz’s belief in a universal digital coding embodied his principle of maximum diversity: infinite complexity from finite rules.

What Gödel (and Turing) proved is that formal systems will, sooner or later, produce meaningful statements whose truth can be proved only outside the system itself.

Wiener had launched his mathematical career with a theory of Brownian motion—the random trajectory followed by a microscopic particle in response to background thermodynamic noise. He was thus prepared for the worst case possible: an aircraft that changes course at random from one moment to the next.

Wiener’s theory, strengthened by Bigelow’s experience as a pilot, held that the space of possible trajectories (equivalent to the space of possible messages in communications theory) was constrained by the performance envelope of the aircraft and the physical limitations of the human being at the controls.

Almost all combat flying, Bigelow observed, was composed of curves, not straight lines. Straight-line extrapolation of a flight path was a reliable predictor only of where the...

The goal, as Bigelow described it, was an anti-aircraft director that kept the signal (the aircraft’s flight path) separate from the noise: noise introduced both by a pilot attempting to behave unpredictably, and by observation and processing errors along the way.

“To re-separate the signal from the soup in these last two terms is no cinch, and in the case of random or Brownian noise with no simple spectrum it is quite impossible to do the filtering perfectly,” noted Bigelow. “Result: lost ground.”

Bigelow describes the situation as one of people “who had to think about what they were trying to do” objecting to people “who seemed to know what they were trying to do.”

It was like looking into a very accurate mirror with all unnecessary images eliminated, only the important details left.

Booth, who believed that “if mathematics isn’t useful, it’s not worth doing,” left Cambridge to pursue physics, engineering, and chemistry on his own terms.

“They would work till eight or nine o’clock, go out to dinner for two hours and then go back to work again,” remembers Thelma Estrin, an electrical engineer who arrived, with her husband, engineer Gerald Estrin, in June 1950, in the middle of the final push to finish the machine. “Sometimes they would work all night.”

All computations were run twice, and accepted only when the two runs produced duplicate results. “I have now duplicated BOTH RESULTS how will I know which is right assuming one result is correct?” asks an engineer on July 10, 1953. “This now is the 3rd different output,” notes the next log entry. “I know when I’m licked.” Someone running a hydrogen bomb code from 2:09 a.m. to 5:18 a.m. on July 15, 1953, signs off: “if only this machine would be just a little consistent.”

The part that is stable we are going to predict. And the part that is unstable we are going to control. —John von Neumann, 1948

After the war, Richardson published a detailed report, Weather Prediction by Numerical Process, so that others might learn from his mistakes. At the end of his account, he envisioned partitioning the earth’s surface into 3,200 meteorological cells, relaying current observations by telegraph to the arched galleries and sunken amphitheater of a great hall, where some 64,000 human computers would continuously evaluate the equations governing each cell’s relations with its immediate neighbors, maintaining a numerical model of the atmosphere in real time.

“By varying the input continuously and observing the output one could determine how most efficiently to modify the input to produce a given output.

“The atmosphere,” he explained in the next progress report, “is composed of a multitude of small mass-elements, whose behavior is so interrelated that none can be dissociated, even in effect, from all the rest.” The problem was how to translate the analog computation being performed by the atmosphere into a digital computer and speed it up.

“All the meteorologists were great fun, hard drinkers,” says Hungarian topologist Raoul Bott, who arrived, with a degree in engineering, as a protégé of Hermann Weyl’s in 1949. “We had tremendous wild parties,” he remembers. “It was a high point in my life.”

Monte Carlo

From this he drew the conclusion, with implications perhaps not yet fully appreciated, that “one sense of Gödel’s theorem is that some properties of these games can be ascertained only by playing them.”

“After examining the possible histories of only a few thousand, one will have a good sample and an approximate answer to the problem.”