Turing's Cathedral Read online

  He evidenced “a chameleon-like ability to adapt to the people he was with,” remembers Herman Goldstine, and never claimed he could not explain something to someone who did not understand the math. “It was just like being out on glass, it was so smooth. He somehow knew exactly how to get you through the forest. Whenever he gave a lecture, it was so lucid, it was like magic, it all seemed so simple you didn’t need to take notes.” Nicholas remembers his brother returning to Budapest to deliver a lecture on quantum mechanics and giving a nontechnical summary to the extended family before his talk. “The light theories of Dirac are not easy to explain,” Nicholas points out.24

  “The first thing that struck me about him were his eyes—brown, large, vivacious, and full of expression,” remembers Stan Ulam, who first met von Neumann in Warsaw in 1935. “His head was impressively large. He had a sort of waddling walk.” Ulam found him congenial, lighthearted, and “far from remote or forbidding,” but noted that “he was ill at ease with people who were self-made or came from modest backgrounds. He felt most comfortable with third- or fourth generation wealthy Jews.” Despite his sense of humor, “there seemed to be some sort of thin screen, or veil, a kind of restraint between him and others,” noted Colonel Vincent Ford, von Neumann’s attaché on the Air Force Strategic Missiles Evaluation Committee, and assistant while he was confined to Walter Reed Army Medical Center during the last year of his life. “He seemed to be a part of this world in one sense … and not a part of it in another.”25

  The frontline engineers on the computing project, who were treated as warmly by von Neumann as they were treated coolly by the other professors at the IAS, were nonetheless intimidated when he visited them in the machine room or at the bench. “The likelihood of getting actual numerical results was very much larger if he was not in the computer room, because everybody got so nervous when he was there,” says Martin Schwarzschild. “But when you were in real thinking trouble, you would go to von Neumann and nobody else.”26

  “We can all think clearly, more or less, some of the time,” says fellow Hungarian American mathematician Paul Halmos, “but von Neumann’s clarity of thought was orders of magnitude greater than that of most of us, all the time.” His was a calculating, logical intelligence, and “he admired, perhaps envied, people who had the complementary qualities, the flashes of irrational intuition that sometimes change the direction of scientific progress,” Halmos adds. “Perhaps the consciousness of animals is more shadowy than ours and perhaps their perceptions are always dreamlike,” physicist Eugene Wigner recalled in 1964. “On the opposite side, whenever I talked with von Neumann, I always had the impression that only he was fully awake.”27

  Von Neumann compensated for these superhuman abilities with an earthy sense of humor and tireless social life, and tried, with mixed success, to blend in on a normal human scale. “You would tell him something garbled, and he’d say, ‘Oh, you mean the following,’ and it would come back beautifully stated,” says his former protégé Raoul Bott. “He couldn’t tell really very good people from less good people,” Bott adds. “I guess they all seemed so much slower.”28

  In 1914, at age ten, von Neumann entered the Lutheran Gymnasium, one of Budapest’s three elite high schools that offered competing eight-year curricula and that supported a small group of serious mathematicians who combined teaching with original research. He drew the attention of the legendary mathematics teacher László Rátz, who, according to classmate (and future economist) William Fellner, “expressed to Johnny’s father the opinion that it would be nonsensical to teach Johnny school mathematics in the conventional way.” Rátz had a gift for identifying mathematical talent and encouraging it to grow. “How can you know that this precocious 10-year-old will someday become a great mathematician?,” asks Eugene Wigner. “You really cannot. Yet somehow Rátz did know this. And he knew it very quickly.”29

  Under the guidance of Joseph Kürschák at the University of Budapest, and the private tutoring of Gabriel Szegö, Michael Fekete, and Leopold Fejér, as well as Rátz, John began serious training in mathematics at the age of thirteen. His first published paper was written (with Fekete as coauthor) at the age of seventeen, and by the time of his high school graduation in 1921, he was recognized as a mathematician of professional rank. Yet his father doubted that mathematics alone offered a viable career path.

