News Feed - People in Maths - John Nash





It has been widely reported that John Nash has passed away tragically in a car accident at the age of 84 today. Nash's most popular mathematical contribution was to the field of Game Theory, in particular, his observation of an equilibrium point in analysis of competitive dynamics - which is known by economists as the "Nash Equilibrium".

Made popular in broader society due to Russell Crow's portrayal of his life in the Hollywood film, A Beautiful Mind, there is more to the Nash story than many people care to know.

Nash, a mathematical prodigy, was a fierce problem solver and accomplished mathematician. Following a mental breakdown in his early 30s, Nash dropped out of the establishment for 20 years as he grappled with the inner working of his mind.

Nash abandoned his medications and managed to recover in older age by applying logical thought and intellect to his prodigious yet at times psychotic thought patterns.

Nash talks about his recovery and the real mix between enlightenment and madness that his thought patterns allowed him to tap into at the following link:

John Nash on a Beautiful Mind (the movie)

I also post a nice article on Nash's relationship with fellow Princeton academic von Neumann. Enjoy.

The following is an extract on John Nash's relationship with the father of Game Theory John von Neumann

The 2001 movie "A beautiful mind" on John F. Nash's life was a huge success, winning four Academy Awards. It is based on the unauthorized biography with the same title, written by Sylvia Nasar, and published in 1998. There is one amazing scene in the book, which unfortunately was missing in the movie. It is where John Nash visited John von Neumann in his office to discuss the main idea and result of his Ph.D. thesis in progress. Nash was at that time a talented and promising student in mathematics at Princeton University, about to get his Ph.D. degree. Von Neumann was professor at the Institute of Advanced Studies in Princeton. A member of a very select group of just a handful of mathematicians with highest distinction, with Albert Einstein as his colleague there. According to the description of this meeting in the book, which is based on an interview with Harold Kuhn, the visit was short and ended by von Neumann exclaiming "That's trivial, you know. That's just a fixed point theorem" [Nasar 1998].

Reading this scene, one cannot help seeing two men, one of them eager to impress with his theorem, the other making every effort not to be impressed. Actually both men were rather competitive, even more than what may be normal in a field like Mathematics. Although few may be able to answer a difficult mathematical question, once someone attempts an answer, many can tell right from wrong answers. For instance, solving an equation may be hard, but checking whether a number is a solution is easy. For this reason, mathematics contests are frequent. In the 16th century, mathematicians used to challenge each other, and pose questions to show their superiority. In our days, there are many mathematical contests for students, Putnam competition, national and international mathematical olympiads, and others. And for adult mathematicians, there is the game to challenge each other with conjectures, and to beat each other by being the first to prove it.

For this reason successful mathematicians often carry the image of a lone wolf. But even wolves need their pack. After a long time, where mathematicians were physically isolated and only able to exchange their ideas in letters, at the end of the 19th century mathematical centers emerged, first in Berlin and Paris, later also in Göttingen and in many European capitals, like Budapest, Warsaw, Vienna. These centers attracted many world-class mathematicians and students eager to learn from them. Of course there were classes, but a lot of this exchange went on during informal activities like extended walks, private seminars at the home of some professor, or regular meetings in coffee houses. The main subject of discussion was always mathematics. 

But why do mathematicians need their pack? Can't mathematics be done with paper and pencil (and nowadays, a computer) in splendid isolation? And of course, it is nice to learn from the best, but couldn't one just read a book or paper written by them? In my opinion, the necessity of these mathematical centers, and also of conferences in mathematics, places where mathematicians meet and exchange informally, demonstrates that, different to the common perception, mathematics has many "soft" features which require close human interaction. Facts can be learned from books, but underlying ideas are often not so clearly visible in them---actually over centuries mathematicians tried hard not to display their ideas in the papers, just the polished results. But of course, when meeting informally communication of ideas is a main topic. What ideas and approaches have been used, which ones were successful, which one were not. Different to the Natural Sciences, in Mathematics failed attempts are never published, but is still very important to know what has been tried unsuccessfully. Maybe the most important reason for the necessity of a large number of mathematicians to meet and exchange is the formation and communication of paradigms driving research. The community decides, just by talking to each other, in which direction a field should and will develop.

