The Honors Class: Hilbert's Problems and Their Solvers

Rating:
Summary: A great work
Review: Due to rapid development of mathemtics in the last century, now one cannot master all subfects of mathematics. This is also true for those historians. Most of the boods of " History of Mathematics " end in the beginning of 20th century. So we know very little about the conteporary mathematicians. This book can be described as a gap for it. After readiming this book, not only you have a knowledge about the life of the great mathemaitcians, you also get the period in World War II how Nazis forced those mathematicians out of Germany and the reason why U. S. A. is now the leading centre of mathematics.

Rating:
Summary: A page turner
Review: Fascinating historical comments, lively portraits of mathematicians, and their times. While the narrative is about the lives of some great mathematicians, it sucessfully outlines main ideas in the subject,--the personal and scientific context. The author does a great job in sharing his fascination with the rest of us. The book covers roughly the past hundred years. It is a great service to the mathematics community,-- and especially, it is an enjoyment for everyone.
It reads like a novel, fast paced, and it is hard to put down. I meant to look at it before going to sleep, but instead read it to the end, finishing in the morning. As a professional mathematician, I am often saddened by how little our work is perhaps understood and appreciated. Books like this can do a lot of good. I can now tell my children that dad does stuff like that.
The author brings the events and the mathematical people to life, and he has a story to tell. This book is and will be a success for a long time to come.

Rating:
Summary: Best since "Fermat's Enigma"
Review: In 1900, David Hilbert gave an address to the International Congress of Mathematicians that outlined the twenty three most important unsolved problems of mathematics, as he saw them. In "The Honors Class", Benjamin Yandell describes the problems and the very remarkable people who worked on them. More than a century later, there are still a few that remain unsolved, and some of those that have been successfully attacked withstood assault for many decades. I was familiar with many of the names in book because they are associated with equations that I have used and that I teach my students about. It was not until reading this attractive and well-written history that I was able to put those names and their contributions into context. This is the best popular book about mathematics that I have read since "Fermat's Enigma" by Simon Singh.

Rating:
Summary: Almost a biography of math itself
Review: 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.

Rating:
Summary: Useful and insightful
Review: The book is well written with the right mix of anecdote and theory. What I do like about it is the fact that we find out a little more about the lives of mathematicians, and they are portrayed as people rather than idols.

Where the book falls down is that it goes into a little too much detail of the theorems, something which the non-mathematician will undoubtedly find hard to follow.

Rating:
Summary: great historical research and entertaining accounts
Review: 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.

Rating:
Summary: The Honors Class: Hilbert's Problems and Their Solvers
Review: The task of explaining Hilbert's problems and their solutions for perhaps a general audience is not an easy one. William Yandell has done a wonderful job in explaining the development of some significant mathematics as a by product of reviewing the work on Hilbert's problems. Along the way he gives us some insights in how mathematicians in other societies and periods have existed in a social context.

The interesting thing about books such as this is that they actually help you put things in context. One of the hardest things in mathematics is that professionals tend to launch into "There exists a t>0 such that for all ....etc" mode as though the motivation has come from outer space (the motivations are of course appreciated by those in the field). Books such as this actually bring the motivations together in an accessible way for others.

Overall a very good read.