Product Description
David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. It lays the groundwork for his later work with Bernays.This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Gödel's completeness proof for the predicate calculus has been updated.In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic. ... Read more

Customer Reviews (2)

Good old book
I learned logic from this book, and so I am very fond of it.As a presentation of what was known at that time, it cannot be beaten.The only problem with it is that a lot more has been discovered, so a modern treatment of the subject is better for the beginner who wants to be properly informed.For this I suggest the books by Copi or even Kleene.But if you don't care about modernity, or have an interest in the way things used to be done, I strongly recommend this book.Also, I might mention that Hilbert's "Geometry and the Imagination" is good even for the modern mathematician.

classic
Brief though it is, _Priniciples_ manages to cover not only the usual topics (sentential calculus, first-order predicate calculus, completeness, decidability), but also:the monadic predicate calculus in relation toAristotelian logic; second-order logic; set theory and the Fregean conceptof number; and the theory of types (logics of higher order).You might saythat Hilbert covers the same ground in 160 pages that Russell and Whiteheadlabor over for 3 volumes.The bottom line:a treat for anyone interestedin logic, especially in the period from Frege to Godel. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Customer Reviews (7)

The Foundations of Geometry - Forgotten Books edition

The FORGOTTEN BOOKS edition of Hilbert's Foundations of Geometry isn't Hilbert's Geometry. Notice the number of pages (which I didn't when ordering it). This publication contains ONLY the diagrams in large format (with a very few absent) from the text of Hilbert's Geometry. There is no title page or author listed, but this is in fact what the content is from. It is clearly a scan from an old book, so there must be some historical context for it. Maybe someone can clarify the mystery. I give it 5 stars because these comments will probably show up among the reviews of Hilbert's full text and I don't want to skew the star rating of the book, but this particular reprint I don't find of any actual value, except that it's from Hilbert and there may be some interesting reason why it occurs as an independent publication.

Along with this reprint, I also ordered the FB Classic Reprint of Elements of Geometry and Trigonometry by Charles Davies. These two books are the first reprints I've purchased from any of the reprint publishers selling on amazon. For more on the quality of Forgotten Books reprints, see my review of Davies' book. The mysterious Hilbert-diagrams text they sell under the title of Hilbert's Foundations of Geometry is, I suspect, an anomaly. Besides, their honest page-count should raise questions about the content. Now you know what that content is.

Buy the Open Court edition - a better translation into English
For the best translation into English (not Townshend's translation) see Hilbert's "Foundations of Geometry" as extended by Paul Bernays, Open Court Publishing Co, second edition in English, 1971.(This should be a translation of the 10th and final edition in German by Bernays, which dates from 1968.)Bernays was Hilbert's assistant at Göttingen beginning in 1917 and his co-author of "Grundlagen der Mathematik" (1934-39).
Hardcover ISBN: 0875481639
Paperback ISBN: 0875481647

Incomplete
This is the first book ever to present the axiomatic foundations of euclidean geometry. The first edition appeared in the nineties of the nineteenth century.

Most of the book can be read and appreciated by someone who is mature in elementary euclidean geometry (in fact the material was originally conceived to be used in a summer school for mathematics teachers in Germany). If you expect to find a treatment that will fill up all the gaps in the elementary books you will be disappointed, it does not. If you are looking for a text that does fill all the gaps try to get a copy Forders' book The foundations of Euclidean geometry,.

This edition is not based on the last German edition that is available and does not contain the appendices by Hilbert and thesupplements by Paul Bernays, so as a text on the foundations of euclidean geometry it is not useless but it is surely crippled.

I do not dare to give a book with Hilberts name on it less than five stars.

