Customer Reviews

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

5 star:		(6)
4 star:		(4)
3 star:		(2)
2 star:		(1)
1 star:		(1)

 

 

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.

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.

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.

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.
With this volume, Casti has done more than his share of the professional duty of mathematicians. It is one more brick to be used in the construction of a mathematics that is cherished by all people.

Published in Journal of Recreational Mathematics, reprinted with permission.

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.

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 mistakes in the form of claims that are false. A example of this is the pervasive confusion between closed and compact, a mistake which makes the relevant theorems wrong as stated. A less damaging problem is the tangental to disconnected material in each section. These tangents are hardly related to the topic.

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, or ones that are out of practice. His presentation of real world applications of some of the more abstract rules of mathematics is rather interesting.

On a warning note, though, if you are not at least conversant with modern mathematical theory, you may get very lost. It's been a while since I've studied mathematics seriously and occasionally I lose track of the thread of the topic.

Still, it's a good read and very informatative.

Neat idea, but look elsewhere. I love maths, and I didn't like the book. Other books on modern maths are better. A good example is "Strategic Thinking" - by Dixit and Nalebuff. Dixit's book analysed the same concepts as those in the first chapter of this book. Dixit's book is clearer, more exciting, and better explained. On these dimensions Casti's book fails, badly. Don't buy.

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 of the chapters in order to make any sense out of it. I understood the Minimax and Halting Theorem chapters because I'm familiar with the material, but the rest of it was a little over my head for the amount of time I was willing to invest. I think that if you are really into math, this book will seem too simple, and if you aren't, it won't seem valuable. Three stars because the effort was there.

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 essays. I've never seen anything comparable. Casti's presentation of game theory and topology are audacious. I've spent a lot of time with the 'Halting Problem' and disagree with a variety of conclusions Casti draws, but the clarity of presentation makes it a 'must read'.