The Hilbert Challenge

Rating:
Summary: An excellent history of Hilbert and his problems
Review: 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.

Rating:
Summary: all you want to know about hilbert and his problems
Review: 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!