Enjoyable
Hilbert gives his new system of axioms and studies their consistency, independence and necessity. Consider for example the theorem that the angle sum in any triangle cannot be greater than two right angles. We can prove it as follows. Consider a triangle ABC with the angles labelled so that ABC<=ACB. Let D be the midpoint of BC. Draw AD and extend it to E so that AD=DE. By SAS, ACD=BDE, so that angle CAD=angle BAE and angle DBE=angle ACB. Thus ABC has the same angle sum as ABE. ABC<=ACB means that AC=BE<=AB, so angle BAE<=angle AEB, so angle BAE<=angle BAC/2. In other words: for any angle A in any triangle we can construct a new triangle with equal angle sum that has as one of its angles A/2. By repeating this process we can make the angle A as small as we like. Thus, if the angle sum of some triangle was greater than two right angles, and we applied this procedure, we would get a new triangle where two of the angles are greater than two right angles, which is impossible. The "as small as we like" part gives away the fact that we are relying on Archimedes' axiom, which is necessary. "The investigation of this matter which [Max] Dehn has undertaken at my urging led to a complete clarification of this problem. ... If Archimedes' axiom is dropped then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less than two right angles. Moreover, there exists a geometry (the non-Legendrian geometry) in which it is possible to draw through a point infinitely many parallels to a line and in which nevertheless the theorems of Riemannian (elliptic) geometry hold. On the other hand there exists a geometry (the semi-Euclidean geometry) in which there exists infinitely many parallels toline through a point and in which the theorems of Euclidean geometry still hold. From the assumption that there exist no parallels it always follows that the sum of the angles in a triangle is greater than two right angles." Another interesting topic is the connection between laws of algebra and the theorems of Pappus (which Hilbert calls Pascal's) and Desargues. Geometrically, we can multiply two numbers a and b using only the axioms of projective geometry as follows. We choose a line to be the "x-axis" and call one of its points the origin O and another of its points the unit 1. Mark Oa and Ob on this line. Draw another line, the "y-axis", through O. Pick some point i on the y-axis. Connect 1 and i, and draw the parallel to this line through b, meeting the y-axis at b' (as usual, "parallel to l" means: meets l at an arbitrarily designated line called the line at infinity). Connect a and 1 and draw the parallel to this line through b'. In Euclidean geometry this line cuts the x-axis at ab. In general, then, we may define multiplication in this way. The algebraic identity ab=ba now becomes a geometric theorem. This is the beautiful part: ab=ba is not just any old geometric theorem, it is in fact equivalent to Pappus's theorem: the construction of ab consisted of the line connecting 1 and i and three more lines, the construction of ba consists of the line connecting 1 and i and three more lines, each of which is parallel to one of the lines from the ab construction. Therefore, deleting the line connecting 1 and i, Pappus applies and says ab=ba. Similarly, Desargues is equivalent to a(bc)=(ab)c.

Available for Free
This historic book is available for free from Project Gutenberg http://www.gutenberg.org. Search for Geometry. This book is one of a few books available. This is the complete Open Court text. It is available both as a pdf file and a TeX file. ... Read more

Product Description
This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer--after more than half a century! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians.

"Hilbert and Cohn-Vossen" is full of interesting facts, many of which you wish you had known before, or had wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.

One of the most remarkable chapters is "Projective Configurations". In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.

A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!

The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry.

It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the "pantheon" of great mathematics books. ... Read more

Customer Reviews (8)

l'imagination au pouvoir
Ce livre est magnifique, on sent le génie de Hilbert dans ce livre. Parfois un peu difficile à suivre, il est bon de complété sa lecture avec d'autres livres.
Il donne une introduction à a peu près toutes les parties de la géométrie et va parfois très loin dans le sujet.
Un plaisir...

A classic on Geometry
A pearl! Anyone interested in Geometry shouldn't miss the lucid presentation of the great Hilbert.

Many beautiful things
This is a marvellous book. I will illustrate by one sample from each chapter (except chapter 1 on "the simplest curves and surfaces" which is the least exciting chapter).

