Meta Maths: The Quest For Omega, by Gregory Chaitin, Pantheon Books, New York, New York, 2005. 224 pp. ₤17.99 (hardbound). ISBN 9781843545248.
Three of five stars
The point of this book is to define and state the significance of a real number called omega (Ω). It is related to the halting probability and is defined on page 129 in the following way:
Given the set of all possible computer programs you select one at random (p) and run it on a specific computer. Each time the computer requests the next bit of the program, you flip a fair coin to generate it. The computer then must decide by itself when to stop reading the program. You sum for each program that halts the probability of getting precisely that program by chance. In terms of a formula, it is
Ω = ∑ 2-(size in bits of p)
program p halts
program p halts
Since the definition is based on a process halting, it is necessary to mention Hilbert’s stating of the halting problem and Turing’s brilliant solution. Cantor’s work on the different levels of infinity and the contributions of Kurt Gödel in the area of what can be proven are also mentioned.
The path from the opening statement to the definition of Ω and the mention of its significance is a convoluted one. There is a section on computer programming and the language used in the explanation is LISP. At times Chaitin descends into a lot of I/me discussion, not to the point of being insufferable, but certainly more than is necessary.
One point of specific historical interest appears in chapter three, where the Leibniz versus Newton debate over the origin of calculus is rekindled. According to Chaitin “Leibniz was such an elevated soul ...,” “... Leibniz was good at everything,” and “In fact you can only really appreciate Leibniz if you are at his level.” Whereas, Newton is protrayed as a dark man that derived joy from destroying people. Needless to say there is historical disagreement over this point.
This is a book of popular mathematics that is designed to prove the existence and significance of the halting probability. It is different from most other popular mathematics books in that it is not as linear in organization, starting from one point and following a well defined path to the conclusion. This will make it difficult for the non-technical person to understand.