Birds and Frogs Freeman Dyson ome mathematicians are birds, others skill as a mathematician. In his later years he hired are frogs. Birds fly high in the air and younger colleagues with the title of assistants to survey broad vistas of mathematics out do mathematical calculations for him. His way of to the far horizon. They delight in con- thinking was physical rather than mathematical. cepts that unify our thinking and bring He was supreme among physicists as a bird who Stogether diverse problems from different parts of saw further than others. I will not talk about Ein- the landscape. Frogs live in the mud below and see stein since I have nothing new to say. only the flowers that grow nearby. They delight in the details of particular objects, and they solve Francis Bacon and René Descartes problems one at a time. I happen to be a frog, but At the beginning of the seventeenth century, two many of my best friends are birds. The main theme great philosophers, Francis Bacon in England and of my talk tonight is this. Mathematics needs both René Descartes in France, proclaimed the birth of birds and frogs. Mathematics is rich and beautiful modern science. Descartes was a bird, and Bacon because birds give it broad visions and frogs give it was a frog. Each of them described his vision of intricate details. Mathematics is both great art and the future. Their visions were very different. Bacon important science, because it combines generality said, “All depends on keeping the eye steadily fixed of concepts with depth of structures. It is stupid on the facts of nature.” Descartes said, “I think, to claim that birds are better than frogs because therefore I am.” According to Bacon, scientists they see farther, or that frogs are better than birds should travel over the earth collecting facts, until because they see deeper. The world of mathemat- the accumulated facts reveal how Nature works. ics is both broad and deep, and we need birds and The scientists will then induce from the facts the frogs working together to explore it. laws that Nature obeys. According to Descartes, This talk is called the Einstein lecture, and I am scientists should stay at home and deduce the grateful to the American Mathematical Society laws of Nature by pure thought. In order to deduce for inviting me to do honor to Albert Einstein. the laws correctly, the scientists will need only Einstein was not a mathematician, but a physicist the rules of logic and knowledge of the existence who had mixed feelings about mathematics. On of God. For four hundred years since Bacon and the one hand, he had enormous respect for the Descartes led the way, science has raced ahead power of mathematics to describe the workings by following both paths simultaneously. Neither of nature, and he had an instinct for mathematical Baconian empiricism nor Cartesian dogmatism beauty which led him onto the right track to find has the power to elucidate Nature’s secrets by nature’s laws. On the other hand, he had no inter- itself, but both together have been amazingly suc- est in pure mathematics, and he had no technical cessful. For four hundred years English scientists have tended to be Baconian and French scientists Freeman Dyson is an emeritus professor in the School of Cartesian. Faraday and Darwin and Rutherford Natural Sciences, Institute for Advanced Study, Princeton, were Baconians; Pascal and Laplace and Poincaré NJ. His email address is [email protected]. were Cartesians. Science was greatly enriched by This article is a written version of his AMS Einstein Lecture, the cross-fertilization of the two contrasting cul- which was to have been given in October 2008 but which tures. Both cultures were always at work in both unfortunately had to be canceled. countries. Newton was at heart a Cartesian, using 212 NNOTICESOTICES OFOF THETHE AMAMSS VOLUMEOLUME 56, NUMBERUMBER 2 pure thought as Descartes intended, and using wave mechanics in 1926. Schrödinger was a bird it to demolish the Cartesian dogma of vortices. who started from the idea of unifying mechanics Marie Curie was at heart a Baconian, boiling tons with optics. A hundred years earlier, Hamilton had of crude uranium ore to demolish the dogma of unified classical mechanics with ray optics, using the indestructibility of atoms. the same mathematics to describe optical rays In the history of twentieth century mathematics, and classical particle trajectories. Schrödinger’s there were two decisive events, one belonging to idea was to extend this unification to wave optics the Baconian tradition and the other to the Carte- and wave mechanics. Wave optics already existed, sian tradition. The first was the International Con- but wave mechanics did not. Schrödinger had to gress of Mathematicians in Paris in 1900, at which invent wave mechanics to