# From Nash to Ahmes

By now, everyone knows who got what Oscars last week. There are therefore no prizes for knowing that A Beautiful Mind got Best Picture and other awards, and that the leading actor Crowe portraying John Forbes Nash, Jr. didn’t get Best Actor. I like watching the Oscars, but this time, I missed all the TV slots and I do not have a VCR machine that works. I like in particular listening to those acceptance speeches, which are always good and more than entertainment.

Most show biz characters are larger than life, and many have left us rather interesting quotes and food for thought. For example, the Oscar-winning director Billy Wilder who died last week at 95 had this to say about making movies. He said making movies is like walking into a dark room. Some people stumble across furniture, others break their legs, but some see better in the dark than others. The ultimate trick is to convince and persuade. “Every single person out there is an idiot, but collectively they are a genius.” He was the first filmmaker to win three Academy Awards in one year, and he knew less than 100 words of English when he first went to Hollywood where he started a career which spanned six decades. Billy Wilder would probably be best remembered for his 1959 film “Some like it hot” starring Marilyn Monroe whom Wilder said was very tough to work with, but an excellent dialogue actress.

Film Director Cameron Crowe had published a series of interviews with Wilder and had said his movies “are a world-wide language of love, intelligence and sparking wit.” And Ron Howard, the director of the Best Picture this year has this to say, “His characters ran the spectrum as far as their moral standards were concerned, but they were all human beings and therefore relatable, which made the movies very, very entertaining.”

Back to John Nash Jr., which was what I set out to discuss. I have never been very good at mathematics. I heard about the Game Theory in my undergrad days from my friends reading pure mathematics. It was the late Sixties and there were good brains in Pokfulam Road. Most of them graduated with First Class Honours and were snapped up by reputable universities abroad. I was then studying the chromosomes of a fruit fly, copying the experiments of Nobel laureates and under the supervision of a much feared and highly disciplined lecturer. Nobel prizes fascinated me and still do.

I have no stomach to discuss or debate the life of John Nash in the fashion Sylvia Nasar would and has or in the manner that proponents of would be Oscars had bashed him in a bid to promote the chances of their principals. Suffice it to say that I find it rather unnecessary and something the protagonist would not and probably never seek to invite. But the actual life of Nash could have been more moving and touching than what was portrayed in A Beautiful Mind though there was no doubt that the film, aided by Howard and interpreted through Crowe had helped to romanticize and dramatize an already full and real human being with a beautiful mind.

The Nobel Prize in Economic Sciences awarded to John Nash in 1994 actually went jointly to three scientists for their pioneering analysis of equilibria in the theory of non-cooperative games. Apart from Nash, they were Professor John Harsanyi of University of California, Berkeley, USA and Professor Dr. Reinhard Selten of Rheinische Friedrich-Wilhelms-Universitat in Bonn, Germany. All three had written lengthy autobiographies covering their life, their research and their work on aspects of the Game Theory. Professor John Harsanyi died in 2000.

The Game Theory actually takes it rise from studies of games such as chess and poker where players need to think ahead and develop strategies based on expected countermoves from other players. The foundations of the theory went back to a 1944 study by John Neumann and Oskar Morgenstern on the Theory of Games and Economic Behaviour. The 1994 citation by the Royal Swedish Academy of Sciences noted that the three laureates constituted a natural combination through their contributions to equilibrium analysis in non-cooperative game theory, with Nash providing the foundations for the analysis, Selton developing it with respect to dynamics, and Harsanyi with respect to incomplete information.

John Nash is indeed a gifted mathematician. In his autobiography, he said he read the classic “Men of Mathematics” by E T Bell in high school and succeeded in proving the classic Fermat theorem on integers and prime numbers. He was less straightforward about the Riemann Hypothesis, except to say that while he was at M.I.T., he managed to solve a classical unsolved problem relating to differential geometry which would be relevant to geometric questions arising in general relativity.

Now, Bernhard Riemann was a German mathematician and had originated Riemannian geometry. He was the son of a Lutheran pastor. He studied theology to please his father before he studied mathematics under Gauss to please himself. Karl Freidrich Gauss was known for his number theory. At 33, Riemann became professor of mathematics in Gottingen. He died in 1866 of tuberculosis. He was 39. His friend spoke of his gentle mind and of his faithful service to God, just like his father, except in a different way.

Riemann had written few but important papers with profound consequences and influence in mathematics and physics. He had laid down a new approach to the theory of complex functions based on theoretical physics and geometry and he developed the concept of Riemann surfaces. He produced new definitions for distance and curvature in n-dimensional space and extended the work of his mentor Gauss on non-Euclidian geometries. He was said to have provided the mathematical tools for Einstein to construct his general theory of relativity.

Mathematics must be one of the earliest subjects of civilization. A number of its branches, notably, logic, arithmetic, geometry and astronomy, were regarded as the original disciplines of liberal Arts and Sciences. The earliest known writer of mathematics was an Egyptian scribe called Ahmes living at circa 1650BC. He was said to have copied an earlier text on handling fractions and solving arithmetical problems. However, for at least 1000 years before that, the scribes of Egypt and Mesopotamia were developing ways of representing numbers and solving problems of a mathematical nature. Geometry is a branch of mathematics to do with shapes and sizes, originally as tools for measuring things that can be seen and for estimating spatial relationship between the Earth and other heavenly bodies. Man had begun the study of the universe and of oneself since time immemorial. Considerable progress has been made in the former, but not so in the latter.

Gauss and Riemann’s non-Euclidean geometries were developed from Euclid’s fifth axiom which can be stated as follows – through any one point there can be drawn one and only one straight line parallel to a given straight line – or in many variations. Euclid taught mathematics around 300BC in Alexandria. By then Plato’s Academy had been around for 70 years, though both he and Aristotle had already died. It is interesting that Aristotle had spoken on many subjects and was a well-respected philosopher in his days and even now, but neither he nor Plato was into mathematics. Maybe it was simply too difficult.