  Theodore von Kármán—the Hungarian aerodynamicist who established the Jet Propulsion Laboratory in Pasadena, built the first supersonic wind tunnel, assumed the first chairmanship of the Air Force Scientific Advisory Board, and “invented consulting,” according to von Neumann—remembers how “a well-known Budapest banker came to see me with his seventeen-year-old son.… He had an unusual request. He wanted me to dissuade young Johnny from becoming a mathematician. ‘Mathematics,’ he said, ‘does not make money.’ ”

  “I talked with the boy,” von Kármán continues. “He was spectacular. At seventeen he was already involved in studying on his own the different concepts of infinity, which is one of the deepest problems of abstract mathematics.… I thought it would be a shame to influence him away from his natural bent.”30 A compromise was reached where von Neumann enrolled in the chemical engineering program at the Eidgenössische Technische Hochschule (ETH) in Zurich “to prepare him for what was then a reasonable profession,” while simultaneously enrolling as a student of mathematics at both the University of Berlin and the University of Budapest. For the next four years he divided his time between Zurich and Berlin, attending classes in chemistry while working independently in mathematics and returning to Budapest at the end of each term for examinations that he passed without attending class. He received his degree in chemical engineering from the ETH in Zurich in 1925, followed by his doctorate in mathematics from Budapest.

  His thesis, on the axiomatization of set theory, was the result of work begun in his freshman year. Abraham Fraenkel, editor of the Journal für Mathematik in 1922–1923, remembers receiving “a long manuscript of an author unknown to me, Johann von Neumann, with the title ‘Die Axiomatisierung der Mengenlehre’ (‘The Axiomatization of Set Theory’). I don’t maintain that I understood everything, but enough to see that this was an outstanding work and to recognize ‘ex ungue leonem’ (the claw of the lion).”31 The paper was published in 1925 under the title “Eine Axiomatisierung der Mengenlehre” (“An Axiomatization of Set Theory”) and expanded in 1928 with the An changed back to The.

  Axiomatization is the reduction of a subject to a minimal set of initial assumptions, sufficient to develop the subject fully without new assumptions having to be introduced along the way. The axiomatization of set theory formed the foundations, mathematically, of everything else. An ambitious previous attempt, by Bertrand Russell and Alfred North Whitehead, despite 1,984 pages extending across three volumes, still left fundamental questions unresolved. Von Neumann started fresh. “The conciseness of the system of axioms is surprising,” comments Stan Ulam. “The axioms take only a little more than one page of print. This is sufficient to build up practically all of the naive set theory and therewith all of modern mathematics … and the formal character of the reasoning employed seems to realize Hilbert’s goal of treating mathematics as a finite game.”32

  The mathematical landscape of the early twentieth century was dominated by Göttingen’s David Hilbert, who believed that from a strictly limited set of axioms, all mathematical truths could be reached by a sequence of well-defined logical steps. Hilbert’s challenge, taken up by von Neumann, led directly both to Kurt Gödel’s results on the incompleteness of formal systems of 1931 and Alan Turing’s results on the existence of noncomputable functions (and universal computation) of 1936. Von Neumann set the stage for these two revolutions, but missed taking the decisive steps himself.

  Gödel proved that within any formal system sufficiently powerful to include ordinary arithmetic, there will always be undecidable statements that cannot be proved true, yet cannot be proved false. Turing proved that withi
n any formal (or mechanical) system, not only are there functions that can be given a finite description yet cannot be computed by any finite machine in a finite amount of time, but there is no definite method to distinguish computable from noncomputable functions in advance. That’s the bad news. The good news is that, as Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the noncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance.

