A=B

content better than 98%

writing better than 98%

Not very often do I get to recommend a math book that covers recent research advances as a great read. But this book is different, in many interesting and delightful ways.

The first thing to say about this book is that it is compelling reading. The part called Background grabbed me, as a mathematician, and as a mathematics instructor, and as a person who works with computers all day long. I have reread that part many times, and reflected on it at length. It has a great deal to say, and says it in a clear and compelling way.

Because I am a logician, I am interested in what computers can and cannot do, in theory. Because I sit at a keyboard all day long, I am interested in what these boxes can and cannot do in practice, also. This book speaks to both interests. It develops (in, as I said, a way that raises many interesting issues) a means for computers to solve a wide class of problems. I will quote the key paragraph (p. 19).

The problem of discovering whether or not a given hypergeometric sum is expressible in a simple ``closed form'' and if so, finding that form, and if not, proving that it is not, is a task that computers can now carry out by themselves, with guaranteed success under mild hypotheses about what a ``hypergeometric term'' is ... and what a ``closed form'' is ... .

That is, if you've ever tried to read some algorithm analysis in, say, Knuth's _Art of Computer Programming_, but been staggered by the difficult summations involving binomial coefficients, etc., there is good news. You can give such a problem to the computer. You give the computer the summation formula and it will tell you whether there is any answer at all, and if there is an answer, it will tell you that answer.

The development of the solution takes up the majority of the book. It is well-written, and mathematically beautiful. It is, however, hard for a person who isn't a professiona mathematician. (Please note that there is an on-line errata page that I found most useful.)

But even if you are not going to go through that development, the implications of the existence of such a solution must be clear. For instance, should mathematicians and instructors of mathematics shift emphasis from problem solution to problem statement? The book takes some of these issues up. As I said earlier, it does so in a way that, while balanced and scholarly, is a simply great read.

I highly recommend this book. Certainly to understand the issues discussed a reader needs to have enough experience in mathematics or computer science to have proved some reasonably hard things (say, a `Math of CS' course which covers induction, proof by contradiction, etc.). Better would be if the reader is a senior math major doing an independent study. But the writing is clear, the issues are worthwhile, and it speaks to a centeral intellectual question of our time: the role of computers. It is a fine book.

The above review is copyrighted by its author, and is copylefted under the following license:
Open Publication License 1.0 without options A or B

The contents of this web page, except the parts contributed by members of The Assayer, are copyright (c) 2000 by Benjamin Crowell, and are copyleft licensed under the Open Publication License 1.0, without options A or B.