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

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.

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'.

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. The presentation is great, as I have come to expect from the publisher (Wiley).The content is very good. Note that these are from applied mathematics rather than pure mathematics. The author also makes this clear in the preface. This is an approach that I (sometimes) prefer, but just be forewarned that it is not the best of both worlds. Each topic is well explained and has many modern-day examples (e.g., Cuban Missile Crisis) worked into the text, making it all the more interesting.Good reading and recommended.

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.

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.