The leading mathematician of the 20th century, David Hilbert liked to quote "an old French mathematician" saying "A mathematical theory should not be considered complete until you have made it so clear that you can explain it to the first man you meet on the street". By that standard, this book by Hilbert was the first to complete several branches of geometry: for example, plane projective geometry and projective duality, regular polyhedra in 4 dimensions, elliptic and hyperbolic non-Euclidean geometries, topology of surfaces, curves in space, Gaussian curvature of surfaces (esp. that fact that you cannot bend a sphere without stretching some part of it, but you can if there is just one hole however small), and how lattices in the plane relate to number theory.

It is beautiful geometry, beautifully described. Besides the relatively recent topics he handles classics like conic sections, ruled surfaces, crystal groups, and 3 dimensional polyhedra. In line with Hilbert's thinking, the results and the descriptions are beautiful because they are so clear.

More than that, this book is an accessible look at how Hilbert saw mathematics. In the preface he denounces "the superstition that mathematics is but a continuation ... of juggling with numbers". Ironically, some people today will tell you Hilbert thought math was precisely juggling with formal symbols. That is a misunderstanding of Hilbert's logical strategy of "formalism" which he created to avoid various criticisms of set theory. This book is the only written work where Hilbert actually applied that strategy by dividing proofs up into intuitive and infinitary/set-theoretic parts. Alongside many thoroughly intuitive proofs, Hilbert gives several extensively intuitive proofs which also require detailed calculation with the infinite sets of real of complex numbers. In those cases Hilbert says "we would use analysis to show ..." and then he wraps up the proof without actually giving the analytic part.

If you find it terribly easy to absorb Hilbert's THEORY OF ALGEBRAIC NUMBER FIELDS and also Hilbert and Courant METHODS OF MATHEMATICAL PHYSICS, then of course you'll get a fuller idea of his math by reading them--but only if you find it very easy. Hilbert did. And that ease is a part of how he saw the subject. I do not mean he found the results easily but he easily grasped them once found. And you'll have to read both, and a lot more, to see the sweep of his view. For Hilbert the lectures in GEOMETRY AND THE IMAGINATION were among the crowns of his career. He showed the wide scope of geometry and finally completed the proofs of recent, advanced results from all around it. He made them so clear he could explain them to you or me.

There might be less than 10 mathematics books in the world that I am glad to put under my pillow when I go to sleep. And this book is one of the top three.

This is one of the best books on Mathematics ever written. The author is arguably the best mathematician of the century. Here he treats geometry, including topology, in an elementary, though profound, way, with no formalism. A work of art. Books like this shouldn't ever become "out-of-print".

This is a lovely little book. While the subject may be relatively specialized within mathematics, the presentation in the book is lively, and the book is likely to engage readers who are generally interested in math,-- even if they might not, at the outset, recognize the words in the title of the book. And readers who are not already familiar with the subject, are likely to be surprised by the universality, the beauty, and power of the underlying idea. If they thought at first that the writing might be dry, they will instead find a page-turner (--at least relative to other specialized math books). The present authors do take great care in explaining the ideas in plain English. The co-authors are all masterful expositors who engage their readers. They are equally engaging and popular as speakers at confernces and workshops in mathematics.

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.

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.

"Hilbert" is justly famous as one of the best mathematical biographies around. Constance Reid, who also wrote a biography of Hilbert's student Courant, initially ran into some resistance from Hilbert's associates when she started work on this book. Max Born was not keen on the idea of a woman, who was neither German nor a mathematician, writing a study of Hilbert's life. Born was enthusiastic about the final product, however, and it has become a classic.

Hilbert took over from Poincare the title of the most famous mathematician in the world. His mathematical achievements are numerous and varied; Reid does a good job of providing an overview of the impact Hilbert had on many different fields, and of his style; his strengths and weaknesses. There is a good deal of coverage of the famous twenty-three Hilbert problems, presented to the Second International Congress of Mathematicians in Paris in 1900, including a large section of the talk Hilbert gave.

Reid paints a vivid picture of the mathematical circle at Gottingen, a luminous collection of talents. Minkowski and Hilbert were close friends; Klein was the director of the institute there; Emmy Noether was there; Hurwitz; Zermelo; Landau; the list is long and impressive. It's all the more sad to read about the way the Institute was destroyed by the Nazis in the name of racial purity. Almost without exception the leading mathematicians emigrated, one by one, to America. Hilbert, who had retired in 1930 (retirement at age 68 was mandatory) was forced to watch as the work of decades was dismantled. The last years, of age, fading memory and the privations of war, are mercifully given less than a dozen pages.

Hilbert's life leads from the great days of the mid-nineteenth century to the Nazis and the atomic bomb. Reid has done a wonderful job of capturing the feel of Germany over his long life, and the mathematic impact and importance of his work. A compulsory book for those interested in modern mathematical history.

Very well written. Gives a great feel for who Hilbert was as a person. It also does a good job of placing his achievments amoung the other mathematians of his time. I have attempted reading a few biography's of other mathematicians that focus on the math and not the person. They are almost unreadable. This book does not make this mistake. I have just bought "Foundations of Geometry" because of this book. I highly recommend it. PS - 4 stars is as high a rating as I give. I like having room to move for the truely fantastic.

David Hilbert was arguably one of the greatest mathematicians
ever!. He contributed to several branches of mathematics,
including functional analysis, mathematical physics,
calculus of variations, and algebraic number theory.
(Everyone knows what a Hilbert space is right!)

At the turn of the 20th century, Hilbert enumerated
23 unsolved problems of mathematics that he considered worthy
of further investigation. To this day, very few of these, including
the 10th problem, on the finite solvability of Diophantine
equations, have been resolved! (thanks to
Yuri Matiyasevich, Martin Davis and Julia Robinson!).
Besides, Hilbert was also a character (read the section
when Norbert Weiner of cybernetics fame, came to give
a talk at Gottingen, and .... :-)).

