*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 GĂ¶del
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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