  In his axiomatization of set theory, “one can divine the germ of von Neumann’s future interest in computing machines,” says Ulam, speaking with hindsight from 1958. “The economy of the treatment seems to indicate a more fundamental interest in brevity than in virtuosity for its own sake. It thereby helped prepare the grounds for an investigation of the limits of finite formalism by means of the concept of ‘machine.’ ”33

  Von Neumann’s style was now set. He would approach a subject, identify the axioms that made it tick, and then, using those axioms, extend the subject beyond where it was when he showed up. “What made it possible for him to make so many contributions in so many different parts of mathematics?” asks Paul Halmos. “It was his genius at synthesizing and analyzing things. He could take large units, rings of operators, measures, continuous geometry, direct integrals, and express the unit in terms of infinitesimal little bits. And he could take infinitesimal little bits and put together large units with arbitrarily prescribed properties. That’s what Johnny could do, and what no one else could do as well.”34

  Along with his doctorate in 1926—after an oral examination at which David Hilbert was reported to have asked a single question: “In all my years I have never seen such beautiful evening clothes: pray, who is the candidate’s tailor?”—von Neumann received a Rockefeller Fellowship to work with Hilbert at Göttingen, a lifeline from America at a time when positions in Europe were scarce.35 He published twenty-five papers in the next three years, including a 1928 paper on the theory of games (with its minimax theorem proving the existence of good strategies, for a wide class of competitions, at the saddle point between convex sets) as well as the book Mathematical Foundations of Quantum Mechanics, described by Klári as his “permanent passport to the world of science,” and, eighty years later, still in print. In 1927 he was appointed a Privatdozent (or associate professor) at the University of Berlin, and transferred to the University of Hamburg in 1929.

  By this time Nazism was on the rise in Europe and depression was descending over the United States. Oswald Veblen was recruiting for the Princeton University mathematics department in preparation for the move to new quarters in Fine Hall, and, as Klári puts it, “in his search for talent he found Johnny … and used every means of persuading first the University, then the Institute, into appointing this young, relatively unknown Hungarian.”36 The initial invitation, from the university, was for a visiting lectureship to be shared with Eugene (Jeno) Wigner, with the two Hungarians dividing their time between Europe and the United States. To the gatekeepers at Princeton, half of two Hungarians was more acceptable than hiring one Hungarian full-time.

  “One day I received a cable offering a visiting professorship at about eight times the salary which I had at the Institute of Technology in Berlin,” Wigner recalls. “I thought this was an error in transmission. John von Neumann received the same cable, so we decided that maybe it was true, and we accepted.” The pay was $3,000 for the semester, with $1,000 for travel—a small fortune at the time.

  Newly married to Mariette Kovesi, the daughter of a prominent physician who was the director of the Jewish Hospital of Budapest, von Neumann arrived in Princeton in February of 1930 and, according to Wigner, “felt at home in America from the first day.” Upon arrival in New York City, Wigner (then Wigner Jeno) and von Neumann “agreed that we should try to become somewhat American: that he would call himself ‘Johnny’ von Neumann, while I would be ‘Eugene’ Wigner.”37

  Von Neumann received a permanent appointment to Princeton University in 1931. “Johnny was among the very early birds to leave,” says Klári, “voluntarily resigning his excellent academic position long before the Nazis had the power to force him out.” His decision was made on economic as well as political grounds. “The economical crisis in Germany is very acute now,” he wrote to Veblen in January of 1931, “and as people do not like to be alone in their miseries, there is much talk about the bad state of things in America. Is anything of this kind true?”38

  In January of 1933 (using funds that had been earmarked for an indecisive Hermann Weyl), Abraham Flexner offered von Neumann a professorship at the Institute, where he joined Oswald Veblen, Albert Einstein, and James Alexander, who were already settled in Fine Hall, the Institute’s interim home. The starting salary was $10,000 (higher than at the university), with benefits that included help with purchasing a house (or building one on Institute land that was being subdivided along Battle Road at the edge of Olden Farm). The Institute term, beginning in October and ending by early May, left time for a return to Europe for the summer months. The Flexners had a summer retreat at Magnetawan, two hundred miles north of Toronto, in the Canadian woods, and the Veblens had theirs in Maine. Einstein spent summers sailing on Long Island Sound, and Alexander, a passionate mountaineer who had soloed Alexander’s Chimney on the East Face of Long’s Peak in Colorado in 1922, spent his summers in the American West.

