Turing's Cathedral

by George Dyson

  “The matter should not be left in the hands of the European gangsters,” they warned. Acknowledging that “some effort, not entirely successful, has been made to enlist the help of the United States Government,” they urged Aydelotte to bring the prospect of an atomic bomb (“which we have had on our minds for several months, without knowing what, if anything, to do about it”) to the attention of the Rockefeller Foundation, “which would be in a position to act in a simple and direct manner.”33 The Foundation responded with emergency funding to help quietly bring key European nuclear physicists—among them Wolfgang Pauli and the brothers Niels and Harald Bohr—to the safety of England and the United States. When the Manhattan Project launched in 1942, critical talent was in place.

  The United States finally entered the war in December 1941. “At long last [Johnny] could effectively vent his spleen,” says Klári. “At the same time, he was also using this perfectly honorable, patriotic excuse to shake off the self-imposed yoke of pure mathematics and get into more applied fields, with which he had a secret flirtation long before he openly admitted his steadily increasing interest in it.”34 Von Neumann would never return to pure mathematical work.

  Klári became pregnant, and Johnny’s customary signature line to the Ulams—“from house to house”—was revised to include “best greetings from both of us, and (½)2 unknown.” Klári, now thirty-one, suffered a miscarriage on June 16, 1942, and Johnny was increasingly absent from Princeton for defense-related work. His assignment to Britain on behalf of the navy in early February 1943 was both secretive in purpose and indeterminate in duration. All communication was censored. On April 13, 1943, he cabled Klári from London: “Congratulations on statistics very impressed stop Boske visiting here all very well very much love.” The telegram was intercepted. “Will you please be so kind as to furnish this office with a complete explanation of the text of this message,” the Office of Censorship asked.35

  With censors looking over his shoulder, von Neumann’s correspondence lost its passionate tone. “The recent monotonous style in your letters infuriates me,” Klári wrote on May 15, 1943. “What on earth is the matter with you?” Klári took a wartime job, full time, with Princeton University’s Office of Population Research, under the auspices of the Rockefeller Foundation and the Woodrow Wilson School. Frank W. Notestein’s population research group was looking at both historical trends in human population and a series of future “what ifs”—for instance, what would happen to a reconfigured postwar Europe, a centrally planned Soviet Union, or a proposed Jewish state in the Middle East? Klári was swiftly promoted, and offered an academic position in 1944, which she declined.

  In July, von Neumann was recalled from England and began disappearing under ever more secretive circumstances, which led, in September, to Los Alamos, where “Project Y” was now under way. When not in residence at Los Alamos, he spent much of his time on the West Coast, returning occasionally to Princeton and making regular visits to Chicago, Oak Ridge, Philadelphia, Aberdeen, and Washington, D.C. At Los Alamos he was able to get cigarettes—preferably Lucky Strikes—at the PX, hoarding them for Klári in Princeton. “Whenever he came home, we usually spent most of the night talking,” Klári remembers, “his pent-up tension was pouring out in a flow of words which, as a rule, he kept strictly to himself.”36

  On October 19, 1943, the Institute added additional coverage for “extra-hazardous activities” to von Neumann’s insurance policy under his contract with the Office of Scientific Research and Development, a sign that he was taking a more than theoretical interest in weapons research. When the surrender of Germany was announced, he was on a field assignment at Los Alamos, and it was twelve hours before he heard the news. “Well, it’s over,” he wrote to Klári the next morning. “How do you feel?” The cigarette situation improved, and the scientists kept working on the bomb. “Since May 3, inclusive, I am getting an average of about 2 packs of Luckies per day,” he reported to Klári on May 11. “Whadayasay?”37

  The next six months brought intense activity: the Trinity test, Hiroshima, Nagasaki, the surrender of Japan, and, behind the scenes, the completion of the ENIAC, the first H-bomb calculations, and the launching of the computer project at the IAS. “This exposure to such a marvelous machine,” recalled Nicholas Metropolis, regarding his first visit to reconnoiter the ENIAC, “coupled in short order to the Alamogordo experience was so singular that it was difficult to attribute any reality to either.” The same day that a copy of the Trinity bomb was dropped on Nagasaki, Edward Teller cabled von Neumann in Princeton: “Stan and Nick can now act openly as coming from Los Alamos.”38 Stan Frankel and Nick Metropolis were already at the Institute for Advanced Study to begin preparing the first H-bomb codes.

