Wednesday, August 6, 2008

Wednesday Math, Vol. 33: John Von Neumann

To write about John Von Neumann, I feel as though I'm bursting at the seams. It is like having friends visit your city for the first time and you want to show them the cool stuff. Where do you start?

I've started with these two pictures. The melancholy picture of Von Neumann is easily found on the Net. It's his official badge picture from Los Alamos lab when he worked on the Manhattan Project. Many people assume that he knew the end result before the project started. I also included a picture of him smiling, because he was famous also for throwing parties and generally enjoying life on a grand scale.

There are a lot of people who worked on The Manhattan Project. There are even more who enjoy the finer things in life. I write about Johnny Von Neumann because he was one of the greatest mathematicians of all time. In earlier posts, I talked about the Original List, the three mathematicians that were supposed to be the greatest, Archimedes, Newton and Gauss. Those three would only take us to 1855, and nowadays most people would include my buddy Leonhard Euler as belonging on the pre-1855 list. I also put David Hilbert, who did much of his great work in the late 19th Century and early 20th Century on the list, as I think most mathematicians have seen the breadth of his work by now and concede he belongs there too. If most would agree Hilbert belongs, there would be even less argument about Von Neumann.

There were two shows on the BBC in the 1970s about the history of culture, Sir Kenneth Clark's Civilization about art and Jacob Brownowski's The Ascent of Man about the sciences. Jacob Bronowski's depth of knowledge became famous, and was even turned into a joke by Monty Python.* While Bronowski was lauded by many for his intellect, he modestly stated that a genius is a person who has two great ideas in a lifetime, and he readily admitted that the greatest genius he ever met was Johnny Von Neumann.

Janos Von Neumann was born in 1903 to a wealthy Jewish family in Hungary. His parents were not observant Jews, and the family mixed in both Jewish and Christian traditions into family life. The most difficult time the family had with anti-Semitism was after World War I when a Communist government run by Béla Kun took power in Hungary. The affluent Von Neumann family fled, and returned only after the Kun government failed, but because Kun and his allies were largely Jewish, all Jews took the brunt of the Hungarian unhappiness with the regime, even Jews like the Von Neumanns who fled and returned.

Like many gifted young men, Janos burned through his studies. By his early twenties, he was used to being pointed out at conferences as "the young genius". He got his degree in chemistry, and one of his early great works is the mathematical basis for quantum mechanics. His main mathematical interest was in logic, and he came up with the idea of classes, a generalization of the mathematical idea of sets, which many had hoped to be the solid basis for the foundation of all mathematics until some very unusual paradoxes were discovered. Von Neumann's class theory did away with those paradoxes, but in many ways the damage was already done.

So with the pure mathematical ideas of logic and the applied mathematics of quantum mechanics, Von Neumann meets Bronowski's criterion of two great ideas. Many mathematicians do their best work in their twenties and more or less coast through the rest of their careers. Not Von Neumann. As great as these works were, he is best known for his great ideas from his late thirties and early forties. In the 1940's he proves a conjecture of Borel's called the minimax theorem, and from this proof produces The Theory of Games and Economic Behavior, co-written with Oskar Morgenstern. Besides game theory, he is considered one of the fathers of computer science, and is given credit for the idea that some of the memory in a computer should be filled with instructions for what the computer should do, which is to say the idea of software belongs to Von Neumann. He also championed the idea that the bit - a single switch - should be the basic component of the computer.

Some people feel that a great mathematician should have invented practical objects. Here, von Neumann fills the bill many times over. He has the patent on the trigger for the hydrogen bomb, a small atomic explosion that triggers an even larger atomic explosion. He invented the direct dial system for AT&T on a plane ride from Washington D.C. to Baltimore, a flight so short no airline even offers it anymore.

The speed at which Von Neumann's mind worked is the topic of many of the stories told about him. One of his teachers, the great Hungarian mathematician George Pólya, was quoted as saying, "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper."

Much of the mathematics Von Neumann worked on is too complicated to be simply encapsulated by a mathematical slowpoke like me on a blog, most notably the Von Neumann algebras, one of the first great works of his life which he called "rings of operators" and was named for him after his death. He died at the age of 54 of cancer. Like with Mozart, to consider all he did in as little time as he had is mind boggling to us mere mortals. To read more, you can go the great mathematical biography page at St. Andrews University.

*Two of the Python old ladies, who they called The Pepper Pots, are having a heated discussion about the penguin on top of their television set, and this exchange ensues.
"How should I know? I'm not Jacob bloody Bronowski!"
"How would he know?"
"He knows everything!"
"Don't think I'd like that. It would take all the mystery out of life."


ken said...

So, are von Neumann's classes connected with category theory?

Matty Boy said...

Good question, Ken. Category theory is about sets and mappings of sets, either a set to itself or a set to another set. The thing is that the mappings constitute a set, and you can have things called functors that map maps to each other. Are there things that can map functors to functors? You damn betcha! And the fun and functors never end.

My favorite professor Stu Smith said some called category theory "abstract nonsense".

It can be described with sets without resorting to classes, to the best of my knowledge.

Anonymous said...

I'm a lightweight at Math but I enjoyed this post.