In the nineteenth century, mathematics disembodied itself from the real world. Troubled by inconsistencies in Newton’s calculus, mathematicians looked back to math’s foundations to put its techniques on surer ground. Rather than relying on geometric intuitions, theorists sought to ground this new mathematics in arithmetic axioms. These invited no ambiguity and allowed the field of analysis to flourish beyond where simple geometry could take it.
The zeal of the new mathematics pervaded the whole of the sciences. On the grandest of scales, Einstein’s theory of relativity recast the geometry of the universe and finally perfected the laws of gravity governing the motion of planets, which had so fascinated Newton and Kepler. The advance of technology followed suit. Within a lifetime, man would go from building the first primitive airplane to stepping foot on the moon. This is to say nothing of incredible advances in computing, biology, economics, and the rest of the sciences emboldened by mathematics’ progress.
Deeply and essentially involved with more of these fields than seems possible, John von Neumann has a case for being the smartest man who ever lived. Born Neumann János Lajos into a generation of Hungarian minds whose otherworldly intellects earned them the moniker of “Martians,” von Neumann was an exception among the exceptional. “Martian” and Nobel prize-winning physicist Eugene Wigner would later deny the existence of any general “Hungarian phenomenon”—attributing his generation’s supposed brilliance entirely to his childhood friend von Neumann.
Two recent books have grappled with von Neumann’s story—in many ways an origin story for our present science. In Benjamín Labatut’s novel The Maniac, the arc of von Neumann’s life and work is driven by his angst over a tragedy at the heart of mathematics. In his biography of von Neumann, The Man from the Future: The Visionary Ideas of John von Neumann, Ananyo Bhattacharya wades further into von Neumann’s life, finding light deep in the abyss of tragedy.
In a predictable setup for tragedy, twentieth-century mathematics began with the unbridled optimism of the nineteenth century’s progress. One of the greatest mathematicians of the age, David Hilbert, hoped to use simple logical rules, called axioms, to describe all of mathematics. He yearned for a perfect mathematics with no contradictions and no unprovable truths. His credo, levied against skeptics during his life and after, reads on his gravestone, “Wir müssen wissen. Wir werden wissen.”—“We must know. We will know.” Mathematics would fail him on both counts, but not before he enlisted to his cause the young von Neumann, eager to join the great mathematical mission of his time.
From the start, Hilbert’s program had to contend with inconsistencies in the still-new axiomatic method. The theory of sets—the study of collections of objects—was in particular rife with problems. For a while, it was difficult to differentiate tautologies from genuine, if mind-bending, developments. While many mathematicians considered Georg Cantor’s different sizes of infinity to be patently absurd, it is recognized now as a deep, nontrivial insight. On the other hand, Bertrand Russell’s quandaries surrounding sets containing sets—Russell’s paradox—were resolved with some useful axioms provided by a teenage von Neumann working under Hilbert.
Though von Neumann’s early contributions to set theory remain important, a young mathematician named Kurt Gödel was about to change the field forever, banishing the hopes of completeness that Hilbert and von Neumann entertained. As a throwaway remark at the end of a talk he gave at Königsberg in 1930, Gödel mentioned what would become known as his incompleteness theorems. Gödel’s theorems showed that any reasonable list of axioms sufficient to describe the natural numbers is insufficient to prove all truths about the natural numbers. Additionally, he showed that it would be impossible for a system of axioms to prove that they cause no contradictions. As fate would have it, von Neumann was in the audience that day. After the talk, he pressed Gödel for details on his new theorems, perhaps the only one who understood the full implications of Gödel’s aside.
Gödel’s theorems are a tragedy in the truest sense: they refute Hilbert’s arrogance and man’s attempt to build an edifice of pure reason. They warn of monsters lurking deep in mathematics: theorems that will never be proven or contradictions impossible to resolve. In The Maniac, Labatut indulges the drama latent in this tragedy. Labatut has von Neumann’s childhood friend Eugene Wigner remark on von Neumann’s “sudden sense of emptiness, [which] did not remain limited to his ideas about mathematics, but began to permeate his entire worldview, which became darker and darker as the years went by.”