Incidentally the author Constance Reid is the sister of
Julia Robinson (of Hilbert's 10th problem fame!),
hence there can no one better to write about
Hilbert!. Besides Constance Reid is a well known chronicler
of mathematicians lives (this one is a tour de force and
her best!).

No one can can call himself/herself a mathematician without
having Reid's book on his/her bookshelf. Strongly
recommended!

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.

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!

David Hilbert's "Grundlagen der Geometrie" is a work of great significance for anyone interested in mathematical foundationalism, the history of geometry, and intellectual history and philosophy in general. Sadly, however, the translation of this edition is extremely poor --- not simply akward, or rough, but careless to the point of making the text unreadable. If I did not have access to the German original, I would have long ago given up on making sense of the translation. In Theorem 7, for example, it speaks of "points which are not on the plane alpha." The German is extremely ambiguous, but mathematically it only makes sense if you interpret the sentence as referring to the "line a." On page 31, the translator commits the unpardonable error of mistaking "nun" (now) for "nur" (only). At the end of theorem 34, and entire equation was left out, and the meaning of the sentence completely bungled. Most extraordinary is Theorem 35, where what should be translated as "It follows from Theorem 22 that the sum of two angles of a triangle is less than two right angles" becomes "the sum of the angles of a triangle is less than two right triangles." In the very next sentence, "mithin" is interpreted as "hence," implying a direct logical entailment where there is none. It should have been rendered simply as "of course." Finally, in the next paragraph, it reads "where epsilon denotes any angles." The German has "irgendeinen Winkel" --- unambiguously singular.

Given the tremendous importance of Hilbert's Foundations, it is quite sad that there is not a quality translation available.

Euclid's initial work on Geometry held firm for two thousand years and then Hilbert wrote the "Foundations of Geometry." Hilbert's work is the cornerstone of modern geometry and its simple, elegant, rigor has had a profound impact in many other areas of modern science.

Compare Hilbert's use of definition and axiom to Euclid's. Euclid defines "point" as the cryptic "that which has no part" whereas Hilbert dives straight for gold and says "between any two points there is just one straight line." There are issues with this approach too and the fun value of this book is to grok it sufficiently that you come to know not only the geometry contained in it but also the reason why it is generally considered superior to Euclid's work and just what the issues in the respective formal systems are.

The translation from the original German is, I understand, poor.

Unlike other books of geometry , the author of this book constructed geometry in a axiomatic method . This is the feature which differ from other books of geometry and the way I like . Let's see how the author constructed axiomization geometry . Intuition and deduction are two powerful ways to knowledge . The axioms are the intuitive principles which are needless to be proved . The theorems are the demonstrated propositions which are deduced from axioms . Although axioms are intuitive , they may have the demonstrated propositions called theorems which contradict . If they do , the system of the axiomization geometry would break down . Because it has some false propositions if you think the contradictory ones as truth , and vice versa . There are all the discussions of the problems above in chapter 2 called consistency which is very important in an axiomatic system .

This book gives a nice introduction to the mathematical formalism behind quantum physics and the logic of measurement. The first chapter gives an introduction to measure theory with emphasis on probabilities of measurement outcomes. The author is careful to point out that the calculation of the Lebesgue integral presents more difficulties than in the Riemann integral case, since the fundamental theorem of calculus does not apply to Lebesgue integrals.

This is followed by an elementary introduction to Hilbert space in Chapter 2. This is standard material and most of the proofs of the main results are omitted and left to the reader as projects.

Chapter 3 is more controversial, and attempts to formulate a logic of experimentation for "non-classical" systems. This is done by use of what the author calls a "manual", which is viewed as an abstraction of the experimenters knowledge about a physical system. A manual is a collection of experiments, and an "event" is a subset of an experiment. Orthogonality of events is defined, along with the notion of a collection of events being "compatible", meaning that there is an experiment that contains all of these events. A manual is called "classical" if every pair of events is compatible. The author then exhibits systems that are not classical via the double-slit and Stern-Gerlach experiments. A logic of events is then developed in the next section, where quantum logic is defined explicitly. The author defines a pure state that is not dispersion-free as a state of ontological uncertainty as opposed to "epistemic" uncertainty. Quantum systems have states that are ontologically uncertain according to the author. The author chooses not to engage in the debate about the actual existence of these states and, accordingly, no real-world experiments are given to illustrate the relevance of the concepts and definitions.

The next chapter covers the geometry of infinite-dimensional Hilbert spaces. The structure of the collection of these subspaces is defined in terms of the quantum logic defined earlier. This is followed by a discussion of maps on Hilbert spaces, as preparation for defining observables in quantum systems. The important Riesz representation theorem is stated but the proof left to the reader. Projection operators are defined also with the eventual goal of relating them to the compatibility of two propositions.

Gleason's theorem is discussed in Chapter 6, along with a discussion of the geometry of state space. The proof of Gleason's theorem is omitted, the author emphasizing its difficulty. The proof in the literature is non-constructive and thus the theorem is suspect according to some schools of thought.

The spectral theorem, so important in quantum physics, is discussed in the next chapter. Once again the proofs are left to the reader for most of the results. The spectral theorem allows the author to define another notion of compatibility in terms of the commutativity of two Hermitian operators.

The books ends with a overview of the EPR dilemna and is naturally more controversial than the rest of the book. This topic has provoked much philosophical debate, and the author gives the reader a small taste of this in this chapter.

The book does serve its purpose well, and regardless of one's philosophical position on quantum physics, the mathematical formulations of quantum physics and measurement theory are nicely expounded in this book.