’His mathematical intelligence was unparalleled’

The mathematical skills of John von Neumann (in the background on the left) were
The mathematical skills of John von Neumann (in the background on the left) were already legendary during his lifetime: while at Princeton, he engaged in dialogue with Albert Einstein. (Photograph: Princeton University / Palmer Lab Researchers and of Physics Department Faculty)
John von Neumann was one of the most important mathematicians and computer pioneers of the 20th century - and an ETH alumnus. He began his studies in chemistry here one hundred years ago. ETH Professor Benjamin Sudakov pays tribute to a mathematical legacy at a symposium.

ETH News: Let’s start with an anecdote. John von Neumann’s mental arithmetic skills were legendary. It is said that he managed to solve even the most complex problems in his head at lightning speed.
Benjamin Sudakov: This is true: The first thing that people recall about John von Neumann is his phenomenal speed of thought. He didn’t have to remember things; he computed them. If he was asked a question and didn’t know the answer, he would think for three seconds and produce a response. Yet, fast thinking was not his most outstanding characteristic. He was also very deep. It is the breadth of his scientific heritage that amazes me the most.

Today, John von Neumann is recognised as being one of the most important mathematicians of the 20th century, not to mention a computer pioneer.
Benjamin Sudakov: He was a visionary when it came to the first computers. His interest in computers was motivated by the applied problems he sought to solve. He also had an incredible impact on the development of modern mathematics. Just to understand the sheer breadth of his heritage: even highly influential scientists are usually associated with at most three to perhaps seven major scientific achievements. If you look at John von Neumann, you will find he made more than 100 significant contributions to various subjects in different scientific fields. I think that in terms of mathematical intelligence, he was virtually unparalleled.

When John von Neumann began his studies at ETH Zurich, he was enrolled in chemistry.
Benjamin Sudakov: Von Neumann studied chemistry for two reasons: On the one hand, it was a compromise with his father, a wealthy banker, who insisted that he study something that would allow him to earn a real income later. On the other, he had a great affinity for applications throughout his life, which manifested itself not only in his involvement in designing the first computers but also in chemistry. In the end, he graduated with a PhD in Chemical Engineering from ETH Zurich in 1926 while at the same time completing a PhD in Mathematics in 1926 in Budapest.

During his studies, ETH Zurich was also a global focal point for mathematical research. Did he have contact with ETH mathematicians?
Benjamin Sudakov: He actually engaged in dialogue with them. He often talked with another outstanding Hungarian mathematician, George Pólya, who was a professor at ETH Zurich during von Neumann’s time as a student. Pólya had a very famous saying about John von Neumann. He said that the only student he was ever afraid of was Johnny - as he used to call him. If Pólya mentioned an unsolved problem during the course of a lecture, the chances were that von Neumann would come to him as soon as the lecture was over with a few scribbles on paper on which he had come up with the complete solution.

What are von Neumann’s greatest achievements in mathematics?
Benjamin Sudakov: It is amazing how easy it was for him to switch from one area of mathematics to another. He not only solved existing fundamental problems but also created completely new fields. Among the areas of research to which he made a fundamental contribution at a very early stage, or was even the first to do so at all, are quantum physics, ergodic theory, game theory and the Monte Carlo method.

What was his contribution to quantum physics?
Benjamin Sudakov: When von Neumann left ETH Zurich, physicists had already come up with various concepts and insights into the workings of quantum mechanics. However, they lacked a stringent language and rigorous mathematical foundations to describe them. Von Neumann developed a whole mathematical language and methodology of quantum mechanics in very short time. His book on the mathematical foundations of quantum mechanics paved the way for a new field of research. This was such a major contribution to science that he would have been remembered for this alone.

What about his other major contributions?
Benjamin Sudakov: He was also one of the founding fathers of ergodic theory. This theory usually applies to the long-term behaviour of various physical systems. For example, it relates the movements of individual molecules to the behaviour of gas as a whole. Here again, von Neumann laid the foundations for mathematically exact analyses.

What about Monte Carlo?
Benjamin Sudakov: He was working at Los Alamos as part of the Manhattan Project when the US was building the first atomic bomb. Together with other mathematicians, he carried out various very complicated computations related to nuclear chain reactions. The scientists developed a method of computation based on random measurements that enabled them to understand very complicated phenomena. Today, the Monte Carlo method is a very powerful tool that cannot be overestimated. It has proven useful in all areas of science, as it allows estimates to be drawn up for processes that are difficult to calculate empirically.

Why is it called Monte Carlo?
Benjamin Sudakov: John von Neumann and his partners Stan Ulam and Nicolas Metropolis were obliged to use a code word to keep their research secret. That’s when Metropolis remembered how his uncle used to gamble at the casino in Monte Carlo. Since what they were doing was very similar to gambling, they decided to call it the Monte Carlo method.

Regarding gambling, John von Neumann invented game theory, which is broadly used in economics and social sciences today. This theory also describes the logic of deterrence in the Cold War. What is its value in mathematics?
Benjamin Sudakov: Game theory was invented by von Neumann with the help of economist Oskar Morgenstern. One interesting output is the Minimax theorem, which is one of the key mathematical ideas in game theory. This theorem describes a zero-sum game that, put simply, states that my loss and my opponent’s gain balance each other out. What is mathematically interesting is that this theorem can also be used as the basis to determine how efficient a randomised algorithm can be. This also plays a role in computer science.

So, the question remains, why didn’t he get a Nobel Prize? Was his field of research too broad?
Benjamin Sudakov: You know, the Nobel Prize is awarded for applied aspects of science. Mathematics is not on the list of disciplines eligible for the Nobel Prize. It is also usually awarded to people who are at a late stage in their career. Von Neumann died very young in 1957. The Nobel Prizes in Economics for game theorists were awarded much later.
Florian Meyer