Customer Reviews

The Honors Class: Hilbert's Problems and Their Solvers

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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.

As a career scientist for over 50 years, I am versed in mathematics but not exactly a mathematician. I bought it to become familiar with Hilbert's problems. I quickly realized that Yandell's book was more about the attempters and solvers than about the problems. Yet the problems are described too, in considerable and certainly sufficient detail.

What was ultimately fascinating was the web Yandell weaves throughout the book. Those famous mathematicians and their colleagues, their personal lives, those famous problems, and all integrated so cohesively.

When I started reading I knew I was in for a long adventure. In fact, it took me over a year to read - of course, only an hour or so every few days. What extended it was the temptation to go back and reread, again and again. Finally, a week ago, I turned the last page. With great reluctance I put it on my bookshelf. I had a strong urge to start all over again from the very beginning, and I knew if I succumbed I was in for another year with it.

As I reflect, partly it was the subject - those difficult problems in such vastly different fields. Partly it was those mathematicians - many of them already heroes of mine. But mostly it was Yandell's skill in putting together this riveting accounting. His love of and fascination with mathematics, and his desire to share his romanticism with others, comes through so clearly. It is sad that he died, at the young age of 53, a scant two years after writing this book. Of a heart attack and multiple sclerosis. What a tragic loss. He was a gem.

Even if you are only mildly interested in mathematics, its history and personalities, you will absolutely love this accounting of it.

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.

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.

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.

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.

First, my background: I am not a mathematician, but an academic with fair knowledge of college math and even some advanced materials. I do greatly enjoy reading books about mathematics and mathematicians.

This book is obviously a work of great effort by the author. My difficulty probably came from the work's ambitious premise: offering mathematical and biographical history of Hilbert's problems. There are simply too many ideas and persons (some well known, others a bit obscure to lay reader like myself) to cover in one book. The author dutifully and honestly gives references to his sources. My impression is that the author collected as much material as he could about each problem and solvers, and tried to squeeze the information as compactly as possible into the pages.

The result: the narrative is very methodically told - explanation of the problem, some necessary ideas introduced, who the major solvers were, then a short biography of each solver, when and where they were born, who their parents were, where they went to school, who they married and so on; then another cycle begins. Halfway into the book, I began to get bored.

I can imagine mathematicians enjoying a quick review of and glimpses into their discipline and heroes, but lay readers much beware. I recommend lay readers to check out a few pages carefully online or at your local library to see whether you like it. I certainly didn't hate it but did not like it as much as I expected.

I was hoping for something more like The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, an explanation of the problems and how they were solved, at a reasonably accessible level. This book appears to be just chat about the funny people who become math professors. Sort of interesting, but not what I wanted.