  In the spring of 1933 the Nazis began dismissing Jewish professors from German universities, and von Neumann resigned his position in Berlin, followed by his membership in the German Mathematical Society in January of 1935. “He, more than anyone I know,” adds Klári, “took it as a most personal affront that any nation, group of people, or individuals, could possibly prefer the base and unsophisticated philosophy of Nazism or any other ‘ism’ to such minds as Einstein, Hermann Weyl, Wolfgang Pauli, Schrödinger and many, many others including, last but not least, himself.”39

  According to Klári, “during the Thirties Johnny had criss-crossed the Atlantic at least twenty times,” until the doors closed in 1939. “There is not much happening here except that people begin to be extremely proud in Hungary, about the ability of this country to run its revolutions and counter-revolutions in a much smoother and more civilized way than Germany,” he reported to Veblen in April of 1933 from Budapest. “I did not hear anything about changes or expulsions in Berlin, but it seems that the ‘purification’ of universities has only reached till now Frankfurt, Göttingen, Marburg, Jena, Halle, Kiel, Königsberg—and the other 20 will certainly follow.”40 A line began to form at the exit toward the United States. The emergency Immigration Restriction Act of 1921 and the National Origins scheme of the Reed-Johnson Act of 1924 limited immigrants from Hungary to an annual total of only 869. Exemptions were available to teachers or professors with full-time appointments, but full-time appointments were in short supply even for those already in the United States.

  Veblen devoted all available resources—from the Bambergers, the Rockefeller Foundation, Princeton University, and a network of mathematics departments—to rescuing as many mathematicians as he could. Although von Neumann left for America ahead of the flood of refugees, and could easily have obtained a position elsewhere, he credited Veblen for his chance at a new life. “There was a very real affection between those two,” says Klári. “Johnny, who lost his father when he was quite young, somehow transferred his filial affections to Veblen. He was convinced that had it not been for Veblen, he would have perished in the European mess.”41

  “His hatred, his loathing for the Nazis was essentially boundless,” Klári adds. “They came and destroyed the world of this perfect intellectual setting. In quick order they dispersed the concentration of minds and substituted concentration camps where many of those who were not quick enough … perished in the most horrible ways.” Von Neumann bore the traces of this for life. “There wa
s a surface there of a very convivial sort,” his daughter, Marina, explains, “which overlay what was fundamentally a rather cynical and pessimistic view of the world.”42

  “I feel the opposite of a nostalgia for Europe, because every corner I knew reminds me of the world, of the society, of the excitingly nebulous expectations of my childhood—and by childhood I mean childhood, which ended when I was 19 or 22 or something like that—of a world which is gone, and the ruins of which are no solace,” von Neumann reported after his first postwar visit to Europe, in 1949. “My second reason for disliking Europe is the memory of my total disillusionment in human decency between 1933 and September 1938.”43

  Princeton was four and a half thousand miles away. The Institute for Advanced Study, an enclave within the enclave of Fine Hall, had been founded just in time. There were no quotas at the Institute, and Flexner, Veblen, and the Bambergers extended invitations as far as their budget could be stretched. Kurt Gödel was brought to Princeton (from Vienna) on a stipend of $2,400 for 1939 (although prevented from leaving until 1940); Stan Ulam (from Warsaw), as a temporary visitor for $300; and Paul Erdos, at $750 for one term (from Budapest).

  Fine Hall, with its library, common room, and massive fireplaces, was both living room, study, and in some cases the only home left in existence to the mathematicians camped out in nearby rooming houses in Princeton’s overcrowded downtown. Von Neumann found himself at the center of a thriving mathematical community, assuming the role that Hilbert had played in the Göttingen of 1926. “I would come to Fine Hall in the morning and look for von Neumann’s huge car,” remembers Israel Halperin, a student in 1933, “and when it was there, in front of Palmer Lab, Fine Hall seemed to be lit up. There was something in there that you might run into that was worth the whole day. But if the car wasn’t there, then he wasn’t there and the building was dull and dead.”44