A walk through Johnny von Neumann's garden

Foundations of Mathematics Johnny von Neumann left behind him six massive volumes of collected works, assembled and edited by Abraham Taub [1]. The collected works are his garden, containing a large and heterogeneous set of objects that he planted. Each of them grew from a seed, from an idea or a problem that came into his head. He developed the idea or solved the problem and then wrote it down and published it. He wrote fast and published fast, so that the flowers are still fresh. For my talk this morning I decided to take a walk through the garden and see what I could find. Luckily only two of the papers are in Hungarian. He wrote mostly in German until he came to live permanently in the United States at the age of thirty, and after that in English. Johnny was educated at the famous Lutheran High School in Budapest from age ten to age eighteen. There he had excellent teachers and even more excellent schoolmates. One of the schoolmates was Eugene Wigner, who became an outstanding physicist and a lifelong friend. But Johnny’s father understood that the Lutheran High School was not giving Johnny everything he needed. Johnny had a passion for mathematics going far beyond what the school could teach. So his father hired Michael Fekete, a mathematician from the University of Budapest, to work with Johnny at home. The first flower in Johnny’s garden is a paper, “On the position of zeroes of certain minimum polynomials” [2], published jointly by Fekete and von Neumann when Johnny was eighteen. The style of the paper is dry and professional, following the tradition set by Euclid two thousand years earlier. Almost everything that Johnny wrote as a mathematician is in the Euclidean style, stating and proving theorems one after another with no wasted words. Although the subject of his first paper was probably suggested by Fekete, the style is already recognizable as Johnny's. Johnny’s unique gift as a mathematician was to transform problems in all areas of mathematics into problems of logic. He was able to see intuitively the logical essence of problems and then to use the simple rules of logic to solve the problems. His first paper is a fine example of his style of thinking. A theorem which appears to belong to geometry, restricting the possible positions of points where some function of a complex variable is equal to zero, is transformed into a statement of pure logic. All the geometrical complications disappear and the proof of the theorem becomes short and easy. In the whole paper there are no calculations, only verbal definitions and logical deductions. The next flower in the garden is Johnny’s first solo paper, “On the introduction of transfinite numbers” [3], which he published at age nineteen. This shows where his strongest interests lay at the beginning of his career when he was a young bird ready to leave the nest and stretch out his mathematical wings. His dominating passion then and for the next five years was to understand and reconstruct the logical foundations of mathematics. He was lucky to arrive on the scene at the historical moment when confusion about the foundations of mathematics was at a maximum. In the nineteenth century, Georg Cantor had greatly enlarged the scope of mathematics by creating a marvelous theory of transfinite numbers, giving precise definitions to a vast hierarchy of infinities. Then, at the beginning of the twentieth century, Freeman Dyson is professor emeritus at the Institute for Advanced Study, Princeton. His email address is dyson@ ias.edu.