Chapter 2 on "regular system of points" contains a beautiful derivation of Leibnitz' series pi/4=1-1/3+1/5-1/7+... If we draw a large circle centred at the origin then of course a good measure of its area is the number of integer points it contains. Now, for any such point, x^2+y^2 is an integer less than r^2. So the number of such points can be obtained by going through all integers less than r^2 and counting how many times it can be written as the sum of two squares. But this is a classical problem in number theory and the solution is known. So this number theoretic result essentially tells us the area of a large circle, so it implies an expression for pi, namely Leibnitz' series.

Chapter 3 is on projective geometry. We go through many projective configurations that are not seen very often today, but still the classics are the best, such as Desargues' theorem. If we have a triangular pyramid and cut it with two planes to get two triangles then the three points of intersection of the extensions of corresponding sides will or course be on a line (the intersection of the two planes), which is the three-dimensional Desargues' theorem. But by projecting the triangles onto one of the walls of the pyramid we get two projectively related plane triangles and the theorem holds for them also. All we have to do to prove the plane Desargues' theorem is to prove that all such configurations can be obtained in his way (i.e. that one can always erect an appropriate pyramid based on two projectively related plane triangles) which is practically obvious.

Chapter 4 is on differential geometry. The fundamental concept of differential geometry is curvature, which is a number that indicates how curved a surface is at a given point. It may be defined as follows. We draw a little circle around the point on the surface and consider all the normals to the surface at these points. Take these normals and put them with their origin at the center of a sphere; then they will sweep out a section of the surface of the sphere. The curvature is the ratio of the area enclosed on the surface and that on the sphere as the circle is taken infinitesimally small. This quantity is seen to be invariant under bending by triangulating the surface; then the the circles are polygons with fixed angles and the theorem follows from the fact that the area of a spherical triangle is determined by its angles (proof omitted here; see any Stillwell geometry book for Harriot's beautiful proof (a.k.a. "Euler's proof")). Now, there are two fundamentally different types of points. Either the surface bends in the same direction in every direction, as on a sphere, or it bends in different directions like a saddle. In the first case the boundary on the sphere traced out by the normals has the same orientation as the boundary on the surface; in the second case the orientation is reversed. So, using signed area, the second type of points have negative curvature. A typical surface will have areas of positive curvature and areas of negative curvature and in between there will be lines of zero curvature. An absolutely wonderful, although perhaps not entirely successful, application of this concept is Klein's Apollo Belvidere hypothesis that the curves of zero curvature on a human face determine beauty.

Chapter 5 on kinematics contains a determination of the curve that "we may observe ... every day in cups and tin cans when the light shines on them", i.e. the coffee cup caustic. With the sun at x=-infinity, the radius that makes an angle theta with the x-axis will point to a point where the angle of reflection is also theta. Consider a concentric circle of half the radius, and another circle with the other half of the radius as its diameter. The arc cut out of the inner circle by the radius and the x-axis is equal to the arc cut out of the outer circle by the radius and the reflected ray (arc with central angle theta in the big circle = arc with central angle 2*theta in the small cirlce). The shape of the caustic follows by rolling the outer circle on the inner. The reflected light rays are tangent to this curve since they are perpendicular to the line connecting the generating point with the center of motion (intersection of the two circles).

From chapter 6 on topology one nice result is that any continuous mapping of a disc onto itself has a fixed point. For suppose it did not. Then any point in the circle can be connected with its image by an arrow. Now consider the point on the boundary. The arrow direction varies continuously as we walk once around the circle, and it end up where it started so it must have made an integer number of revolutions. But there is also a tangent at each point, and the tangent of course make one revolution as we walk once around. The arrows always point to some point in the disc so they could never point in a direction parallel to the tangent so the arrows in fact have to make one revolution also (they would have to be parallel to the tangent for a moment to overtake it, and if they stood still they would be parallel to the tangent "at six o'clock" so to speak). But if we consider the same situation for a concentric circle inside the disc then it too must have arrows making one revolution because the number of revolutions can not make jumps since the new circle is obtained by continuous shrinking of the circumference circle. But as we shrink this circle to infinitesimal radius then all its arrows point in the same direction, so they don't make one revolution, so we have a contradiction. One sees similarly that a continuous mapping of the sphere onto itself also has a fixed point. Since the projective plane is the sphere with diametrically opposite points identified this proves that any projective transformation has a fixed point.

