A BEAUTIFUL MATH:
BY TOM SEIGFRIED
On the surface this book is about John Nash (1), the Princeton mathematician most famous for his portrayal in the movie 'A Beautiful Mind' (2)about a mathematical genius cursed with schizophrenia. Nash won the 1994 Nobel prize for his pioneering work in the branch of mathematics called "game theory", but his contributions to algebraic geometry were what actually won him the most fame within the mathematics community.
The subtitle of this book is 'John Nash, Game Theory and the Modern Quest for a Code of Nature', but the author ranges far and wide across a number of theorists, fields of inquiry and ideas. From Asimov to Lan Zhou, from 'The Age of Reason' to 'Zero Sum Games', it all plays out on these pages.
Siegfried is an award winning science journalist who has taken on a rather grandiose project in this book. While Nash is indeed important to the development of game theory and its applications as diverse as evolutionary biology, information theory and experimental economics one can't help but feel that the title and cover were designed more to capitalize on the success of the movie rather than to describe the contents of the book. In his introduction the author tries to give a brief overview of game theory and how it touches on such fields as those mentioned above and others such as neurophysiology, anthropology and even, according to the author, quantum physics. Nash himself is rather peripheral to the central thread of the book, that game theory may be the sort of "psychohistory" that Sci-Fi author imagined in his 'Foundation Trilogy', ie a mathematically precise theory that can describe the changes and stases in society in the same sort of statistical but testable way that statistical mechanics describes the behavior of such things as gases even if the behavior of each and every molecule is inaccessible to analysis. Asimov put it as "the science of human behavior reduced to mathematical equations", but Siegried makes much larger claims for the utility of this branch of mathematics, some of them already being played out and some of them quite frankly speculative.
Molly has to admit to a certain amount of scepticism regarding such claims. While there is little doubt of the utility of game theory in evolutionary biology and in experimental economics some of the other claims are the purview of the fringes of certain fields. But...I'm reviewing this book as I read it, so many my scepticism will be overcome by the time I reach the index. For now...
Chapter One: Smith's Hand: Searching for the Code of Nature:
There's an old libertarian book, written I believe by Jerome Tuccille (3), entitled 'It Usually Begins With Ayn Rand' . In this case it begins way before that. Siegfried goes back a lot further to a much more respected figure, the economist Adam Smith(4), to begin his story. Chapter one is all about the theories of Adam Smith, with special reference to his "invisible hand" and how it was a precursor to the sort mathematical science of society that he sees in game theory. Along the way he gives a brief biography of Smith, how he was influenced by the French physiocrats, especially Francois Quesnay, and of how he came to formulate his theories in reaction to theirs, especially in regards to the source of wealth which Smith believed was labour rather than land. Both Quesnay and Smith believed that most (certainly not all in the case of Smith) government interference with the economy disrupted a natural process of economic interaction that usually produced results much superior to those produced by government action. Both authors opposed the prevailing merchantilist theories of the day that encouraged ceaseless government action to produce a favourable balance of trade, and both held to a free-trade "laissez-faire" attitude towards most economic questions.
Where this connects with the 'Code of Nature' that Siegfried sees in game theory is that Smith's was the first systematic "inquiry" that tried to build a theory of society that examined how the efforts of individuals could produce "macro" effects such as the changes in price that result from a competitive economy. This was the "invisible hand", and it described in at least a partial way how an equilibrium resulted from actions taken by individuals that have no such goal in mind. Along the way the author corrects a lot of misconceptions about Smith. Adam Smith was not the dogmatic advocate of free markets in all things that moderns tend to think he was. He saw at least a limited role for government. He also wrote on "moral sentiments" and did not believe that "rational selfish calculation" explained more than a subset of human society and its developments. What Smith, however, pointed to was that there could be a "natural order" of society that could be investigated by scientific methods. The present day experimental economists carry out this tradition investigating the often messy and even non-rational ways that choices are made by real people in the real world.
The chapter concludes with a brief subheading on 'The Origin of Darwinism'. In 'The Structure of Evolutionary Theory' Stephen J. Gould (5)has traced innumerable influences on Darwin's intellectual development. Smith was one of them, but it was not 'The Wealth of Nations' with which Darwin was familiar but rather another of Smith's works, 'The Theory of Moral Sentiments' in which Smith argues in a manner quite different from what he is usually portrayed as today.
What Darwin took from the general culture, of which was Smith was an illustrious part, is not some fantastic turn of political opinion. What he took was the general idea that small actions could produce-statistically- large effects which were part of an 'emergent order' not immediately implied by the actions themselves. If individual competition can produce an economy with regular laws then natural selection can produce the origin of species.
More on this book later,
1) See also John Nash's home page at http://math.princeton.edu/jfnj and his Nobel prize address at http://nobelprize.org/nobel_prizes/economics/laureates/1994/nash-auobio.html which gives the account of his life in his own terms.
2) The official site for the movie is at http://abeautifulmind.com . The PBS Network did a much better biography on Nash that corrects many of the misstatements and omissions of the movie. this can be accessed at http://www.pbs.org/wgbh/amex/nash . The PBS site includes many more items on Nash including something of a "Game Theory for Dummies" guide.
3)Ahhh....Jerome Tuccille. In addition to the above link you can see his home page at http://jerometuccille.com . Tuccille is a long time libertarian, and one of the most amusing authors I have ever read. I can remember his 'It Usually Begins With Ayn Rand' from decades ago. At the time I read it as a leftist anarchist who had never met a libertarian in his life. The times have changed. At the time libertarianism was purely !!!! an American ideology. I had no end of pleasure of reading his account of attempting to form an alliance between the anti-statist realms of both the left and right at that time (the early 70s). I probably agree with him now about as much as I did then ie "only in a very limited sense", but his writing style was entertaining beyond belief, and I was as much gratified then as I am now by his descriptions of the "nuts on the right" and how it consoled me in my "hope" that insanity and creepiness was not just confined to my own position as a leftist. I can recommend Tuccille beyond all other writers with whom i disagree merely because of the skill of his description.
4) While researching this blog I came upon the on-line library of the 'Library of Economics and Liberty'. This site contains the online writings of a great number of economists, including Adam Smith. So, if you want to read the classics of economics go there. Everything from 'The Wealth of Nations' to Marx's 'Capital' is available there. The site is conservative in political orientation, but it is still the best resource for economics "in the original" that I have yet to come upon.
5)See also the Stephen J. Gould Archive at http://www.stephenjaygould.org, a collection of many of his writings and a brief biography at http://www.annonline.com/interview/961009/biography.html
6)One thing that Molly became aware of during the writing of this blog is the general impression that great mathematicians have a much greater incidence of insanity than others, especially other scientists. The list of famous mathematicians who were demonstrably insane or at least unstable enough to commit suicide is long and impressive. It began with many Greek mathematicians who committed suicide when their theories were proven wrong and continues on to the modern day with people such as Cantor, Godel, Turing, Boltzmann and Eherenfest. Even that supreme example of a failed human being, the 'Unibomber' apparently studied mathematics during his academic days, though he can hardly be counted as a "great mathematician" as even an "adequate one" as he was as much a failure there as he was in the rest of his life. Is it the subject matter and the talent needed for same that leads to this connection between mathematics and insanity ? Is the connection real or merely a widely believed myth ? If it is not the subject matter is there perhaps something about the "culture of mathematics" that drives successes to insanity or suicide or failures(both professional and personal) such as the Unibomber to murder ? All that is the matter of another blog entirely.