universe, so both believed, is rationality and meaning. In its way, it was a religious faith as formidably fecund as the Jewish monotheism of the Torah from which it ultimately stemmed, and it found its liturgy in the logic of mathematics.
Einstein’s greatest attainments, general relativity and the equivalence of energy and mass (E = MC2), were expressions above all of his monotheistic faith that the entire universe epitomized a profound inner consistency and logic, embodied most purely in the aesthetic beauty and wholeness of mathematics.
This faith, the algorithmic faith, ultimately would save the West. When Budapest collapsed into anti-Semitic furies, von Neumann escaped harm mostly through his supreme prowess in mathematics. Introduced by an inspiring teacher, Laszlo Ratz, to the great mathematician Michael Fekete, von Neumann by his senior year in high school already had published a significant original paper on the set theory of Georg Cantor. Above the abstraction of particular numbers themselves, set theory addresses a higher level of abstraction in algebraic symbols of numbers, then moves on up to yet a third level of abstraction: groups of numbers with common logical characteristics. Algorithms on algorithms, their study illuminates issues of the foundations of mathematics.
This set theory paper demonstrated that by the age fourteen, von Neumann was already delving beyond the superficial craft and processes of mathematics toward the ultimate truths beyond. He then went to Germany to study chemical engineering, for protective coloring in a practical science useful to the Reich, while keeping a position in the embattled PhD program in math in Budapest.
In Germany, von Neumann became a protege of the venerable David Hilbert at the legendary University of Gottingen, a relationship that would shape von Neumann’s first great ambition and achievement. Between 1772 and 1788, Joseph Lagrange had translated Isaac Newton’s mechanics into coherent mathematics. As Heims explained: “It was von Neumann’s deep insight in 1926 that if he was to be the Lagrange of quantum mechanics” it would be his task to extend to physics the axiomatic regime Hilbert was imparting to mathematics.”
Reaching back to his own previous work in axiomatic set theory, von Neumann succeeded, providing a unified axiomatic foundation for all forms of quantum mechanics, showing how quantum theory reflected a deeper stratum of mathematical logic.
With his childhood friend Eugene Wigner, he elaborated on his insights, writing four important papers extending quantum theory from the simple lines of the hydrogen spectrum pioneered by Niels Bohr to the thousands of lines of more complex atoms. This feat was regarded by atomic physicist Hans Bethe as von Neumann’s supreme achievement.
In this pioneering work on quantum theory as throughout his later career, wherever he operated in the domains of logical systems, von Neumann triumphed in part because of his embrace of Godel’s incompleteness law. Whether in Cantor sets of pure numbers, quantum mechanics, logical systems, pure games, computer science, or information theory, every system, algorithm, computer, or information scheme would depend on assumptions outside its particular system and irreducible to it. Mathematics ultimately would repose on a foundation of faith. As atheist economist Steven Landsburg puts it: “Mathematics is the only religion that can prove it’s a religion.” The universe rests on a logical coherence that cannot be proven but to which all thinkers must commit if they are to create.
Early in his life, von Neumann recognized those realms in which science could achieve completeness and where it could not, how logic could be embodied in machines, and what were its limits. These insights into the powers and borders of axiomatic thinking made von Neumann at once the most visionary and the most practical of scientists and leaders.
Von Neumann’s contribution to quantum theory was made at the age of twenty-three. He followed a long passage from that moment of intellectual preeminence in physics through a role as protagonist in several other disciplines. In the end, von Neumann’s genius was part of a movement of mind that rescued Western civilization from the chaos and violence of his lifetime.
The decisive event for the Allied victory in World War II sprang from the Manhattan Project. The Manhattan Project produced the bomb that brought Japan to its knees and ended WWII. If Germany, Japan or any of the Axis or Communist powers had been the first to acquire nuclear weapons, the ultimate triumph of the West would have been impossible.
Von Neumann was a major player in the creation of the bomb. As Kati Marton wrote in
Von Neumann’s most direct contribution to the creation of the atomic bomb was to solve “the plutonium problem.” Because the separation of fissile uranium-2 3 5 from uranium-2 3 8 was a slow process performed by hand, this single source could sustain the creation of only one bomb, which no one could be sure would work, or, if it could work, how well, and, finally, whether the Japanese would believe that there was only one such bomb in the Allies’ possession. To build more bombs would entail using the more readily-available element plutonium. But no one knew for sure how to trigger a reaction in plutonium. Von Neumann’s proposal was an implosive process. Using any available computing equipment to calculate the complex non-linearities, von Neumann ended up specifying the process for unleashing a shock wave optimally shaped to compress a fissile mass. Spurring the project by some twelve months, this breakthrough enabled the team to produce the 60 percent more powerful Nagasaki bomb in time to end the war in the Pacific theater.
The Manhattan Project imposed the essence of the Israel test. Capable of anti-Semitic sneers, particularly toward the mercurial Leo Szilard, Brigadier General Leslie Groves, in charge of the Manhattan Project, seems an improbable hero in this story. A stiff and conventional military man and a rigid Christian of a sort disdained by intellectuals, Groves represented the “authoritarian personality” that critics of bourgeois capitalism such as Theodor Adorno believed explained the rise of the Nazis in Europe.
As Kati Marton explained, however, “General Groves, a deeply suspicious person, trusted Johnny von Neumann more than he did most of the other scientists and relied on him for advice that went well beyond mathematics and physics to the strategic.” Perhaps, as some have speculated, Groves did not regard von Neumann as a Jew. A superb judge of men, however, Groves was alert to genius. In selecting a director for the Manhattan Project, Groves faced a choice between the stolid and conservative Nobel laureate Ernest Lawrence, who was widely favored for the job, and Robert Oppenheimer, with all his Communist associations. In an act that may well have been decisive in the war, Groves chose Oppenheimer. The general explained, “While Lawrence is very bright, he is not a genius… J. Robert Oppenheimer is a real genius… he knows about everything.” By relying on Oppenheimer and von Neumann, Groves passed his Israel test. He enabled Los Alamos to assemble and compress in the desert a critical mass of genius and ingenuity that propelled the Manhattan Project to triumph.
As always von Neumann’s vantage point was the algorithmic realm, the center of the sphere, from which opportunities open up in all directions. This was also the vantage point of Einstein, who famously refused to contemplate the empirical data until he had deduced and perfected the logical structure of his findings. Neither was religious in any traditional way, but both reflected the Jewish insight of monotheism: a universe ruled by a single mind lending it order and significance.
Heims explains von Neumann’s strategy: “It became [von Neumann’s] mathematical and scientific style to push the use of formal logic and mathematics to the very limit, even into domains others felt to be beyond their reach” regarding the empirical world, “probably even life and mind, as comprehensible in terms of abstract formal structure.”
Bottom-up induction, stemming from empirical measurements alone, occurs not at the center but on the surface of the sphere. Induction requires theories — every experiment entails a concept to guide it — but the theories at the heart of scientific progress are often unacknowledged and mostly undeveloped. This inductive approach has ruled much of late twentieth-century science, driving it to an inexorable instinct for the capillaries. The unacknowledged governing idea is that the smaller an entity, be it particle or string — and the larger and more costly the apparatus needed to conjure it up — the more important the entity is. By rejecting this approach von Neumann left as his greatest legacy the most ubiquitous, powerful, adaptable scientific “apparatus” humanity has ever known — and it made a new world.
