From Philosophy to Program Size

Abstract

Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. I'll discuss how one can use this complexity measure to study what can and cannot be achieved by formal axiomatic mathematical theories. In particular, I'll show (a) that there are natural information-theoretic constraints on formal axiomatic theories, and that program-size complexity provides an alternative path to incompleteness from the one originally used by Kurt Godel. Furthermore, I'll show (b) that in pure mathematics there are mathematical facts that are true for no reason, that are true by accident. These have to do with determining the successive binary digits of the precise numerical value of the halting probability Omega for a "self-delimiting" universal Turing machine. I believe that these meta-theorems (a,b) showing (a) that the complexity of axiomatic theories can be characterized information-theoretically and (b) that God plays dice in pure mathematics, both strongly suggest a quasi-empirical view of mathematics. I.e., math is different from physics, but perhaps not as different as people usually think. I'll also discuss the convergence of theoretical computer science with theoretical physics, Leibniz's ideas on complexity, Stephen Wolfram's book A New Kind of Science, and how to attempt to use information theory to define what a living being is.