Five Golden Rules: Great Theories of 20th-Century Mathematics--and Why They Matter 1st Edition

by John Casti (Author)

14 customer reviews

ISBN-13: 978-0471002611

ISBN-10: 0471002615

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An accessible and engaging treatment of the world's greatest mathematical theorems. Devoted to one theorem, each chapter introduces the mathematicians who created the theorem and the history of math at that time. Explains, in simple terms, what the theorem says and how it was proved. Numerous diagrams, cartoons and examples illustrate how the theorem works and bring the math to life.

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

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Product Details

Hardcover: 256 pages
Publisher: Wiley; 1 edition (August 25, 1995)
Language: English
ISBN-10: 0471002615
ISBN-13: 978-0471002611
Product Dimensions: 6.4 x 0.9 x 9.5 inches
Shipping Weight: 1.2 pounds
Average Customer Review: 4.1 out of 5 stars See all reviews (14 customer reviews)
Amazon Best Sellers Rank: #2,649,394 in Books (See Top 100 in Books)
- #529 in Books > Science & Math > Mathematics > Reference
- #1913 in Books > Science & Math > Mathematics > History
- #21153 in Books > Textbooks > Science & Mathematics > Mathematics

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Customer Reviews

4.1 out of 5 stars

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3 star		14%
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1 star		7%

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28 of 28 people found the following review helpful

contemporary mathematics but may be too technical for the lay person

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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22 of 22 people found the following review helpful

Lay some math on me, baby!

By Adam Rutkowski on August 6, 2000

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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14 of 14 people found the following review helpful

Understandable but not dumbed down

By Kyle Maxwell on March 1, 2001

Format: Paperback Verified Purchase

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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10 of 10 people found the following review helpful

Examples of the value of mathematics

By Charles Ashbacher HALL OF FAMETOP 500 REVIEWERVINE VOICE on January 2, 2004

Format: Hardcover

Mathematics is a discipline that has an essential niche in modern society. From the early days of civilization where the plotting of fields for taxation required geometry to the modern international interconnection of computers, mathematics has been essential. People also constantly bet their lives on the accuracy of mathematical models. When we climb into an airplane, we are entrusting our lives to the accuracy of the mathematical models used to design that plane. And yet hardly a day goes by in the life of this reviewer without someone commenting in some way on the limited worth of mathematics. A combination of fear, insecurity and uncertainly lead to common misconceptions concerning the value of mathematics. All mathematicians have a professional duty to counter these problems any way they can. In doing this, we are not just engaging in activities to boost our job security, but in every real sense helping to insure the survival of civilization as we know it.
In this book, Casti chooses five of the most significant modern developments in mathematics and gives detailed explanations as to why they make a difference in our lives. The topics are:
1. The Maximum Theorem of Game Theory.
2. The Brouwer Fixed-Point Theorem of Topology.
3. Morse's Theorem of Singularity Theory.
4. The Halting Problem of Computer Science.
5. The Simplex Method of Optimization Theory.
Targeted at the general reader, formulas and theorems are kept to a minimum. Applications, some literally life and death, are used to emphasize the value of these concepts. As someone who experienced the apparent irrationality of the Cuban Missile Crisis of 1962, it is comforting to see it reduced to a simple "game" matrix.Read more ›

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Most Recent Customer Reviews

Unique source for exposition on mathematics

Casti really knows his stuff and is able to convey the underpinnings and applications of 5 interesting and fundamental topics. Read more

Published on March 12, 2013 by randy patton

The first section makes the book worthwhile

I think in designing the book he should have switched the weakest section ( the last) with the first. Read more

Published on October 1, 2007 by Roger Bagula

If you love math and have free time...

While I think that this is one of the best introductory texts with regard to twentieth century mathematics, a person has to be willing to spend some significant time studying some... Read more

Published on September 10, 2007 by Nathan W. Smith

golden rules is golden

this book has a lot to offer. i have gone through it several times and even discussed the ideas with family.

Published on August 19, 2005 by D. H. Maxwell

Inspiring dash through 20th century breakthroughs

I found the text an inspiration. On one level, the material is fascinating. At another, you have to be impressed with Casti's ability to compress complex subjects into short... Read more

Published on November 1, 2004 by Mark Mills

A brave attempt but thematically and mathematically flawed

It is probably impossible to make the topics accessible to readers without mathematical training. This is a brave attempt, however the exposition contains many mathematical... Read more

Published on July 30, 2004 by Tychonoff

Good Reading

This is a good book for those that have a degree of mathematical curiousity. Not necessarily expertise, just a decent interest in it.
The reading is easy. Read more

Published on January 14, 2004 by Randy Given

defective

My problem with this book lies not with the writing, but with the printing: missing fifty pages, the book repeats the first quarto. Read more

Published on October 30, 2001 by Joanna White

Interesting book for methematicians

I personally haven't completed this book but it is actually a great text for mathematicians: either ones that are just starting out, ones that have been in the field for a while,... Read more

Published on April 28, 2000