Don't expect to find it "easy."
I agree that this book, co-authored by the co-greatest mathematician of the first quarter of the twentieth century, is a masterpiece to be treasured and kept in print, as other reviewers have stated.

However: The Preface states: "This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."

All I can say is that if you read this and find it "easy," then you have terrific mathematical talent! Yes, the drawings and the intuitive descriptions are helpful, but much of the book is so obscure that I have been told that one of the world's leading geometers is working on an annotated edition explaining what the authors were talking about. On topics which I had already studied elsewhere, I found the presentation illuminating.

I still recommend this book.

Beautiful, Rewarding, and Deep.
I have some 47 books in the geometry section of my shelves.If I had to discard 40 of these, Geometry and the Imagination would be among the 7 remaining.

Geometry is the study of relationships between shapes, and this book helps you see how shapes fit together.Ultimately, you must make the connections in your mind using your mind's eye.The illustrations and text help you make these connections.This is a book that requires effort and delivers rewards. ... Read more

Product Description

Volume 5 has three parts, dealing with General Relativity, Epistemological Issues, and Quantum Mechanics. The core of the first part is Hilbert’s two semester lecture course on ‘The Foundations of Physics’ (1916/17). This is framed by Hilbert’s published ‘First and Second Communications’ on the ‘Foundations of Physics’ (1915, 1917) and by a selection of documents dealing with more specific topics like ‘The Principle of Causality’ or a lecture on the new concepts of space and time held in Bucharest in 1918. The epistemological issues concern the intricate relation between nature and mathematical knowledge, in particular the question of irreversibility and objectivity (1921) as well as the subtle question whether what Hilbert calls the ‘world equations’ are physically complete (1923). The last part deals with quantum theory in its early, advanced and mature stages. Hilbert held lecture courses on the mathematical foundations of quantum theory twice, before and after the breakthrough in 1926. These documents bear witness to one of the most dramatic changes in the foundations of science.

Product Description
If the life of any 20th century mathematician can be said to be a history of mathematics in his time, it is that of David Hilbert. To the enchanted young mathematicians and physicists who flocked to study with him in Goettingen before and between the World Wars, he seemed mathematics personified, the very air around him"scientifically electric." His remarkably prescient proposal in 1900 of twenty-three problems for the coming century set the course of much subsequent mathematics and remains a feat that no scientist in any field has been able to duplicate. When he died, Nature remarked that there was scarcely a mathematician in the world whose work did not derive from that of Hilbert.

Constance Reid's classic biography is a moving, nontechnical account of the passionate scientific life of this man-from the early days in Koenigsberg, when his revolutionary work was dismissed as "theology," to the golden years in Goettingen before Hitler came to power and within a few months destroyed the entire Hilbert school. ... Read more

Customer Reviews (8)

a masterpiece biography that brings david hilbert and göttingen university to life
i almost cried when i got up to the death of minkowski, hilbert's closest friend and a great mathematician in his own right. such is the completely engaging power of constance reid's biography of hilbert. reid did an amazing job integrating the details of professor hilbert's life with the german zeitgeist, all the while providing some exposition of the mathematics for a general audience.

