Meta Math! The Quest for Omega

  • Gregory Chaitin
Pantheon Books: 2005. 223 pp. $26 0375423133 | ISBN: 0-375-42313-3

The surprise of Kurt Gödel's incompleteness theorem of 1931 lay not so much in the incompleteness itself, but that it was found in so simple a mathematical theory as first-order arithmetic. It follows, then, that any richer mathematical or logical theory, which is most of them, is also incomplete. In addition, Gödel's method of proof, elaborating what became known as recursive functions, was fruitful in its own right. In particular, it helped in Alan Turing's creation in 1936 of computable numbers and his finding that it cannot be decided in a finite number of steps whether or not a computer can calculate some given number, and whether or not any formula expressible in this system is also a theorem of it.

Turing is Gregory Chaitin's hero, to the extent of being credited with the more general result that it cannot be decided whether or not a computer will complete a given task in some finite number of steps. This extension of Turing's conception, which creates the ‘halting problem’, actually seems to be attributable to Martin Davis in the early 1950s. In Meta Math, Chaitin has extended this line of thought by using the notion of the computer program to study incomplete mathematical theories. Written for the general reader, the book consists of a main text of about 160 pages, followed by reprints of two earlier papers of a more technical character.

A key notion in this book, inspired by biology, is the ‘complexity’ of a program, specified in terms of the smallest program(s) (in bit size) that can produce some given output. Such a program is irreducible, or incompressible: “So you can define randomness as something that cannot be compressed at all,” according to Chaitin. Thence follows the Omega number, an infinitely complex positive real number specifying the halting probability. AsChaitin puts it, it is defined from all the programs chosen by chance to run on a fixed computer and also to continue to run by random operator decisions until the computer “must decide by itself when to stop reading the program”; if a program halts after k bits, then it contributes 1/2k to the number. As stated, the number depends on the computer used; presumably this machine is Turing-powerful enough to run any program installed on it. Drawing upon a discussable claim that any mathematical theory can be encoded in programming terms, Chatin concludes that the Omega number “marks the current boundary of what mathematics can achieve”.

Chatin's investigation is attended by several skirts around paradoxes, especially those involving naming. For example, to qualify as a program contributing to the Omega number, the program has to be able to say how large it is when it halts. This type of concern also owes much to Gödel's 1931 theorem, where new standards were imposed on distinguishing logic from metalogic. It is a pity that Chaitin never states that theorem precisely, and once even states it quite wrongly.

The scope of the author's meta-programme (as it were) is impressive: essentially straightforward assumptions and steps lead to some wide-ranging consequences and claims about mathematics, logic and computing science. The account is nicely signposted by the frequent use of information boxes containing the main definitions, steps or relationships. As the book is intended for a wide audience, it might have been enriched by some comments on concurrent developments that have used versions of the main notions; for example, (non-biological) complexity with A. N. Kolmogorov in the 1960s, or the realm of intelligent activity lying beyond computability, as debated by Roger Penrose and others in recent times.

The style of writing throughout is better suited to an internet chatroom than to a book (“Discours de Métaphysique — that's the original French” is only one such example) and has exclamation marks spread liberally. Instead of properly referencing works that are precisely cited in the text, “I decided to concentrate mostly on recent books that caught my eye,” says the author. The list lacks, among other key works, J. W. Dawson's Logical Dilemmas, The Life and Work of Kurt Gödel (A. K. Peters, 1997) and The Essential Turing, edited by B. J. Copeland (Oxford University Press, 2004).

Many historical remarks are made, but are seemingly free of knowledge of the figures involved and their importance. For example, “the nearly-forgotten 17th-century genius Leibniz”, “Newton's incomprehensible Principia — written in the style of Euclid's Elements”, or “it was Cantor's obsession with God's infiniteness and transcendence that led him to create his...theory of infinite sets and infinite numbers”. The reader should be ready to add their own exclamation marks to such passages.

It is nice to have popular books on modern mathematics, logic and science. But it is nicer if they are prepared with care.