How has computer science changed mathematical thinking? In this first ever comprehensive survey of the subject for popular science readers, Arturo Sangalli explains how computers have brought a new practicality to mathematics and mathematical applications. By using fuzzy logic and related concepts, programmers have been able to sidestep the traditional and often cumbersome search for perfect mathematical solutions to embrace instead solutions that are "good enough." If mathematicians want their work to be relevant to the problems of the modern world, Sangalli shows, they must increasingly recognize "the importance of being fuzzy."
As Sangalli explains, fuzzy logic is a technique that allows computers to work with imprecise terms--to answer questions with "maybe" rather than just "yes" and "no." The practical implications of this flexible type of mathematical thinking are remarkable. Japanese programmers have used fuzzy logic to develop the city of Sendai's unusually energy-efficient and smooth-running subway system--one that does not even require drivers. Similar techniques have been used in fields as diverse as medical diagnosis, image understanding by robots, the engineering of automatic transmissions, and the forecasting of currency exchange rates. Sangalli also explores in his characteristically clear and engaging manner the limits of classical computing, reviewing many of the central ideas of Turing and Godel. He shows us how "genetic algorithms" can solve problems by an evolutionary process in which chance plays a fundamental role. He introduces us to "neural networks," which recognize ill-defined patterns without an explicit set of rules--much as a dog can be trained to scent drugs without ever having an exact definition of "drug." Sangalli argues that even though "fuzziness" and related concepts are often compared to human thinking, they can be understood only through mathematics--but the math he uses in the book is straightforward and easy to grasp.
Of equal appeal to specialists and the general reader, The Importance of Being Fuzzy reveals how computer science is changing both the nature of mathematical practice and the shape of the world around us.
"I know quite a few books on the fuzzy set theory and neuro-fuzzy systems, but this work is unique. No other book have I found so pleasant to read and, at the same time, no other book seems so informative for the fuzzy or soft computing newcomer. . . . [T]he book presents an excellent, clear written, and easy-to-understand advertisement for fuzzy computing."--H. Toth, Computing Reviews
"[Sangalli's] goal is to describe this recent work to a broad audience; he succeeds quite admirably. The Importance of Being Fuzzy is clear and easy to read, and yet it provides enough mathematical detail to give some appreciation for the meaning behind the metaphors."--Mark Johnson, The Mathematical Association of America Online Book Review
"A fascinating tour of the computing of the next century. Arturo Sangalli has beautifully described the three main branches of 'soft' computing: fuzzy logic, neural networks, and genetic algorithms, in addition to reviewing the main ideas and philosophy of traditional computing. Anyone who wants to find out what the mainstream of computing will look like thirty years from now (and perhaps much sooner) must read this fascinating book."--Doron Zeilberger, winner of the Leroy P. Steele Prize for Mathematics
Table of Contents:
To the Reader
Pt. 1 Blurred Visions 1
Ch. 1 Classes with Uncertain Borders 3
Ch. 2 Fuzzy Does It 19
Pt. 2 Limits 47
Ch. 3 The Limits of Classical Computing 49
Ch. 4 The Limits of Formal Reasoning 76
Pt. 3 Natural Solutions 93
Ch. 5 Net Gains 95
Ch. 6 Solutions via Evolution 126
App. 1 Fuzzy Inferences 159
App. 2 The Functions of Natural Numbers Cannot Be Enumerated 162
App. 3 The Halting Problem Is Unsolvable 164
App. 4 Learning with the Back-Propagation Algorithm 166
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