Meta math! : the quest for omega
(Book)

Chaitin, an eminent and renowned mathematician, offers an excellent account of the nature and philosophy of mathematics, presented in an easy-to-understand, accessible style. Bertrand Russell, a great philosopher and mathematician, discovered some disturbing paradoxes in logic that attracted much attention in the mathematical community. In the beginning of the 20th century, Hilbert suggested formalizing completely all mathematical reasoning, and wanted to develop one formal axiomatic system (FAS) for developing new mathematics. In 1931 Austrian mathematician Kurt Godel showed that Hilbert was not on the right path and that this could never be done. Godel was able to prove convincingly the incompleteness of mathematics, absolutely devastating to many mathematicians. Then came Alan Turing's idea of uncomputability, showing that no formal axiomatic system can be complete. Chaitin was not completely satisfied with Godel's or Turing's ideas and put forward the idea of randomness in pure mathematics; using this idea, it is concluded that Godel's incompleteness is natural rather than mysterious. The Omega number, introduced by Chaitin, is a very complex representation of unknowability in mathematics. A very readable, thought-provoking, and interesting book, transporting readers to the very frontiers of scientific thinking. ^BSumming Up: Highly recommended. Upper-division undergraduates through professionals. D. V. Chopra Wichita State University

Note the exclamation point: Chaitin is on fire about math and is unable to restrain his enthusiasm. No mere number cruncher, he is renowned for finding another proof of Kurt Godel's incompleteness theorem and another for Alan Turing's halting problem in computation. Chaitin explains these two achievements here, in prose that is difficult for general readers to follow, but the spirit he brings to his subject will be apparent to all. Chaitin radiates his zeal like a preacher seeking converts. His asides often directly speak to students who might want to become professional mathematicians, stoking their fire, for example, with the vulnerability of even ancient theorems to new analysis (he sketches two ways, in addition to Euclid's, to prove the infinity of prime numbers). Chaitin's freewheeling expressions of mathematical creativity will be this work's lasting impression. --Gilbert Taylor Copyright 2005 Booklist

Note the exclamation point: Chaitin is on fire about math and is unable to restrain his enthusiasm. No mere number cruncher, he is renowned for finding another proof of Kurt Godel's incompleteness theorem and another for Alan Turing's "halting problem" in computation. Chaitin explains these two achievements here, in prose that is difficult for general readers to follow, but the spirit he brings to his subject will be apparent to all. Chaitin radiates his zeal like a preacher seeking converts. His asides often directly speak to students who might want to become professional mathematicians, stoking their fire, for example, with the vulnerability of even ancient theorems to new analysis (he sketches two ways, in addition to Euclid's, to prove the infinity of prime numbers). Chaitin's freewheeling expressions of mathematical creativity will be this work's lasting impression. ((Reviewed September 15, 2005)) Copyright 2005 Booklist Reviews.