as a student and teacher of mathematics, i went into this book expecting to learn more about one of my heroes, the legendary david hilbert, perhaps best known to most for the famous twenty-three hilbert problems. hilbert did research in an impressive number of areas within mathematics, as well as branched out to physics and the philosophy of mathematics. hilbert's breadth and depth is what gave him the right to influence the course of twentieth century mathematics through the hilbert problems. suffice it to say, hilbert is very important in the mathematical community and i was more than a bit wide-eyed even before reading page one of reid's biography. after turning the last page, i feel like i understand more of the man actually standing on the well-deserved pedestal. my respect for hilbert has only grown knowing of his human flaws and what he has accomplished in spite of them.

it should be made explicit that this book is not just for math nerds. any intelligent reader with any interest in mathematics should be able to enjoy and benefit from reading reid's biography. perhaps you've heard of hilbert or the hilbert problems from some newspaper article somewhere and you wondered how mathematicians really lived. if so, that one spark of interest will be rewarded by reid's thorough biography.

of particular interest to me was the other side of the professorial life: teaching. research is the most important part of a mathematician's life, but i'm glad that reid even goes into details on hilbert's teaching style. apparently, hilbert didn't sufficiently prepare for lectures and would often get confused while teaching, needing the help of his assistants to get out of trouble. these mishaps were because of hilbert's strong desire to offer the students the most important points of the lessons, and the details would sometimes suffer because of this emphasis. hilbert would also repeat things as much as five times on purpose in order to get students to remember because he didn't have confidence in their abilities. half of his lecture time would also be spent reviewing the material from the previous lecture, again in order to really ingrain the material in the students' minds. i found these pedagogical details quite fascinating.

outside of hilbert's life, reid does a fantastic job transporting the reader to germany. the germans have a rich history and complex culture, but many folks can only think of the "n" word when it comes to germany. it's a pet peeve of mine that a large number of people cannot distinguish between "nazi" and "german." at göttingen university, mathematicians of all backgrounds, and even from different countries, were able to come together and work on mathematics under the leadership of david hilbert. göttingen was the epicenter of mathematical thought and it was spiritually destroyed by the nazis in their misguided quest for racial purity. this human story of political madness turning its sword on the purist of realms, mathematics, should hopefully dispel some negative german stereotypes. it is hoped that the reader will appreciate the tragedy of german identity a little more afterward.

those looking for a similar style of biography meshed with some explanations of higher mathematics for a non-specialized audience should also check out benjamin yandell's "the honors class: hilbert's problems and their solvers." yandell covers the hilbert problems in more depth than they are in reid's book. finally, reid also wrote "courant," a biography of one of hilbert's students. "courant" is regarded as the sequel to "hilbert" and should be read to complete the story. all three of these books are excellent and will immeasurably enrich the curious reader.

For non-mathematicians by a non-mathematician author
Constance Reid is a non-mathematician author, so she is the best person who can explain the 'abstract' modern math to the curious non-mathematicians. By following the book on the Greatest mathematician in 20th AD, the readers can understand the major development of Modern Math evolved around Hilbert and all the world's top mathematicians gathered in Gottingen before WWII.
Most of us learn abstract math without knowing the background from which these abstract concepts were derived. In this book (chapter VI: Changes) I learn from Reid the simple yet revealing explanation of 'Ideals' being born out of conflict of 'Algebraic Number Field' with the 'Fundamental Law of Arithmetics', and Kummer's Ideal Number, Kronecker and Dedekind's complicated 'Ideal Primes', and finaly David Hilbert's great contribution in the 'Ideal Primes' theory.

David Hilbert
A excellent biography of the German mathematician David Hilbert.Particularly poignant is the loss of Minknowski and the decline of mathematics at Gottingen following the Nazi prosecutions.

David Hilbert, one of the greatest mathematicians ever
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!

