" My approach to understanding the full implications of Gödel's work is mathematically analogous to the ideas of thermodynamics and Boltzmann and statistical mechanics. You might say, not completely seriously, that what I'm proposing is 'thermodynamical epistemology'!"
- G. Chaitin

Abstract
We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871–1956).

Introduction
Experimental physicists know how difficult accurate measurements are. No physical quantity has ever been measured with more than 15 digits or so of accuracy. Mathematicians, however, freely fantasize with infinite-precision real numbers. Nevertheless within pure math the notion of a real number is extremely problematic.

We’ll compare and contrast two parallel historical episodes:
1. the diagonal and probabilistic proofs that reals are uncountable, and
2. the diagonal and probabilistic proofs that there are uncomputable reals.

Both case histories open chasms beneath the feet of mathematicians. In the first case these are the famous Jules Richard paradox (1905), Emile Borel’s know-it-all real (1927), and the fact that most reals are unnameable, which was the subject of Borel, 1952, his last book, published when Borel was 81 years old James, 2002. In the second case the frightening features are the unsolvability of the halting problem (Turing, 1936), the fact that most reals are uncomputable, and last but not least, the halting probability , which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason Chaitin, 2005.

In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete Smolin, 2000 and perhaps even a giant computer Fredkin, 2004, Wolfram, 2002. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken.

Gregory Chaitin has devoted his life to the attempt to understand what mathematics can and cannot achieve, and is a member of the digital philosophy/digital physics movement. Its members believe that the world is built out of digital information, out of 0 and 1 bits, and they view the universe as a giant information-processing machine, a giant digital computer. In this book on the history of ideas, Chaitin traces digital philosophy back to the nearly-forgotten 17th century genius Leibniz. He also tells us how he discovered the celebrated Omega number, which marks the current boundary of what mathematics can achieve. This book is an opportunity to get inside the head of a creative mathematician and see what makes him tick, and opens a window for its readers onto a glittering world of high-altitude thought that few intellectual mountain climbers can ever glimpse.

"The first uninteresting positive whole number"