Abstract: I'll discuss how Gödel's paradox "This statement is false/unprovable" yields his famous result on the limits of axiomatic reasoning. I'll contrast that with my work, which is based on the paradox of "The first uninteresting positive whole-number", which is itself a rather interesting number, since it is precisely the first uninteresting number. This leads to my first result on the limits of axiomatic reasoning, namely that most numbers are uninteresting or random, but we can never be sure, we can never prove it, in individual cases. And these ideas culminate in my discovery that some mathematical facts are true for no reason, they are true by accident, or at random. In other words, God not only plays dice in physics, but even in pure mathematics, in logic, in the world of pure reason. Sometimes mathematical truth is completely random and has no structure or pattern that we will ever be able to understand. It is NOT the case that simple clear questions have simple clear answers, not even in the world of pure ideas, and much less so in the messy real world of everyday life.


Gregory Chaitin is at the IBM Watson Research Center in New York. In the mid 1960s, when he was a teenager, he created algorithmic information theory, which combines, among other elements, Shannon's information theory and Turing's theory of computability. In the three decades since then he has been the principal architect of the theory. Among his contributions are the definition of a random sequence via algorithmic incompressibility, and his information-theoretic approach to Gödel's incompleteness theorem. His work on Hilbert's 10th problem has shown that in a sense there is randomness in arithmetic, in other words, that God not only plays dice in quantum mechanics and nonlinear dynamics, but even in elementary number theory. He is the author of five books: Algorithmic Information Theory published by Cambridge University Press; Information, Randomness & Incompleteness and Information-Theoretic Incompleteness, both published by World Scientific; and The Limits of Mathematics and The Unknowable, both published by Springer-Verlag. In 1995 he was given the degree of doctor of science honoris causa by the University of Maine, and he was elected to the IBM Academy of Technology. In 1998 he was named visiting professor at the University of Buenos Aires.