Printer-friendly versionArticles by IAS Faculty
The Prisoner's Dilemma
By Freeman Dyson
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Groups lacking cooperation are like dodoes, losing the battle for survival collectively rather than individually. |
The Evolution of Cooperation is the title of a book by Robert Axelrod. It was published by Basic Books in 1984, and became an instant classic. It set the style in which modern scientists think about biological evolution, reducing the complicated and messy drama of the real world to a simple mathematical model that can be run on a computer. The model that Axelrod chose to describe evolution is called “The Prisoner’s Dilemma.” It is a game for two players, Alice and Bob. They are supposed to be interrogated separately by the police after they have committed a crime together. Each independently has the choice, either to remain silent or to say the other did it. The dilemma consists in the fact that each individually does better by testifying against the other, but they would collectively do better if they could both remain silent. When the game is played repeatedly by the same two players, it is called Iterated Prisoner’s Dilemma. In the iterated game, each player does better in the short run by talking, but does better in the long run by remaining silent. The switch from short-term selfishness to long-term altruism is supposed to be a model for the evolution of cooperation in social animals such as ants and humans.
Mathematics is always full of surprises. The Prisoner’s Dilemma appears to be an absurdly simple game, but Axelrod collected an amazing variety of strategies for playing it. He organized a tournament in which each of the strategies plays the iterated game against each of the others. The results of the tournament show that this game has a deep and subtle mathematical structure. There is no optimum strategy. No matter what Bob does, Alice can do better if she has a “Theory of Mind,” reconstructing Bob’s mental processes from her observation of his behavior.
Letter from the Director: The Most Successful Route Often Begins with a Short Step to the Side
By Robbert Dijkgraaf
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Robbert Dijkgraaf, IAS Director and Leon Levy Professor, in the Mathematics–Natural Sciences Library in Fuld Hall |
I am honored and heartened to have joined the Institute for Advanced Study this summer as its ninth Director. The warmness of the welcome that my family and I have felt has surpassed our highest expectations. The Institute certainly has mastered the art of induction.
The start of my Directorship has been highly fortuitous. On July 4, I popped champagne during a 3 a.m. party to celebrate the LHC’s discovery of a particle that looks very much like the Higgs boson—the final element of the Standard Model, to which Institute Faculty and Members have contributed many of the theoretical foundations. I also became the first Leon Levy Professor at the Institute due to the great generosity of the Leon Levy Foundation, founded by Trustee Shelby White and her late husband Leon Levy, which has endowed the Directorship. Additionally, four of our Professors in the School of Natural Sciences—Nima Arkani-Hamed, Juan Maldacena, Nathan Seiberg, and Edward Witten—were awarded the inaugural Fundamental Physics Prize of the Milner Foundation for their path-breaking contributions to fundamental physics. And that was just the first month.
Nearly a century ago, Abraham Flexner, the founding Director of the Institute, introduced the essay “The Usefulness of Useless Knowledge.” It was a passionate defense of the value of the freely roaming, creative spirit, and a sharp denunciation of American universities at the time, which Flexner considered to have become large-scale education factories that placed too much emphasis on the practical side of knowledge. Columbia University, for example, offered courses on “practical poultry raising.” Flexner was convinced that the less researchers needed to concern themselves with direct applications, the more they could ultimately contribute to the good of society.
The Symplectic Piece
By Helmut Hofer and Derek Bermel
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This image (produced with a Java applet by Alec Jacobson at http://alecjacobåson.com/programs/three-body-chaos) shows colorful trackings of the paths of satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits. |
I can’t understand why people are frightened of new ideas. I’m frightened of the old ones.—John Cage
Helmut Hofer, Professor in the School of Mathematics, writes:
Last September, the School of Mathematics launched its yearlong program with my Member seminar talk “First Steps in Symplectic Dynamics.” About two years earlier, it had become clear that certain important problems in dynamical systems could be solved with ideas coming from a different field, the field of symplectic geometry. The goal was then to bring researchers from the fields of dynamical systems and symplectic geometry together in a program aimed at the development of a common core and ideally leading to a new field—symplectic dynamics.
Not long before, in my 2010 inaugural public lecture at IAS, “From Celestial Mechanics to a Geometry Based on the Concept of Area,” I had described the historical background and some of the interesting mathematical problems belonging to this anticipated field of symplectic dynamics. The lecture began with a computer program showing chaos in the restricted three-body problem. This problem describes the movement of a satellite under the gravity of two big bodies, say the earth and the moon, in a rotating coordinates system in which the earth and the moon stay at fixed positions. The chaos in the system is illustrated by putting about ten satellites initially at almost the same position with almost the same velocity.
When the system starts evolving, the program shows colorful trackings of the paths of the satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits, once the orbits of the different satellites start separating.
Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity
By Richard Taylor
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In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. Here we have seven “hours” on our clock—arithmetic modulo 7. To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1. |
Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs.
For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. But in one of those strange pieces of serendipity which often characterize the advance of human knowledge, in the last half century modular arithmetic has found important applications in the “real world.” Today, the theory of modular arithmetic (e.g., Reed-Solomon error correcting codes) is the basis for the way DVDs store or satellites transmit large amounts of data without corrupting it. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. You can visualize the usual arithmetic as operating on points strung out along the “number line.”
Morals and Moralities: A Critical Perspective from the Social Sciences
By Didier Fassin
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The Good Samaritan, Rembrandt van Rijn |
Philosophers have always been interested in moral questions, but social scientists have generally been more reluctant to discuss morals and moralities. This is indeed a paradox since the questioning of the moral dimension of human life and social action was consubstantial to the founding of their disciplines.
A clue to this paradox resides in the tension between the descriptive and prescriptive vocations of social sciences: is the expected result of a study of moralities a better understanding of social life, or is the ultimate goal of a science of morals the betterment of society? At the beginning of the twentieth century, the German sociologist Max Weber, following the first line, pleaded for a value-free study of value-judgment, examining, for instance, the role played by the Protestant ethic in the emerging spirit of capitalism. His French contemporary Emile Durkheim, more sensitive to the second option, strongly believed that research on morality would not be worth the labor it necessitates were scientists to remain resigned spectators of moral reality, a position that did not prevent him from proposing a rigorous explanation of why we obey collective rules. This dialectic between exploring norms and promoting them, between analyzing what is considered to be good and asserting what is good, has thus been at the heart of the social sciences ever since their birth.
For anthropology, the problem was even more crucial, since the confrontation with other cultures, and therefore other moralities, led to an endless discussion between universalism and relativism. Given the variety of norms and values across the globe and their transformation over time, should one affirm that some are superior or accept that they are all merely incommensurable? Most anthropologists, from the American father of culturalism, Franz Boas, to the French founder of structuralism, Claude Lévi-Strauss, adopted the second approach, certainly reinforced by the discovery of the historical catastrophes engendered by ideologies based on human hierarchy, whether they served to justify extermination in the case of Nazism, exploitation for colonialism, or segregation with apartheid. This debate was recently reopened with issues such as female circumcision (renamed genital mutilation) and traditional matrimonial strategies (requalified as forced marriages), with many feminists arguing in favor of morally engaged research when it came to practices they viewed as unacceptable.