Meta Maths is a popular foray into the philosophy of mathematics and computation, applying ideas of program size, complexity and algorithmic information to understanding the limits of formal systems. The central concept is the definition of an object's complexity as the smallest program that can generate it. The primary goal is an explanation of the halting probability Omega, a real number which is irreducibly complex in that there is no shorter method of producing its digits than simply listing them. (Omega is also known as Chaitin's Number, since he is the one who discovered it.) All this is connected to the philosophy and history of mathematics, with Chaitin touching on Leibniz, intuitionism, criticisms of real numbers, and connections with physics, among other topics.

Though it is pitched at a general audience, with a chatty tone and informal style, Meta Maths still demands a reasonably robust engagement with mathematics. Early on, for example, Chaitin presents Euler's proof of the infinity of primes using products of infinite series, even if that comes with a warning "this is the most difficult proof in this book. Don't get stuck here". But anyone familiar with Gödel and Turing's key results should find Meta Maths accessible — and a usefully different perspective.

Chaitin talks about himself and his experiences quite a bit in Meta Maths, and makes no pretence of impartiality in his presentation of the history and philosophy. This is sometimes distracting but comes across less as bombastic than as an attempt to communicate his own excitement and his emotional engagement with his subject. It's not clear that anything would be lost by removal of all the exclamation marks, though!

Note: Meta Maths seems to be a popularisation of ideas from Chaitin's book Algorithmic Information Theory (Cambridge University Press 2004).

August 2013

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