David Hilbert, one of the greatest mathematicians ever
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! ... Read more

Product Description
This text is part of a double volume anthology which presents a survey of current philosophical and scientific writings on colour. This second volume provides an overview of colour science and presents work in colour science that is relevant to philosophical thinking about colour. ... Read more

Product Description

The core of Volume 3 consists of lecture notes for seven sets of lectures Hilbert gave (often in collaboration with Bernays) on the foundations of mathematics between 1917 and 1926. These texts make possible for the first time a detailed reconstruction of the rapid development of Hilbert’s foundational thought during this period, and show the increasing dominance of the metamathematical perspective in his logical work: the emergence of modern mathematical logic; the explicit raising of questions of completeness, consistency and decidability for logical systems; the investigation of the relative strengths of various logical calculi; the birth and evolution of proof theory, and the parallel emergence of Hilbert’s finitist standpoint. The lecture notes are accompanied by numerous supplementary documents, both published and unpublished, including a complete version of Bernays’s Habilitationschrift of 1918, the text of the first edition of Hilbert and Ackermann’s Grundzüge der theoretischen Logik (1928), and several shorter lectures by Hilbert from the later 1920s. These documents, which provide the background to Hilbert and Bernays’s monumental Grundlagen der Mathematik (1934, 1938), are essential for understanding the development of modern mathematical logic, and for reconstructing the interactions between Hilbert, Bernays, Brouwer, and Weyl in the philosophy of mathematics.

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
In diesem Buch, erstmals 1924 bzw. 1937 erschienen, sprt man noch wie am ersten Tag die Frische und Inspiration zweier groer Mathematiker und Lehrer. Hilbert kann man mit Fug und Recht als den letzten Mathematiker bezeichnen, der in allen Gebieten seiner Wissenschaft zu Hause war und in den verschiedensten Bereichen der Mathematik grundlegende neue Erkenntnisse gewann. Seine Resultate haben entscheidend die moderne Auffassung vom Wesen der Mathematik geprgt. Sein Schler Courant galt und gilt auch heute noch als ein ausgezeichneter Lehrer, der auch schwierigste Materien verstndlich darstellen konnte. Das bei Springer erschienene Buch von Courant/Robbins: Was ist Mathematik, kann in diesem Zusammenhang als beispielhaft genannt werden. Alles in allem eine groartige Zusammenfassung der mathematischen Hilfsmittel des Physikers, die auch heute noch viele enthusiastische Leser finden wird. ... Read more

Product Description

David Hilbert (1862-1943) was the most influential mathematician of the early twentieth century and, together with Henri Poincaré, the last mathematical universalist. His main known areas of research and influence were in pure mathematics (algebra, number theory, geometry, integral equations and analysis, logic and foundations), but he was also known to have some interest in physical topics. The latter, however, was traditionally conceived as comprising only sporadic incursions into a scientific domain which was essentially foreign to his mainstream of activity and in which he only made scattered, if important, contributions.

Based on an extensive use of mainly unpublished archival sources, the present book presents a totally fresh and comprehensive picture of Hilbert’s intense, original, well-informed, and highly influential involvement with physics, that spanned his entire career and that constituted a truly main focus of interest in his scientific horizon. His program for axiomatizing physical theories provides the connecting link with his research in more purely mathematical fields, especially geometry, and a unifying point of view from which to understand his physical activities in general. In particular, the now famous dialogue and interaction between Hilbert and Einstein, leading to the formulation in 1915 of the generally covariant field-equations of gravitation, is adequately explored here within the natural context of Hilbert’s overall scientific world-view.

This book will be of interest to historians of physics and of mathematics, to historically-minded physicists and mathematicians, and to philosophers of science.

Product Description
Few problems in mathematics have had the status of those posed by David Hilbert in 1900. Mathematicians have made their reputations by solving some of them like Fermat's last theorem, but several remain unsolved including the Riemann Hypotheses, which has eluded all the great minds of this century. A hundred years later, this book takes a fresh look at the problems, the man who set them, and the reasons for their lasting impact on the mathematics of the twentieth century. In this fascinating book, the authors consider what makes this the pre-eminent collection of problems in mathematics, what they tell us about what drives mathematicians, and the nature of reputation, influence and power in the world of modern mathematics. It is written in a clear and entertaining style and will appeal to anyone with interest in mathematics or those mathematicians willing to try their hand at these problems. ... Read more

Customer Reviews (3)

Absolutely horrible book; look elsewhere for Hilbert
In brief, I do not recommend this book to anyone.

