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Five Golden Rules: Great Theories of 20th-Century Mathematics--and Why They Matter

4.1 out of 5 stars 13 customer reviews
ISBN-13: 978-0471193371
ISBN-10: 0471193372
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Editorial Reviews

Amazon.com Review

What do mathematicians do all day, and why in the world should we care? Science writer John L. Casti approaches these questions with characteristic flair in Five Golden Rules, a short work explaining five important 20th-century mathematical theories and their importance in our daily lives. The reader is left with a new appreciation for the men and women who are paid to do little but think and convince their peers that their thoughts are important. Von Neumann's minimax theorem, crucial for modern economics and military strategy, is first up, and Casti expresses its simple elegance in terms that even those with the rustiest high school math can handle. Four other theories, covering topology, computing, optimization, and singularities, get their turn, and each is a work of beauty much like the greatest poems or paintings of our time. Some of the practical applications are surprising (who knew that geometrical analysis can tell us if a joke will be funny?) but more surprising still is our general ignorance of the role math plays in our lives. Though the material gets more involved as the book progresses, elementary algebra and geometry, coupled with a willingness to work things out before proceeding, will suffice for most readers. Casti has once again gifted us with a clear, penetrating book covering a subject still largely uncovered. Five Golden Rules will make math real even to the most hardened number-phobe. --Rob Lightner --This text refers to an out of print or unavailable edition of this title.

From Library Journal

Most people have a very poor idea of what mathematicians do, since in school they are exposed only to centuries-old material. Casti (Complexification, HarperCollins, 1994) has written an excellent exposition of five of the most interesting mathematical theories of the 20th century that are still undergoing significant study. He discusses the mathematicians who discovered each theorem and explains how these theories are applied to real problems. This work is not for everyone; it is aimed at the nonmathematician willing to invest some time and effort to learn about modern mathematics. The serious reader will find it worth the effort; the chapter on "Theory of Communication" is in itself sufficient to recommend the purchase of the book.?Harold D. Shane, Baruch Coll., CUNY
Copyright 1995 Reed Business Information, Inc. --This text refers to an out of print or unavailable edition of this title.

Product Details

  • Paperback: 256 pages
  • Publisher: Wiley (September 22, 1997)
  • Language: English
  • ISBN-10: 0471193372
  • ISBN-13: 978-0471193371
  • Product Dimensions: 6.1 x 0.7 x 9.3 inches
  • Shipping Weight: 11.4 ounces
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (13 customer reviews)
  • Amazon Best Sellers Rank: #1,341,050 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Michael R. Chernick on January 24, 2008
Format: Paperback
Casti writes about 20th Century mathematics for general audiences. As he states, he uses five mathematical theorems that were proven in the 20th Century and shows how they relate to general theory and application. He gives a reasonable set of criteria to show the reader how these five theorems emerge out of the millions of theorems that mathematicians have proven in the mathematical literature of the 20th Century.
He also explains why all the theorems were developed in the first half of the century. Basically, it takes time for the impact and value of a theorem to take effect. While there may be many theorems developed in the later half of the century that will eventually prove to be more valuable than some of the five golden rules, we may not know this clearly for some time.

There seems to be a preference for theorems related to operations research. For example the Brouwer fixed point theorem from topology has applications to game theory. Von Neumann's minimax theorem was developed for game theory and its application to military strategy and economic problems. This one also falls into the realm of operations research. Finally Dantzig's simplex method provides an algorithm to solve linear programming problems and some extensions. This is also clearly in the realm of optimization problems in operations research.

Turing's halting theorem is also presented. This deals with important questions about the limitation of computing machines as it relates to mimicking human intelligence.

Many of the ideas are difficult to present in lay terms and there is a lot of development to try to make the theory understandable to the reader. But it is difficult to do these subjects justice. Casti's emphasis is clearly in applied mathematics and he excels at showing the impact of the results on our society.
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Format: Paperback
There are many important theorems in mathematics, and a great number of them were discovered in the twentieth century. To whittle these down to the five most important ones is no easy feat, but this is exactly what Casti has done. He admits that there is no group of five which are clearly more important than the others (after all, he has written a sequel, Five More Golden Rules!), and that this selection is no doubt very subjective, with any other prominent mathematician likely coming up with a very different selection.
Casti is primarily an applied mathematician, and his choice in theorems reflects this. Because of his expertise, he has been able to see the bounty that each theorem has created, as far as new ideas is concerned. While some of these theorems may seem rather abstract and not terribly useful, it is all of the follow-up work that these theorems created and inspired that makes them so special.
I found this book difficult in places to follow, and I'm no beginner in mathematics. This is probably no fault on the part of Casti, who has done a great job of making these theorems as accessible as possible to the layman. I believe that it is the difficult nature of some of the theorems that makes it difficult to understand. For instance, 40 pages is very little in which to distill the basics of topology, and explain the relevance of Brouwer's Theorem. And given that a large portion of the information in `Godel, Escher, Bach' has been compressed into the 40 pages covering Turing's Halting Problem, it is no surprise that the going is not easy. However, for a good insight into some important areas of twentieth century mathematics, this book is hard to beat.
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Casti's text focuses on five key mathematical theories from the 20th century; this is actually somewhat misleading, since 4 of them are actually proved theorems. It's an interesting survey of applied mathematics; a number of sub-disciplines are covered here.
He does an excellent job of bringing the math down to a reasonable level without dumbing it down. Most of the book can be understood with simple logic and algebra; truly understanding a few of the theorems (not just appreciating them) does take a little calculus. Knowing some more advanced math (like topology) helps, but a reasonably mathematically-inclined person with less formal education will be able to follow it just fine. I'd like to see more popular math books at this level; this is somewhere between the level of Paulos' series of books ("Innumeracy" et al) and an undergrad maths textbook.
It's interesting to see a side of mathematics not often covered in high school maths courses. I really recommend this book to anyone with more than a passing interest in mathematics; it may even rejuvenate your interest to a more active level.
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I am in violent disagreement with the previous reviewer from Australia. I also love math, and I found Casti's presentation and style engaging. Few books on fundamental mathematics are as accessible to the thoughtful reader, and Casti's book strikes just the right balance between precision and comprehension.
For that reason, I put this book in the same category as Feynman's classic monograph QED, which explains the essence of quantum electrodynamics without ever descending into the esoterica of the formal mathematical underpinnings. Casti does the same for (arguably) the five most significant mathematical results of this century.
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