Leibniz, Information, Math and Physics

The information-theoretic point of view proposed by Leibniz in 1686 and developed by algorithmic information theory (AIT) suggests that mathematics and physics are not that different. This will be a first-person account of some doubts and speculations about the nature of mathematics that I have entertained for the past three decades, and which have now been incorporated in a digital philosophy paradigm shift that is sweeping across the sciences. 1. What is algorithmic information theory? The starting point for my own work on AIT forty years ago was the insight that a scientific theory is a computer program that calculates the observations, and that the smaller the program is, the better the theory. If there is no theory, that is to say, no program substantially smaller than the data itself, considering them both to be finite binary strings, then the observations are algorithmically random, theory-less, unstructured, incomprehensible and irreducible. theory = program −→ Computer −→ output = experimental data So this led me to a theory of randomness based on program-size complexity [1], whose main application turned out to be not in science, but in mathematics, more specifically, in meta-mathematics, where it yields powerful new information-theoretic versions of Gödel’s incompleteness theorem [2, 3, 4]. (I’ll discuss this in Section 3.) ∗IBM Research Division, P. O. Box 218, Yorktown Heights, NY 10598, USA, E-mail: [email protected]