The book has three main problems that make it unenjoyable and quite tedious to get through.

The first problem is that the mathematical problems are not sufficiently well explained. In some cases, the problem and it's math are not explained in sufficient depth. In other cases, the problem is not clearly explained and comprehensible. In yet other cases, the problem's application or usefulness is not explained well. Finally, some math is explained so cursorily that it might as well not even have been discussed.

The second problem is that the layout of the book made very little logical sense. First, the author split the problems into various groups. Then, the author split the timeline into various sections: Pre-1900, 1900-1918, 1918-1945 and 1945-1999. He then proceeded to discuss some of the problems in each of the timelines. At each step, you've lost the entire train of thought from the previous section, and it becomes hard to understand what, if any, logical flow or consistency there was in the development of the work on Hilbert's problems or anything that lead from it.

The third and most serious issue against the book is that the author briefly introduces the "23" Hilbert problems and then subsequently expects the reader to understand what "Problem 10" is every time he refers to it, and exactly how it relates to all the other problems. There are quite a few sentences which refer to several problems by number and unless the reader is constantly referring back to the brief summary of all the problems, the sentence might as well have been in martian. After a few dozen pages of playing this "refer to the problem list" game, I became so frustrated that I just gave up and read the book.

Some minor problems include the fact that only 223 pages of the book are by the author; the rest is a translation of Hilbert's 1900 talk. (That is, less than 75% of the book is what the book is supposed to be about.) Another problem for me is that the paper is high gloss. It looks very nice at a glance, but when you try to read it, you are often faced with glare unless you orient the book perfectly with respect to the light source(s) - a problem I ordinarily face only with magazines.

Another minor problem, which may be interpreted as a positive thing by many (and I welcome that interpretation), is that the author spends a fair bit of time discussing social, governmental political and educational political aspects of countries, universities, and mathematicians. Except in a few minor cases, these things seem highly tangential to the work at hand, and although somewhat entertaining, distract from the intent of the book.

All in all, I found myself about half way through finishing it just so I could put it down and start another book, and preferably one completely unrelated to the topic. The author made such a hash of the subject that, despite finding several other interesting books on Hilbert and his problems here on Amazon, I have no interest whatsoever in reading them and actually finding out about things.

all you want to know about hilbert and his problems
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!

An excellent history of Hilbert and his problems
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 willbe 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. ... Read more

Product Description
This scarce antiquarian book is a selection from Kessinger Publishings Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the worlds literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

Customer Reviews (1)

Interesting
The tittle of this book is quite correct, but maybe not in the sense the author meant in 1904. In it elementary geometry is build up from Hilberts' axioms. Only the continuity axiom is missing. So it is not possible for instance to construct figures using compass and ruler. In an appendix this special problem is solved by introducing an axiom that lets you intersect lines and circles. David Hilbert is NOT one of the authors of this book. The author uses some peculiar terminology but for the rest the development of euclidean geometry, in two and three dimensions, is quite straight forward. The book can be downloaded for free, I believe, at the Michigan mathematics book site. ... Read more

Product Description

David Hilbert (1862-1943) was the most influential mathematician of the early twentieth century and, together with Henri Poincaré, the last mathematical universalist. His main known areas of research and influence were in pure mathematics (algebra, number theory, geometry, integral equations and analysis, logic and foundations), but he was also known to have some interest in physical topics. The latter, however, was traditionally conceived as comprising only sporadic incursions into a scientific domain which was essentially foreign to his mainstream of activity and in which he only made scattered, if important, contributions.

