Tuesday, October 11, 2005

Aumann

There is a fascinating interview with Nobelist Bob Aumann. A quote:
What is fascinating about number theory is that it uses very deep methods to attack problems that are in some sense very "natural" and also simple to formulate. A schoolchild can understand Fermat's last theorem, but it took extremely deep methods to prove it. A schoolchild can understand what a prime number is, but understanding the distribution of prime numbers requires the theory of functions of a complex variable; it is closely related to the Riemann hypothesis, whose very formulation requires at least two or three years of university mathematics, and which remains unproved to this day. Another interesting aspect of number theory was that it was absolutely useless--pure mathematics at its purest.

In graduate school, I heard George Whitehead's excellent lectures on algebraic topology. Whitehead did not talk much about knows, but I had heard about them, and they fascinated me. Knots are like number theory: the problems are very simple to formulate, a schoolchild can understand them; and they are very natural, they have a simplicity and immediacy that is even greater than that of prime numbers or Fermat's last theorem. But it is very difficult to prove anything at all about them; it requires really deep methods of algebraic topology. And, like number theory, knot theory was totally, totally useless.

[...]

[F]ifty years later, almost to the day. It's 10 p.m., and the phone rings in my home. My grandson Yakov Rosen is on the line. Yakov is in his second year of medical school. "Grandpa," he says, "can I pick your brain? We are studying knots. I don't understand the material, and think that our lecturer doesn't understand it either. For example, could you explain to me what, exactly, are 'linking numbers'?" "Why are you studying knots?" I ask; "what do knots have to do with medicine?" "Well," says Yakov, "sometimes the DNA in a cell gets knotted up. Depending on the characteristics of the knot, this may lead to cancer. So, we have to understand knots."

I was completely bowled over. Fifty years later, the "absolutely useless"--the "purest of the pure"-- is taught in the second year of medical school, and my grandson is studying it.
Another quote:
You know, sometimes people make disparaging remarks about [game theorist Oskar] Morgenstern, in particular about his contributions to game theory. One of these disparaging jokes is that Morgenstern's greatest contribution to game theory is von Neumann. So let me say, maybe that's true--but that is a tremendous contribution.
And again:
In short, I have serious doubts about behavioral economics as it is practices. Now, true behavioral economics does in fact exist; it is called empirical economics. This really is behavioral economics. In empirical economics, you go and see how people behave in real life, in situations to which they are used. Things they do every day.
More later....

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