complete the unification. Hilbert gave the keynote address, Starting from wave optics as a model, charting the course of mathematics he wrote down a differential equa- for the coming century by propound- tion for a mechanical particle, but the ing his famous list of twenty-three equation made no sense. The equation outstanding unsolved problems. Hil- looked like the equation of conduction bert himself was a bird, flying high of heat in a continuous medium. Heat over the whole territory of mathemat- conduction has no visible relevance to ics, but he addressed his problems to particle mechanics. Schrödinger’s idea the frogs who would solve them one seemed to be going nowhere. But then at a time. The second decisive event came the surprise. Schrödinger put was the formation of the Bourbaki the square root of minus one into the group of mathematical birds in France equation, and suddenly it made sense. in the 1930s, dedicated to publish- Suddenly it became a wave equation ing a series of textbooks that would instead of a heat conduction equation. establish a unifying framework for Francis Bacon And Schrödinger found to his delight all of mathematics. The Hilbert prob- that the equation has solutions cor- lems were enormously successful in responding to the quantized orbits in guiding mathematical research into the Bohr model of the atom. fruitful directions. Some of them were It turns out that the Schrödinger solved and some remain unsolved, equation describes correctly every- but almost all of them stimulated the thing we know about the behavior of growth of new ideas and new fields atoms. It is the basis of all of chem- of mathematics. The Bourbaki project istry and most of physics. And that was equally influential. It changed the square root of minus one means that style of mathematics for the next fifty nature works with complex numbers years, imposing a logical coherence and not with real numbers. This dis- that did not exist before, and moving covery came as a complete surprise, the emphasis from concrete examples to Schrödinger as well as to every- to abstract generalities. In the Bour- body else. According to Schrödinger, René Descartes baki scheme of things, mathematics is his fourteen-year-old girl friend Itha the abstract structure included in the Junger said to him at the time, “Hey, Bourbaki textbooks. What is not in the textbooks you never even thought when you began that so is not mathematics. Concrete examples, since they much sensible stuff would come out of it.” All do not appear in the textbooks, are not math- through the nineteenth century, mathematicians ematics. The Bourbaki program was the extreme from Abel to Riemann and Weierstrass had been expression of the Cartesian style. It narrowed the creating a magnificent theory of functions of scope of mathematics by excluding the beautiful complex variables. They had discovered that the flowers that Baconian travelers might collect by theory of functions became far deeper and more the wayside. powerful when it was extended from real to com- plex numbers. But they always thought of complex Jokes of Nature numbers as an artificial construction, invented by For me, as a Baconian, the main thing missing in human mathematicians as a useful and elegant the Bourbaki program is the element of surprise. abstraction from real life. It never entered their The Bourbaki program tried to make mathematics heads that this artificial number system that they logical. When I look at the history of mathematics, had invented was in fact the ground on which I see a succession of illogical jumps, improbable atoms move. They never imagined that nature had coincidences, jokes of nature. One of the most got there first. profound jokes of nature is the square root of Another joke of nature is the precise linearity minus one that the physicist Erwin Schrödinger of quantum mechanics, the fact that the possible put into his wave equation when he invented states of any physical object form a linear space. FEBRUARY 2009 NOTICES OF THE AMS 213 Before quantum mechanics was invented, classical second commandment says: “Let your acts be di- physics was always nonlinear, and linear models rected towards a worthy goal, but do not ask if they were only approximately valid. After quantum can reach it: they are to be models and examples, mechanics, nature itself suddenly became linear. not means to an end.” Szilard practiced what he This had profound consequences for mathemat- preached. He was the first physicist to imagine ics. During the nineteenth century Sophus Lie nuclear weapons and the first to campaign ac- developed his elaborate theory of continuous tively against their use. His second commandment groups, intended to clarify the behavior of classical certainly applies here. The proof of the Riemann dynamical systems.