Fearless Symmetry: Exposing the Hidden Patterns of Numbers Paperback – Aug 24 2008
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"The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before."--Timothy Gowers, Nature
"The authors . . . outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields."--Science News
"The book . . . does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject."--William M. McGovern, SIAM Review
"Unique. . . . [T]his book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics."--Lindsay N. Childs, Mathematical Reviews
"To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program."--Lindsay N. Childs, MathSciNet
From the Back Cover
"All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different--by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can follow-systematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat's Last Theorem."--Peter Galison, Harvard UniversitySee all Product Description
Top Customer Reviews
While a great deal of the book can be described by the phrase “popular book,” this is not a book on popular mathematics. The third and last part deals with topics that only people with a significant background in mathematics will understand. Other sections in parts one and two would also be difficult for the person not well schooled in mathematics to understand. In the foreword the authors recommend that the reader has studied calculus. I consider this too weak a background, the absolute minimum would be someone well-schooled in calculus.
If you are interested in learning a great deal of the background needed to understand the proof of Fermat’s Last Theorem and have some advanced mathematical background or are able to learn advanced mathematics on your own, this book will allow you to learn the necessary background. However, it will not be easy.
This book was made available for free for review purposes
Most Helpful Customer Reviews on Amazon.com (beta)
The reason I bought this book is that I read Ian Stewert's book on Symmetry and Beauty and found it lacking as it was not very mathematical.
I was not dissapointed in the level of math in this book. If anything, I got overwhelmed by the end.
I call this type of book "drill deep" but not wide. I like that idea.
The author's have a real ambitious goal. It's laid out on pages 11 and 12:
"in this book we explore ..representations...we consider sets, groups, matrices and functions between them. We show you in detail in one particular case that we develop throughout the book that sets us to our goal: mod p linear representations of Galois groups."
THIS IS THE GOAL OF THIS BOOK. They are not kidding this is what the book sets out to do and I belive accomplishes.
The authors are true to this goal in the "drill deep" mode. Example: Chapter 2 is Groups - not everything about Group Theory is presented but enough that is needed for the rest of the book. In a similar manner one chapter is on so called reciprocity laws. Chapter 4 is on Modular Arithmetic a crucial aspect to this book.
One prior reviewer indicated that each chapter is far more difficult than the last; this is sortof the general tenure of the book - but with exceptions if you know that material. Example, Chapter 5, Complex Numbers, for me was a relief sandwiched in between Modular Artimetic and Equations and Varieties. I can attest that for the subject "Complex numbers" - that they treated it at a relativley elementary level and focused on just those aspects needed later on. I am sure that for all subjects like "Quadratric reciprocity" that was the case. However, if you hadn't been exposed to quadratic reciprocity and Legendre symbols it is a tough slog.
For me the high point of the book was Chapter 8, I felt that I understood the difficult concept of the the Absolute group of the field of algebraic numbers by the end of the chapter. It is an infinite group that only elements can really be enumerated - Identity and complex conjugation. It fills in some (but not all) of the points in the number line between the group of rational numbers and the line with no gaps the field of real numbers.
Chapters 13 to 22 my ability to follow went way downhill and I just skimmed to get some highpoints.
I might return to this book in the future. I like the idea of not having to learn every aspect of something like alebraic ring theory , then every aspect of permutation theory etc. but just learning enough to accomplish some higher level of understanding like ultimatley how Fermat's Last Therom was solved.
I would recomend Stwert's book on Symmetry and Beauty first if you feel you want a more general understanding of this subject as opposed to a real math book which this is.
The authors achieved something remarkable: they were able to communicate with accuracy the deepest concepts of arithmetic without the boring style of many mathematics textbooks. The book is very engaging, with nice reflections about the nature of mathematical thought, as well as the motivations behind the concepts.
The authors managed to have a gradual build up of difficulty of topics all the way
to the proof of Fermat's last Theorem. Unlike other introductory texts that let you down
because in their effort to be more engaging, end up too elementary, this one is perfectly balanced. I will also recommend the "Calculus Gallery" as a second outstanding book introducing Analysis.
It would be great if other branches of mathematics like dynamics, algebra, informatics etc had the privilege of such well balanced and insightful introductions.
This book gives a great deal of this insight in the field of Galois theory, the theory of equations, and algebraic number theory. But the reader also gets a taste of such esoteric topics such as etale cohomology and the proof of Fermat's Last Theorem. The authors pull all of this off in 267 pages, an amazing feat considering the nature of the subject matter. The book can be best appreciated by the advanced undergraduate student or graduate student of mathematics, but even professional mathematicians in other fields of mathematics will no doubt find the book helpful in introducing them to the subject. High-energy physicists will love the book, even the parts that are really a review of some elementary linear algebra.
The authors know when to stop when discussing a topic, so as to not lead the unprepared reader into a morass of highly technical argumentation. But they wet the reader's appetite enough to motivate them to consult the references for further reading. This book, and others like it thankfully are becoming more prevalent. Mathematicians are realizing that there is nothing wrong in engaging in a little hand waving in order to explain their ideas. This has enormous didactic power, and one can only imagine the ramifications of a large number of these kinds of books appearing in the next few years. With the deep insights they grant to aspiring mathematicians, this reviewer predicts an enormous explosion of new mathematics in the next decades, even greater than the current rate of progress, incredible as it is.