Around 1930, through a sequence of good luck and donations, Princeton changed from a rather provincial teaching-centered university, where basic knowledge was taught to students, into the "mathematical center of the universe" [Nasar 1998], where research in the hottest area was performed. Generous support from the Rockefeller Foundation allowed Princeton University in the mid-20s to create five Research Professorship. These were filled mostly by Europeans, one of them by the very young and very promising von Neumann. A few years later, the independent Institute of Advanced Studies was created through another generous donation. Einstein, von Neumann and two more researcher got positions there. One should maybe also not underestimate the importance of the construction of Fine Hall in 1931, the mathematics building housing library, comfortable faculty offices furnished with sofas (and some with fireplaces), and, most important, a large common room: Excellent conditions for community building and the opportunity of exchange, as discussed above. [Aspray 1988].

It is probably fair to say that Game Theory, as a mathematical topic, started in Princeton in the 1940s. Yes, there have been attempts on trying to formalize and analyze games before, Zermelo's 1913 theorem on chess, applicable to all sequential games of perfect information, and a little later papers by Emile Borel on poker. Even von Neumann's main theorem in the field dates back to 1928. But only in Princeton, where von Neumann met the economist Oscar Morgenstern, started this collaboration which resulted in the von Neumann-Morgenstern monograph in 1944.

This book got very good reviews and eventually made an enormous impact, but it took a few years before broad research in Game Theory started. After the publication of the book, von Neumann, as was his habit, turned to other research areas. It was not before 1948 that Game Theory became a hot topic in Princeton. This was initiated by Georg Dantzig, who visited von Neumann to discuss Linear Programming with him. Von Neumann reacted impatiently to Dantzig's lengthy description of his Linear Programming setting, but then he immediately noticed the connection between it and his theory of two-player zero-sum games [ARD 1986], [Kuhn 2004]. Consequently a project investigating the connection between Linear Programming and Game Theory of was established in Princeton, which included a weekly seminar in Game Theory. From then on, a lot of the discussion among faculty and students circled around game theoretical ideas [Kuhn 2004], [Nasar 1998]. It was this atmosphere that drew Nash towards Game Theory.

Dantzig's description of the first visit in [ARD 1986] seems to indicate that he didn't enjoy it much. But what happened here, discovering that two areas motivated by different models turn out to be mathematically equivalent, is happening over and over in mathematics and is in fact very healthy to the unity of mathematics. Dantzig must have been grateful that his model turned out to have more applications that he was aware of. Keep also in mind that the work which made Dantzig famous, his invention of the "Simplex Method" was not done yet.

Nash's situation in his meeting with von Neumann was different. He presented his main result and just got von Neumann's "that's trivial" response But he extended the range of games considered. Von Neumann's original research was mainly on zero-sum games, he was initially interested in the case of total competition. Dantzig's model of Linear Programming turned out to be equivalent to the solution concept for these games. In the von-Neumann Morgenstern book, they developed a theory for so-called cooperative games, which are games where negotiations are allowed before the game starts, and the enforceable contracts can be made. What Nash introduced was a very different point of view, a changed paradigm: Looking at non-cooperative games, games without enforceable contracts, maybe even without communication, but in which players don't have completely conflicting interests. Only this made Game Theory applicable to real world situations outside of parlor or casino games, only this changed Game Theory into a branch of Applied Mathematics. Which is ironic, since von Neumann was really interested in applications of Mathematics, but Nash was not.

Now what about the alleged "obviousness" of either Nash's result or his proof? Is it? By all means, no. If you doubt it, try to prove the result, described in the page on Mixed Strategies yourself. But we know that von Neumann was a very fast thinker. Maybe he understood Nash's description very quickly and was able to judge it? This might be possible, but it misses the point. Great Theorems in Mathematics are not necessarily great since they are so difficult to prove or understand. Many of the great theorems, like Euklid's Theorem that there must be infinitely many prime numbers, or even Cantor's Theorems that there are "as many" rational numbers as there are integers, but that there are "more" real numbers than rational numbers (after defining "as many" and "more" appropriately) can nowadays be understood by undergraduates. What makes such theorems great are new ways of thinking, new paradigms. Cantor's definition of a cardinal number is such a new concept. Von Neumann did never carefully look at these non-zero-sum games, and neither did the other mathematicians who were discussing Game Theory at Princeton at that time. But Nash did, and then he proved his beautiful theorem. So, as a conclusion, competition, and centers of mathematical thought are very important for mathematical progress, but also very important is the ability to ask new questions, or to create the right concepts. This is what Nash did in 1950.