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description

Anschauliche Geometrie - wohl selten ist ein Mathematikbuch seinem Titel so gerecht geworden wie dieses außergewöhnliche Werk von Hilbert und Cohn-Vossen. Zuerst 1932 erschienen, hat das Buch nichts von seiner Frische und Kraft verloren. Hilbert hat sein erklärtes Ziel, die Faszination der Geometrie zu vermitteln, bei Generationen von Mathematikern erreicht.

Aus Hilberts Vorwort: "Das Buch soll dazu dienen, die Freude an der Mathematik zu mehren, indem es dem Leser erleichtert, in das Wesen der Mathematik einzudringen, ohne sich einem beschwerlichen Studium zu unterziehen".

Product Description

This volume contains six sets of notes for lectures on the foundations of geometry held by Hilbert in the period 1891-1902. It also reprints the first edition of Hilbert?s celebrated Grundlagen der Geometrie of 1899, together with the important additions which appeared first in the French translation of 1900. The lectures document the emergence of a new approach to foundational study (the ?axiomatic method?), which concentrates on assessing the logical weight of central propositions by exploiting to the full the method of independence proofs by modelling. This culminates in the lectures of 1898/1899 (the immediate precursor of the 1899 monograph) and 1902. The lectures contain many reflections and investigations which never found their way into print.

Customer Reviews (1)

Delightful
This book is of course invaluable for studying the development of Hilbert's Grundlagen. It also contains lectures that are not directly related to foundations, such as excerpts from a more conventional projective geometry course of 1891 (apparently taught to two students and "ein für Geometrie interessierter Mahler") and a few delightful Ferienkurs lectures (including for example three reasons why number theory is the queen of mathematics, pp. 154-156).

The foundations lectures naturally differ from the published Grundlagen in that they contain things that were already well known, such as discussions of non-Euclidean geometry. The first set of lecture notes on the foundations of geometry is from 1893/94. Most aspects of Hilbert's Grundlagen are here already. The story is still thoroughly enjoyable, however, and greatly enhanced by the editors meticulous attention to pointless details (including things like the colour of the ink Hilbert used).

Consider for example the case of the amphibiousness of Desargues's theorem. 3D Desargues is obvious, so Desargues must hold in any sensible 3D geometry. Could it be that Desargues is not only necessary but also sufficient to create a sensible 3D space, i.e., is it true that "if the Planar Desargues Theorem is added as a new postulate to the planar order and incidence axioms, then this will yield spatial consequences, so that, in particular, the full set of spatial incidence and order axioms will hold"? In his 1898/99 lectures Hilbert thinks the answer is probably yes: "Diese Frage ist wahrscheinlich zu bejahen". Apparently he did not have a proof at this stage but he soon found one; in fact, so soon that the proof appears later in the same course, prompting someone to go back to the Lesezimmer notes and cross out "wahrscheinlich" and underline "zu bejahen", "in rough hand, unlike the usual underlining of the Ausarbeitung, which is done carefully with a straightedge."

The fact that we used spatial methods to prove Desargues raises the question of the purity of method ("Reinheit der Methode"): Could Desargues be proved from plane axioms alone? We prove that the answer is no, using a contrived model later replaced by the Moulton plane. In this connection Hilbert's notes read: "It is fashionable to always guarantee the purity of method. In fact, this is appropriate: we are often not satisfied when a proof in number theory uses geometry or geometrical truths of function theory ... [But a detailed study may reveal] a deeper, legitimate basis and beautiful and fruitful connections, e.g. primes and the zeta function, potential theory and analytic functions, etc."

An appendix lists all Hilbert's lecture courses throughout his career. Hilbert was a real, classical professor. Unlike the disgraceful "professors" of today who whine about their teaching "load", he did not disrespect teaching and learning. He taught widely not only in every area of mathematics (including basic calculus courses several times as a full professor) but also in physics (mechanics, hydrodynamics, potential theory, elementary particles, electromagnetism, special and general relativity, quantum mechanics) and the philosophy of mathematics and science. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more