  Von Neumann was now thinking about the next war, and whether it would be fought with nuclear weapons on both sides. “The date of the next war is probably determined by the time it takes the conscious and the subconscious processes of the American people to get into equilibrium with each other,” he wrote to Klári in October 1946. “I don’t think that this is less than two years and I do think that it is less than ten.”39 He had long believed that Soviet Russia would prove to be a greater threat than Germany or Japan. “When the Western troops stopped and even withdrew to let the Russians advance deeper into Germany, Johnny was frantically dismayed,” explains Klári. “His idea was that the Western Allies should have kept going all the way into Russia and abolish in one sweep any dangerous or potentially dangerous form of government that might lead to another war. In the immediate postwar years, Johnny quite openly advocated preventive war before the Russians became too strong.”40

  Klári visited Los Alamos for the first time over Christmas 1945. She headed west by train from Princeton, via Chicago, where she boarded the Super Chief. “Will expect you Lamy Saturday morning,” Johnny cabled on December 15. “Bring riding and skating things if possible opportunities very good.”41 It was love at first sight. The mountains, the horseback riding, the skiing, the Pueblo ruins at Bandelier, the Lodge at the former Los Alamos Boys School (where Johnny, as a VIP, was entitled to stay), the predominance of Europeans (including Hungarians), the frequent, spontaneous parties, the late-night poker games—all evoked memories of Monte Carlo and Budapest. Los Alamos had what Princeton lacked. The sparks between Klári and Johnny were rekindled, and they began collaborating on the codes that would animate the new computer and bring the super bomb to life.

  “The new mathematical tool was not the only experiment that Johnny wanted to try in this connection,” Klári remembers. “He also wanted to see how someone who had none or very little experience in the field, how such a person would take to this novel way of doing mathematics. For this experiment he needed a guinea-pig, preferably a mathematical moron and, unquestionably for this purpose the ideal subject was right there within easy reach—namely me.” Klári had passed her high school examinations in algebra and trigonometry, but only because “my math teacher rather appreciated my frank admission, that I really did not understand a single word of what I had learned.”

  “Long before the machine was finished I became Johnny’s experimental rabbit,” she says. “It was lots and lots of fun. I learned how to translate algebraic equations into numerical forms, which in turn then have to be put into machine language in the order in which the machine has to calculate it either in sequence or going round and round until it has finished with one part of the problem and then go on some definite which-a-way, whatever seems to be right for it to do next.” Klári found programming to be a “very amusing and rather intricate jig-saw puzzle,” and soon “became one of the first ‘coders,’ a new occupation which is quite wide-spread today.”

  “The machine would have to be told the whole story, given all the instructions of what it was expected to do at once, and then be permitted to be on its own until it ran out of instructions,” Klári explains. “There already existed fast, automatic special purpose machines, but they could only play one tune
… like a music box.… In contrast, the ‘all purpose machine’ is like a musical instrument.”42

  There were none of the conveniences programmers take for granted today: compilers, operating systems, relative addressing, floating-point arithmetic. Every memory location had to be specified at every step, and the position of the significant digits adjusted as a computation progressed. “People had to essentially program their problems in absolute,” James Pomerene explains. “In other words, you had to come to terms with the machine and the machine had to come to terms with you.”43

  Klári’s wartime work on population statistics had prepared her for the problems that Johnny was starting to code. The question of whether a given bomb design explodes—and if so, how efficiently—depends on how rapidly its population of neutrons reproduces, and whether mortality and emigration have any moderating effects. “Statistical questions will be amenable to an entirely new kind of treatment,” von Neumann had explained in January of 1945, while the ENIAC was still being built. “It will be possible to answer most questions of this type by performing the actual statistical experiment: by computing hundreds or thousands of special cases and registering their statistical distribution.”44 A statistical approach to otherwise intractable physical problems had been taken up by others, including Enrico Fermi in the 1930s, but it took someone—and that someone was Stan Ulam, assisted by von Neumann and Nicholas Metropolis—to come up with a name for the technique and make it stick.

  At the end of the war, there had been an exodus from Los Alamos. With its remote location and total secrecy no longer necessary, work at the laboratory appeared to be winding down. Those with families to support were advised to leave, if they could. Stan and Françoise Ulam, with their one-year-old daughter, Claire, left for California, where Stan had been offered a teaching job at USC. Before he had settled into the new job, and before Françoise and Claire had even found a place to live, Stan suddenly fell gravely ill, with a case of viral encephalitis that might have killed him without an emergency trepanation at Cedars Sinai Hospital to relieve the pressure on his brain.

