The Honors Class: Hilbert's Problems and Their Solvers [Englisch] [Taschenbuch]

The "Honors Class" is the set of mathematicians who have solved, or heftily contributed to solving, one of the famous 23 problems proposed by David Hilbert a hundred years ago.

Energetically researched, Yandell's book naturally presents numerous morsels of biography, spotlighting the eccentricities, the sobrieties, the childhoods, travails, philosophies (he got me to understand, finally, why the intuitionists cared so much about their program), and politics of the members of the Honors Class. But from all these snippets, what emerges is a biography of mathematics itself in the 20th century; a sense for the marvelous, moving, growing organism that has been the mathematical quest.

Many bright men and women, many geniuses, populate these pages. But with two exceptions (Georg Cantor, the mystical grandfather of modern logic and set theory; and the remarkable Teiji Takagi, who built Japanese mathematical culture, and the class field theory that led to solutions for three of Hilbert's 23, all seemingly with his bare hands) they didn't wield their chalk in solitary splendor. They formed a web made of learning, mentoring, competing, collaborating, inspiring; a web that converged on and spread out from two tumultuous epicenters of the century's math activity: Goettingen in Germany (until Hitler drove out all its best minds), and Princeton's Institute for Advanced Studies.

There are four parts biography to one part math here. That should make the book as approachable for laymen as it is delightful for the math sophisticates who'll get to put faces on all those familiar old names. The address in which Hilbert set out his problems is given in full as an appendix; and those who wish to pursue the technical topics further get a bibliography rich enough to keep them occupied for years.

You'll get only tantalizing tastes, best in the earliest and latest chapters, of the nitty-gritty content of 20th century mathematics. But you will get a doubleplusgood, full-length portrait of what it became as a social and cultural enterprise.

The Honors Class is the collection of mathematicians that individually or in collaboration solved or partially solved at least one of Hilbert's 23 problems. Yandell does a great of gathering up the historical information so that we have an up-to-date account of the progress on each problem and even how some problems evolved because of their vague or incorrect original proposal.
This is a popular math book and is accessible to the nonmathematician such as the fine books by Casti on mathematicians and mathematical developments. It is also similar to Singh's book on Fermat.

I think the historical research and accounting of the mathematics deserves 5 stars. I am a little unsure about how well the technical mathematics is conveyed to the layperson however. Admittedly, this is a very difficult task as much of the mathematics is very abstract, especially the early chapters on the foundational questions. The number theory, geometry and even some of the abstract algebra problems are easier to explain and Yandell does a fine job with them.

As a mathematician who studied algebra, analysis and even some symbolic logic as an undergraduate and graduate student, I still had a hard time feeling that I got the essence of the mathematics associated with some of these problems. Yandell's discussion is at times detailed but is necessarily sketchy on some of the mathematics. This works for me sometimes but not so well at other times. I think it would be much harder for a novice, but I guess it depends on the depth of understanding one is looking for.

I have always found the work of Cantor mysterious and so the ealry chapters that cover Godel and Cohen's amazing results are not the most enlightening for me. I had learned about the axiom of choice in my real analysis classes and was told something about the undecidability of it and its equivalence to the continuum hypothesis but have never really seen the connection or gotten much insight. The material on Paul Cohen is interesting to me because I attended Stanford in the 1970s when he was the buzz of the campus. A younger and less accomplished mathematician compared to many of his famous colleagues in Stanford's prestigious mathematics department, he still was revered because he solved one of Hilbert's problems. Still I am no closer to understanding symbolic logic and the method of deciding whether or not a proposition can be deduced from a set of axioms or can exist independently of the axiom system.

I got hooked on the book with the chapter on the tenth problem. This problem seemed more easily understandable and it was very interesting to see how the many players work together and separately to attack the problem including the very interesting Julia Robinson who was a key player in the middle of alll this.

The lives of these mathematicians, in some cases their suffering and insanity (similar to Nash) is very interesting and entertaining. There is too much here to handle in one reading.

I think this is a book I will go back to again and again. I am interested in reading more on Kolmogorov and want to try to understand some of the abstract algebra and number theory questions in more detail. There is a great deal of commonality in many of the stories. A large number of the members of the Honors Class were from Germany and fled during World War II. Many also traveled through or spent great portions of their career at Princeton University (some at the Institute for Advanced Study).

The book is thorough and gives an account of all the unsolved problems as well providing the insights of the mathematicians who have made attempts at them.

At the dawn of the twentieth century Hilbert proposed of list of problems that he hoped would be solved in the coming hundred years. The list, later expanded, proved more successful than David Hilbert could have imagined: the problems became canonical, and those who solved them became members of the "Honors Class". Yandell has written a charming biography of the problems and the Honors Class. Hilbert had aimed to make his selection comprehensive, and Yandell's book, organized by branch of mathematics rather than by the number of the problem, turns into something like a description of mathematics itself. The writing is lovely, blending personal information with pithy descriptions of the problems. I thought the mathematics was handled at least as well as in Casti's books, and the balance between the technical and anecdotal was well done. I thought it much more engaging than most of the popular books I have read in the last few years. Yandell can fascinate the neophyte without boring or offending those with a math background. I don't understand why this book is not as well marketed as those of Casti or Singh (or Sobel, come to that), and I hope that potential readers will find The Honors Class. It deserves your attention.