The Hilbert Challenge [Hardcover]

Jeremy J. Gray (Author)

Book Description

ISBN-10: 0198506511 | ISBN-13: 978-0198506515 | Publication Date: January 25, 2001 | Edition: 1st

Few problems in mathematics have had the status of those posed by David Hilbert in 1900. Mathematicians have made their reputations by solving some of them like Fermat's last theorem, but several remain unsolved including the Riemann Hypotheses, which has eluded all the great minds of this century. A hundred years later, this book takes a fresh look at the problems, the man who set them, and the reasons for their lasting impact on the mathematics of the twentieth century. In this fascinating book, the authors consider what makes this the pre-eminent collection of problems in mathematics, what they tell us about what drives mathematicians, and the nature of reputation, influence and power in the world of modern mathematics. It is written in a clear and entertaining style and will appeal to anyone with interest in mathematics or those mathematicians willing to try their hand at these problems.

Editorial Reviews

Review

"...the story about the interaction of the personalities and the split between pure and applied mathematics makes very interesting reading. Mathematics is in many ways just another human endeavor, subject to petty spats, nationalistic rivalries and personal biases. While emphasis is on the famous 23 problems, enough effort is expended on what else he did to make the book as much a biography of Hilbert as it is on the problems he posed. That alone would make it well worth reading." -- Charles Ashbacher, Journal of Recreational Mathematics, May 2001

"[R]ecommended to everybody interested in the twentieth century mathematics."--Zentralblatt MATH

"[C]ontains illuminating information about the mathematical development and contributions of Hilbert himself and an expert's panoramic view of the mathematics in this period."--American Mathematical Monthly

About the Author

Jeremy Gray is Senior Lecturer in Mathematics at the Open University. His research interests lie in the history of the mathematics of the 19th and 20th centuries, and in the philosophy and social significance of mathematics. He is the author or editor of nine books, most recently The Symbolic Universe: Geometry and Physics 1890-1930 (OUP 1999).

Product Details

• Hardcover: 328 pages
• Publisher: Oxford University Press, USA; 1st edition (January 25, 2001)
• Language: English
• ISBN-10: 0198506511
• ISBN-13: 978-0198506515
• Product Dimensions: 8.8 x 5.7 x 1 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 3.3 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #2,060,040 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

25 of 26 people found the following review helpful:

4.0 out of 5 stars An excellent history of Hilbert and his problems, April 9, 2001

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: The Hilbert Challenge (Hardcover)

At the turn of the last century, David Hilbert posed 23 problems that he considered to be the most critical ones to be solved in the 20th century. Certainly the best mathematician of his time, the challenge that he put forward has served as a benchmark for progress in mathematics over the last 100 years. This book is a retrospective of what has been done on those problems, a biography of Hilbert and a history of his times. In choosing the problems, he selected only those that he felt would lead to significant mathematics. For example, the recently resolved Fermat's Last Theorem was not on the list.
While many of the problems have been solved, it is a tribute to Hilbert that some are still unsolved and there appears to be no hope that they will be resolved soon. A few of the problems were solved relatively quickly, but most succumbed only after decades of intensive work. All of the problems that he put forward are explained in great detail, and if they were solved, the manner of solution demonstrated. Since these problems are hard, it is not possible to thoroughly describe them without resorting to some advanced mathematics. However, that is kept to a minimum, so it is possible for someone without detailed knowledge to understand most of the explanations.
The German universities were very powerful centers of mathematical progress during Hilbert's lifetime and the story about the interaction of the personalities and the split between pure and applied mathematics makes very interesting reading. Mathematics is in many ways just another human endeavor, subject to petty spats, nationalistic rivalries and personal biases. The saddest part of the book is the description of what happened to the once proud university system when the Nazi party rose to power. An incredible amount of talent was hounded away, which was fortunate for them as most of those who remained and had an incorrect heritage were killed. Hilbert was a firm believer in the value of applied mathematics, so he no doubt would have been frustrated over the split between the pure and applied camps that occurred after the end of the second world war. Given that he was so much of both, I wonder what tone his voice would have had.
Hilbert was an intellectual giant who is known most for his set of famous problems rather than his impressive work on resolving problems. While the emphasis is on the famous 23 problems, enough effort is expended on what else he did to make the book as much a biography of Hilbert as it is on the problems he posed. That alone would make it well worth reading.