  Overwhelmed, Françoise arranged to send Claire back to Los Alamos, in care of David Hawkins (Stan’s collaborator on neutron multiplication) and his wife, Frances (who ran the Los Alamos nursery school). Stan recuperated in Los Angeles, while Claire thrived among the families who had remained on the mesa, where Norris Bradbury, a more down-to-earth administrator than Oppenheimer, had taken the helm. Because the Ulams had lost their government health insurance, and Stan had not started teaching yet, things were looking grim. Then Stan was invited to return to Los Alamos. “The case [of Stan Ulam] is almost unique,” wrote von Neumann, who no doubt had a hand in the invitation, to Carson Mark. “I feel that it is justified for the Los Alamos Laboratory to go any length to keep him there.”45

  Ulam, who had been advised, during his convalescence, to avoid strenuous mental activity, amused himself by playing solitaire. He could not resist a question: What were the chances that a Canfield solitaire with fifty-two cards will play out successfully? “After spending a lot of time trying to estimate them by pure combinatorial calculations,” he recalls, “I wondered whether a more practical method than ‘abstract thinking’ might not be to lay it out say one hundred times and simply observe and count the number of successful plays.” This, he noted, was a far easier way to arrive at an approximate answer than “to try to compute all the combinatorial possibilities which are an exponentially increasing number so great that, except in very elementary cases, there is no way to estimate it.”46

  “This is intellectually surprising, and if not exactly humiliating, it gives one a feeling of modesty about the limits of rational or traditional thinking,” he added. It was characteristic of Ulam to draw deep mathematical conclusions where others would simply consider the immediate problem solved. He observed that mathematical logic itself can be considered as “a class of games—‘solitaires’—to be played with symbols according to given rules.” 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.”47

  Ulam’s attempt to take his mind off serious problems soon brought him back to some of the Los Alamos problems that had been left unresolved. “It occurred to me then that this could be equally true of all processes involving branching of events, as in the production and further multiplication of neutrons in some kind of material containing uranium or other fissile elements,” he recalled. “At each stage of the process, there are many possibilities determining the fate of the neutron.… The elementary probabilities for each of these possibilities are individually known … but the problem is to know what a succession and branching of perhaps hundreds of thousands or millions will do.”48

  Monte Carlo originated as a form of emergency first aid, in answer to the question: What to do until the mathematician arrives? “The idea was to try out thousands of such possibilities and, at each stage, to select by chance, by means of a ‘random number’ with suitable probability, the fate or kind of event, to follow it in a line, so to speak, instead of considering all branches,” Ulam explained. “After examining the possible histories of only a few thousand, one will have a good sample and an approximate answer to the problem.”49 The new technique propagated widely, along with the growing number of computers on which it could run. Refinements were made, especially the so-called Metropolis algorithm (later the Metropolis-Hastings algorithm) that made Monte Carlo even more effective by favoring more probable histories from the start. “The most important property of the algorithm is … that deviations from the canonical distribution die away,” explains Marshall Rosenbluth, who helped invent it. “Hence the computation converges on the right answer! I recall being quite excited when I was able to prove this.”50

  Monte Carlo opened a new domain in mathematical physics: distinct from classical physics, which considers the precise behavior of a small number of idealized objects, or statistical mechanics, which considers the collective behavior, on average, of a very large number of objects, Monte Carlo considers the individual, probabilistic behavior of an arbitrarily large number of individual objects, and is thus closer than either of the other two methods to the way the physical universe actually works. “Because one seems to be getting something for nothing, it is necessary to keep straight the process by which everything comes out all right in the end; the efficiency of the methods in particular cases seems unbelievable,” advised Andrew Marshall in 1954, reviewing Monte Carlo’s first seven years. “The results quite literally have to be seen, and seen through, to be believed.”51

  On von Neumann’s next visit to Los Alamos, Ulam brought up the idea as von Neumann was leaving to catch the train. “It was an especially long discussion in a government car while we were driving from Los Alamos to Lamy,” where the railroad depot was located, Ulam recalls. “We talked throughout the trip, and I remember to this day what I said at various turns in the road or near certain rocks.” Somewhere along the line, with credit usually going to Nick Metropolis, “it was named Monte Carlo,” Ulam explains, “because of the element of chance, the production of random numbers with which to play the suitable games.” The idea was impossible to resist. “Ulam relished the thought of a gambling spree in which the scorekeeping was so designed as to imitate a neutron chain reaction,” Robert Richtmyer recalls. “It’s infinitely cheaper to imitate a physical process in a computer and make experiments on paper, as it were, rather than reality,” Ulam testified at the ENIAC trial in 1971.52

  After the drive to Lamy, von Neumann returned by train to Princeton, working up Ulam’s suggestion during the trip, and then, following a telephone conversation on March 7 with Richtmyer, he typed up an eleven-page letter, fleshing out (for “spherically symmetric geometry,” of either uranium or plutonium) Ulam’s idea. “I am fairly certain that the problem, in its digital form, is well suited for the ENIAC,” he wrote. “Assume
that one criticality problem requires following 100 primary neutrons through 100 collisions (of the primary neutron or its descendants) per primary neutron. Then solving one criticality problem should take about 5 hours.” This would only address the simplified question of “static criticality”—whether the specified assembly would explode, not how well would it explode. Von Neumann estimated what it would take to address this more complex question, involving both hydrodynamics and radiation transport, concluding that “I have no doubt whatever that it will be perfectly tractable with the post-ENIAC device.”53