Meta Maths : The Quest for Omega [Import] [Hardcover]

Gregory Chaitin (Author)

Product Details

• Hardcover: 224 pages
• Publisher: Atlantic Books (2006)
• Language: English
• ISBN-10: 1843545241
• ISBN-13: 978-1843545248
• Product Dimensions: 9.2 x 6.3 x 1.1 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 3.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #2,893,643 in Books (See Top 100 in Books)

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4 of 4 people found the following review helpful:

3.0 out of 5 stars How can the infinite be ..., January 22, 2010

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wiredweird "wiredweird" (Earth, or somewhere nearby) - See all my reviews
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This review is from: Meta Maths : The Quest for Omega (Hardcover)

... well, blah? Chaitin starts with some interesting premises. One is that computation has as much to offer math as the current formal methods do - despite the formal methods' 2000 year head start. It's an interesting approach, one that gets into information theory and complexity theory in a hurry. One measure of complexity that Chaitin proposes, as Kolmogorov has also done, defines a more complex pattern as one requiring more bits in its shortest description. Mathematical pi, for example, looks infinitely complex at first, since its sequence of digits goes on forever without repeating. At first glance, the shortest way to express that pattern is simply to quote it verbatim. But, with a little mathematical insight, pi can also be written very concisely as an infinite series. Its apparent complexity vanishes into the simplicity of the formula for calculating it.

That's true for infinitely many numbers - but, I think, for no more than there are integers. If the formula were written out in text and the finite set of characters treated as digits in a very large number, then the set of all possible formulas (including ill-formed ones) maps easily to the set of integers. The set of integers is small, as infinities go, and the set of real numbers is much larger. Chaitin starts to explore that range of reals, all but an infinitesimal fraction, for which there is no formula. He uses the map provided by the rules of computable arithmetic as the trailhead into unmappable terrain. His Omega uses a somewhat different kind of map, using Turning's halting problem as its central corridor, to name one of those otherwise un-nameable numbers. By writing all possible programs as strings and mapping those strings to integers, Omega summarizes a remarkable characteristic of the set of all possible programs. Since it is infinite, Omega can only ever be approximated, not written out exactly. It has the peculiar property, though, of resisting any effort to determine how good that approximation is. Although an exact number (subject to some initial assumptions), Omega can never be known.

Along the way, Chaitin offers some observations about Liebniz (about whom I want to learn more), creativity (with which I disagree, at least in part), and the role of time in the theory of computability. Oddly, time has no role in the current theories - possibly a simplification for the popular press, since time vs. parallelism can be expressed within the big-O notations that do apply. He also discusses randomness and what the term might mean, especially when applied to a specific, known number.

I consider a book like this successful if it binds together isolated pieces of knowledge I already have, in addition to offering new ideas, but mostly if it energizes me to seek out more in its direction or use it in my daily doings. This connects within and adds to my base of knowledge, but gives me little to put to use and little reason to pursue its topics. I wanted to like this title, but came away underwhelmed.

- wiredweird