John von Neumann
Another important figure in twentieth century mathematics was John von Neumann. Von Neumann was a frog, applying his prodigious technical skill to solve problems in many branches of mathematics and physics. He began with the foundations of mathematics. He found the first satisfactory set of axioms for set-theory, avoiding the logical paradoxes that Cantor had encountered in his attempts to deal with infinite sets and infinite numbers. Von Neumann’s axioms were used by his bird friend Kurt Gödel a few years later to prove the existence of undecidable propositions in mathematics. Gödel’s theorems gave birds a new vision of mathematics. After Gödel, mathematics was no longer a single structure tied together with a unique concept of truth, but an archipelago of structures with diverse sets of axioms and diverse notions of truth. Gödel showed that mathematics is inexhaustible. No matter which set of axioms is chosen as the foundation, birds can always find questions that those axioms cannot answer.
Von Neumann went on from the foundations of mathematics to the foundations of quantum mechanics. To give quantum mechanics a firm mathematical foundation, he created a magnificent theory of rings of operators. Every observable quantity is represented by a linear operator, and the peculiarities of quantum behavior are faithfully represented by the algebra of operators. Just as Newton invented calculus to describe classical dynamics, von Neumann invented rings of operators to describe quantum dynamics.
Von Neumann made fundamental contributions to several other fields, especially to game theory and to the design of digital computers. For the last ten years of his life, he was deeply involved with computers. He was so strongly interested in computers that he decided not only to study their design but to build one with real hardware and software and use it for doing science. I have vivid memories of the early days of von Neumann’s computer project at the Institute for Advanced Study in Princeton. At that time he had two main scientific interests, hydrogen bombs and meteorology. He used his computer during the night for doing hydrogen bomb calculations and during the day for meteorology. Most of the people hanging around the computer building in daytime were meteorologists. Their leader was Jule Charney. Charney was a real meteorologist, properly humble in dealing with the inscrutable mysteries of the weather, and skeptical of the ability of the computer to solve the mysteries. John von Neumann was less humble and less skeptical. I heard von Neumann give a lecture about the aims of his project. He spoke, as he always did, with great confidence. He said, “The computer will enable us to divide the atmosphere at any moment into stable regions and unstable regions. Stable regions we can predict. Unstable regions we can control.” Von Neumann believed that any unstable region could be pushed by a judiciously applied small perturbation so that it would move in any desired direction. The small perturbation would be applied by a fleet of airplanes carrying smoke generators, to absorb sunlight and raise or lower temperatures at places where the perturbation would be most effective. In particular, we could stop an incipient hurricane by identifying the position of an instability early enough, and then cooling that patch of air before it started to rise and form a vortex. Von Neumann, speaking in 1950, said it would take only ten years to build computers powerful enough to diagnose accurately the stable and unstable regions of the atmosphere. Then, once we had accurate diagnosis, it would take only a short time for us to have control. He expected that practical control of the weather would be a routine operation within the decade of the 1960s.
Von Neumann, of course, was wrong. He was wrong because he did not know about chaos. We now know that when the motion of the atmosphere is locally unstable, it is very often chaotic. The word “chaotic” means that motions that start close together diverge exponentially from each other as time goes on. When the motion is chaotic, it is unpredictable, and a small perturbation does not move it into a stable motion that can be predicted. A small perturbation will usually move it into another chaotic motion that is equally unpredictable. So von Neumann’s strategy for controlling the weather fails. He was, after all, a great mathematician but a mediocre meteorologist.
Edward Lorenz discovered in 1963 that the solutions of the equations of meteorology are often chaotic. That was six years after von Neumann died. Lorenz was a meteorologist and is generally regarded as the discoverer of chaos. He discovered the phenomena of chaos in the meteorological context and gave them their modern names. But in fact I had heard the mathematician Mary Cartwright, who died in 1998 at the age of 97, describe the same phenomena in a lecture in Cambridge in 1943, twenty years before Lorenz discovered them. She called the phenomena by different names, but they were the same phenomena. She discovered them in the solutions of the van der Pol equation which describe the oscillations of a nonlinear amplifier, . The van der Pol equation was important in World War II because nonlinear amplifiers fed power to the transmitters in early radar systems. The transmitters behaved erratically, and the Air Force blamed the manufacturers for making defective amplifiers. Mary Cartwright was asked to look into the problem. She showed that the manufacturers were not to blame. She showed that the van der Pol equation was to blame. The solutions of the van der Pol equation have precisely the chaotic behavior that the Air Force was complaining about. I heard all about chaos from Mary Cartwright seven years before I heard von Neumann talk about weather control, but I was not far-sighted enough to make the connection. It never entered my head that the erratic behavior of the van der Pol equation might have something to do with meteorology. If I had been a bird rather than a frog, I would probably have seen the connection, and I might have saved von Neumann a lot of trouble. If he had known about chaos in 1950, he would probably have thought about it deeply, and he would have had something important to say about it in 1954.
Von Neumann got into trouble at the end of his life because he was really a frog but everyone expected him to fly like a bird. In 1954 there was an International Congress of Mathematicians in Amsterdam. These congresses happen only once in four years and it is a great honor to be invited to speak at the opening session. The organizers of the Amsterdam congress invited von Neumann to give the keynote speech, expecting him to repeat the act that Hilbert had performed in Paris in 1900. Just as Hilbert had provided a list of unsolved problems to guide the development of mathematics for the first half of the twentieth century, von Neumann was invited to do the same for the second half of the century. The title of von Neumann’s talk was announced in the program of the congress. It was “Unsolved Problems in Mathematics: Address by Invitation of the Organizing Committee”. After the congress was over, the complete proceedings were published, with the texts of all the lectures except this one. In the proceedings there is a blank page with von Neumann’s name and the title of his talk. Underneath, it says, “No manuscript of this lecture was available.”
What happened? I know what happened, because I was there in the audience, at 3:00 p.m. on Thursday, September 2, 1954, in the Concertgebouw concert hall. The hall was packed with mathematicians, all expecting to hear a brilliant lecture worthy of such a historic occasion. The lecture was a huge disappointment. Von Neumann had probably agreed several years earlier to give a lecture about unsolved problems and had then forgotten about it. Being busy with many other things, he had neglected to prepare the lecture. Then, at the last moment, when he remembered that he had to travel to Amsterdam and say something about mathematics, he pulled an old lecture from the 1930s out of a drawer and dusted it off. The lecture was about rings of operators, a subject that was new and fashionable in the 1930s. Nothing about unsolved problems. Nothing about the future. Nothing about computers, the subject that we knew was dearest to von Neumann’s heart. He might at least have had something new and exciting to say about computers. The audience in the concert hall became restless. Somebody said in a voice loud enough to be heard all over the hall, “Aufgewärmte Suppe”, which is German for “warmed-up soup”. In 1954 the great majority of mathematicians knew enough German to understand the joke. Von Neumann, deeply embarrassed, brought his lecture to a quick end and left the hall without waiting for questions.
Adapted from NOTICES OF AMS 2/2009