Published in Journal of Recreational Mathematics, reprinted with permission.

11 of 12 people found the following review helpful:

5.0 out of 5 stars all you want to know about hilbert and his problems, September 22, 2001

By 

Martijn13Maart1970 (Husavik Iceland) - See all my reviews

This review is from: The Hilbert Challenge (Hardcover)

When I first heard of the hilbert problems, and how important they seemd to be for all of maths that evolved after that, since 1900 until present, I wanted to know more. This book is a very pleasant read for several reasons:
It is easy to read and well explained, even if you don't grasp the full maths, still there is a story around every of the 23 problems that lets you understand the implication, and the full drama of its solution.
It is a nice biography of Hilbert 'the man', intertwined with the 23 problems, so it does not get boring like some biographies do with endless lists of calendar-facts.
There is even a full translation of the original speech he gave in Paris in 1900, which otherwise would be impossible to find.
The problems itself are well explained, as well in the timeframe of 1900, when first posed, as later in our time when maths was ready to solve them. The author did a good job also telling which of the problems really were important, really gave mahts further problems to think about, and which problems didnt give rise to new mathematical areas, and therefore became more or less curiosities after solution.
Reading this book gave me a feeling of how beautiful maths can be, how unexpectedly some problems can and cannot be solved, and evokes some of the drama of the worlds biggest minds at work.
If you are interested in maths and/oir in science and great minds: this is an excellent read!

6 of 8 people found the following review helpful:

1.0 out of 5 stars Absolutely horrible book; look elsewhere for Hilbert, May 11, 2005

By 

Doug201 (New York City, NY) - See all my reviews

This review is from: The Hilbert Challenge (Hardcover)

In brief, I do not recommend this book to anyone.

The book has three main problems that make it unenjoyable and quite tedious to get through.

The first problem is that the mathematical problems are not sufficiently well explained. In some cases, the problem and it's math are not explained in sufficient depth. In other cases, the problem is not clearly explained and comprehensible. In yet other cases, the problem's application or usefulness is not explained well. Finally, some math is explained so cursorily that it might as well not even have been discussed.

The second problem is that the layout of the book made very little logical sense. First, the author split the problems into various groups. Then, the author split the timeline into various sections: Pre-1900, 1900-1918, 1918-1945 and 1945-1999. He then proceeded to discuss some of the problems in each of the timelines. At each step, you've lost the entire train of thought from the previous section, and it becomes hard to understand what, if any, logical flow or consistency there was in the development of the work on Hilbert's problems or anything that lead from it.

The third and most serious issue against the book is that the author briefly introduces the "23" Hilbert problems and then subsequently expects the reader to understand what "Problem 10" is every time he refers to it, and exactly how it relates to all the other problems. There are quite a few sentences which refer to several problems by number and unless the reader is constantly referring back to the brief summary of all the problems, the sentence might as well have been in martian. After a few dozen pages of playing this "refer to the problem list" game, I became so frustrated that I just gave up and read the book.

Some minor problems include the fact that only 223 pages of the book are by the author; the rest is a translation of Hilbert's 1900 talk. (That is, less than 75% of the book is what the book is supposed to be about.) Another problem for me is that the paper is high gloss. It looks very nice at a glance, but when you try to read it, you are often faced with glare unless you orient the book perfectly with respect to the light source(s) - a problem I ordinarily face only with magazines.

Another minor problem, which may be interpreted as a positive thing by many (and I welcome that interpretation), is that the author spends a fair bit of time discussing social, governmental political and educational political aspects of countries, universities, and mathematicians. Except in a few minor cases, these things seem highly tangential to the work at hand, and although somewhat entertaining, distract from the intent of the book.

All in all, I found myself about half way through finishing it just so I could put it down and start another book, and preferably one completely unrelated to the topic. The author made such a hash of the subject that, despite finding several other interesting books on Hilbert and his problems here on Amazon, I have no interest whatsoever in reading them and actually finding out about things.