Review o f The Limits of Mathematics 3 Author : G .J . Chaiti n Published by Springer Verlag in 199 8 Hardcover, $32 .00 160 page s ISBN 981-308-359 X Review by Vladimir Tasi c University of New Brunswick vlad@conway .math .unb .ca If you find pleasure in being baffled by the austerity of logicist incompleteness proofs base d on Berry's paradox, this is not a book for you ; Boolos's minimalist gem (which appeared in th e "Notices of the A .M .S ." a few years ago) is your natural choice . If, on the other hand, you actually want to learn something about the relationship between Berry's paradox, randomness an d incomplete-ness pheno-mena, I recommend "The Limits of Mathematics" . Chaitin has investe d considerable energy into explaining his way of thinking about the topic, from the point of view of algorithmic information theory . This book is primarily concerned with the "why" and th e "how" of limitative results . The ideas are carefully motivated, revisited and reinforced throughout , emphasizing intuitive understanding rather than a dryly formal"theorem-proof" approach . The result is a book that leaves the reader with the feeling of having witnessed one of those rar e events : a good lecture . "The Limits of Mathematics" is not intended to be bed-time reading . It requires active participation of the reader, who is challenged to supply the details and invited t o try out the software that comes along with this course . Admittedly, the fifty pages of code at th e end of the book might appear slightly intimidating to those of us who quit programming upo n encountering COBOL . However, the presentation of algorithmic information theory in terms of a n explicit complexity measure based on a modified version of LISP is one of the key features of th e book . In addition to making possible the hands-on approach which the author suggests, dealin g with a suitably chosen LISP dialect allows Chaitin to establish explicitly some of the constant s that occur in complexity estimates . For example, it is derived that the complexity (in Chaitin 's sense) of the bit-string consisting of the first N bits of the halting probability must be greater tha n N â 8000 . Various other results are made explicit, including the bound on the complexity of the theorems of a formal axiomatic system . This incompleteness theorem is used to make the case fo r a "quasi-empirical" philosophy of mathematics and the use of computers as tools for mathematica l experimentation . I am not a specialist on algorithmic information theory ; having read this book, I feel I understand something about this field . 3 ©1999 Vladmir Tasic
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Book Review: The Limits of Mathematics by G.J. Chaitin (Springer Verlag 1998)
Reviewer-Tasic, Vladimir
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, Volume 30 (1)
Association for Computing Machinery – Mar 1, 1999
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