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.

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