Some of von Neumann’s work in the following years supports this notion of growing nihilism. His treatise on game theory—the first of its kind—would later be used to disturbing effect in American scheming for nuclear war with the Soviet Union. Von Neumann himself reasoned that a nuclear war was inevitable if the Soviets were to develop an arsenal. So before they got the chance, he recommended the United States nuke Moscow first. As he memorably put it, “If you say why not bomb them tomorrow, I say why not today? If you say today at 5 o’clock, I say why not one o’clock?” Von Neumann’s game theory assumed that participants were perfectly rational actors. Labatut speculates that perhaps von Neumann placed such terrifying trust in his theory because he was a perfectly rational actor. Most human beings, on the other hand, “can be highly irrational, driven and swayed by their emotions, subject to all kinds of contradictions.” This unreason is not a fault, but an important feature of the species. Labatut, through von Neumann’s collaborator Oskar Morgenstern, continues: “while this sparks off the ungovernable chaos we see all around us, it is also a mercy, a strange angel that protects us from the mad dreams of reason.”
The greatest fascination of von Neumann’s later life was the burgeoning field of computing. His description of what a computer would need to function—now called a von Neumann architecture—is still the basic theory underlying modern machines. In his unfinished Theory of Self-Reproducing Automata, von Neumann predicted the then-unknown structure of DNA in biology and established the key theories inherited by modern artificial intelligence. Labatut grounds von Neumann’s later work in the totalizing dream of his youth: “perhaps the embers of the fire that Hilbert had kindled in him, his grand hope to arrest the chaotic spinning of this world, were not completely extinguished.” Though thwarted by Gödel’s incompleteness theorems, Labatut imagines that von Neumann still yearned to conjure a lasting edifice out of wisps of reason.
The effect of Labatut’s telling is that von Neumann’s life was a desperate search for the terrible grandeur impossible in mathematics. One leaves it with the impression that von Neumann’s work as a mathematician and Hilbert’s program of axiomatization were nothing more than a waste of time. But a more careful tracing of von Neumann’s life provides a lasting insight into the point of mathematics.
In The Man from the Future, Bhattacharya, who holds a Ph.D. in physics himself, examines von Neumann’s work in greater detail. The underlying connections between von Neumann’s various fields of work come into focus. For example, Bhattacharya argues that von Neumann’s book on quantum mechanics, Mathematical Foundations of Quantum Mechanics, took the disjointed observations of early quantum physicists and laid them out in a cohesive, mathematical framework. Bhattacharya stresses that by “reducing the whole of quantum mechanics to its mathematical essentials,” von Neumann was able to put the science on a surer foundation. And so he could see further and more clearly than anyone else in the field. He was among the first to articulate the “measurement problem” of quantum mechanics, in which one must somehow account for the collapse of the wave function only when it is observed (perhaps more memorably posed by Schrödinger and his cat).
Although von Neumann discovered no new particles or laws of physics, mathematics allowed him to describe the emerging quantum mechanics “as coherently and lucidly as anyone could.” A scaffold of mathematical rigor lifted him up to assess the exotic landscape around him—and spot the strange formations which still confound us today.
One need not delve into every field which von Neumann touched to see that it was always mathematics in the first place which permitted him entry. What is most astonishing, however, is the bridge between von Neumann’s early work on the foundations of mathematics and his later work on computers and self-replicating machines. Bhattacharya shows that the very principles which most seemed to limit von Neumann’s mathematics simultaneously provided the groundwork of theoretical computer science. Even mathematics’ very imperfection held crucial insight.
The tarnished legacy of Hilbert’s utopian program hides the purpose of rigorous mathematics. The point of rigor is not some arcane, world-proving philosophy, but simple clarity. This is how the push for rigor began in the nineteenth century: rigor was needed to clarify what one meant by a “continuous” function or even a function at all. Intuitively, we may quite easily picture what a continuous function looks like, but we can struggle to find the right words for what we really mean without the rigor.
Is the point of mathematical rigor, then, nothing more than to be a tool? It may help first to recognize that mathematics is, at its heart, a peculiar genre of fiction with a difficult-to-get-into reputation. Just like literature, we care about mathematics because it is useful as a metaphor for understanding the world around us. But like fiction, mathematics is never wholly practical. Its most devoted practitioners will insist on the subject’s intrinsic beauty. It is a mistake to consider these two goals as separate. A more beautiful metaphor is a clearer one and a more meaningful story is a prettier one. Here, tragedy too serves a purpose: it brutalizes naïveté with truth and offers catharsis with clarity.