Top critical review
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Very disappointing - book without a clear plan or logic
on 14 October 2013
I want to be frank: this book is a failure.
I do have a extensive and solid mathematics background far beyond high school mathematics, and yet there are many parts in the book that simply do not tell anything, and do not achieve anything for the reader, even one at my level. Some of the reasoning is so stripped down and filled with comments like "You now need to know this but -heck- we cannot explain it here because it is too advanced and would take up too much space if we did". I question there is value in books that bluntly admit defeat and rolls on regardless to the next incomprehensible bare-back statement of defeat.
The book is about the Birch-Dyer conjecture. It sketches the relevance and the content of this conjecture only in the final few pages, and gives a minute (or to be blunt, not even) application of it. The book should have been planned far more extensively and rewritten from scratch.
I would have started out with a few straightforward number theoretic problems, rational or integer arithmetic problems, that are recognisable to any determined reader with high-school mathematics, which is their professed target public. With that context, I would have motivated why high-school mathematics has difficultly handling such problems, either in the positive way (finding the solution) or in the negative way (proving there are no integer or rational solutions, only irrational and/or complex solutions). After creating that context, I would have introduced a plan or a strategy to find out more, and motivate why the strategy stands a chance of developping some meaningful, interesting mathematics. And then go into elliptic curves, group theory, function theory. Without such a framework or a plan, the book just babbles on but doesn't achieve anything, even for the determined reader of more mathematical development of their target public.
It should also have many more examples and many more graphical representations ... it hardly has any! Just google Birch-Dyer conjecture, and you will get a graph on the internet summarising the original work in the 50'ies and 60'ies. Look at WikiPedia and at some of the freely available pdfs, this will show you more compelling examples and more graphical insight and feel for the issues around elliptic curves, Birch-Dyer and other conjectures, related problems in Diophantine equations ...
From the comments on their previous book, "Fearless Symmetry", which I have not purchased, I can glance that that book suffers from the same ills. This book is a failure. In my opinion, authors Ash and Gross should replan and rewrite their popular mathematics books, or bin their ambition to produce any. Surpring that a reputed editor such as Princeton University Press puts this on the market.