## Reviews of David Hilbert's books

We give below extracts from some reviews of some of David Hilbert's books. We list these in order of publication of the first edition. The list is somewhat complicated since there are many editions of Hilbert's books and in recent years further books have been published from lecture notes that were taken during Hilbert's lecture courses. Most confusing of all, one text has been completely rewritten but retains the same title as the original.

**1. Grundlagen der Geometrie (1899), by David Hilbert.**

**1.1. Review by: Julius Sommer.**

*Bull. Amer. Math. Soc.*

**6**(1900), 287-299.

It is our author's aim to lay the proper foundations for Euclidean geometry, and beyond this, for analytic geometry. His system thus finds its conclusion with the final recognition that space can be regarded as a manifold of numbers. Among the more important points in which Professor Hilbert's memoir marks a distinct advance I wish to call particular attention to the following:

(1) The introduction of the axioms of congruence, and the definition of motion as based on these;

(2) the systematic investigation of the mutual independence of the axioms, this independence being proved by producing examples of new geometries which are in themselves interesting;

(3) the principle of not merely proving a proposition in the most simple way but indicating precisely what axioms are necessary and sufficient for the proof;

(4) the theory of proportions and areas without use of the axiom of continuity, and more generally, the proof that the whole of ordinary elementary geometry can be treated without reference to the axiom of continuity;

(5) the various algebras for segments (Streefenrechnungen), in connection with the fundamental principles of arithmetic.

**1.2. Review by: Anon.**

*The Mathematical Gazette*

**2**(38) (1903), 268-269.

This is from a review of the English translation with title

*The Foundations of Geometry*(1902).

A translation of Hilbert's fascinating Grundlagen der Geometrie is heartily welcome in this country, and the volume under notice is further enriched by the author's additions, which appeared in the French translation which M Laugel published some years ago (Gauthier-Villars). It also contains a summary of a memoir embodying Hilbert's latest researches, which has probably already appeared in the Math. Ann. ... As a translation the volume before us cannot be said to be entirely successful. It has been unmercifully and somewhat undeservedly gibbeted by Prof Halsted in Science, Aug. 22, 1902. A sober and detailed criticism by Dr Hedrick of both this and the French translation will be found in the Bull. of the American Math. Soc., Dec. 1902, to which considerations of space compel us to refer the reader, and in which will be found a long list of errors and misprints.

**1.3. Review by: George Bruce Halsted.**

*Science, New Series*

**16**(399) (1902), 307-308.

This is from a review of the English translation with title

*The Foundations of Geometry*(1902).

The merest justice calls for a pointing out of some few among the blemishes in what Professor Townsend puts forth as a translation of Hilbert's beautiful 'Festschrift.' These blemishes are the more indefensible because Professor Townsend had before him, in addition to the limpid original, the admirable French translation of L Laugel.

**1.4. Review by: Henry George Forder.**

*The Mathematical Gazette*

**15**(213) (1931), 397-400.

This is a review of the 7th edition (1930).

Here at last we have a really revised edition of the Grundlagen, for the fifth and sixth editions were reprints of the fourth, which itself differed little from the third, published in 1909; and it is likely that this edition, revised as it is in detail, is definitive. ... It is a matter of regret that the new edition of this great classic-worthy to be ranked with Euclid's Elements-should cost so very much more than the earlier editions. Can nothing be done to stem the tide of rising prices at home and abroad?

**1.5. From the Preface of the 14th edition.**

David Hilbert's

*Grundlagen der Geometrie*were first published in a Festschrift on the occasion of the unveiling of the Gauss-Weber monument in Göttingen in 1899. This work presented a number of new results in a form that even today appears to be admirably well-rounded, and thus contributed to a deepened understanding of the connections between geometric and algebraic structures. Methodically, the work impressed by deliberately dispensing with visualization and geometrical experiment as expressed in the final formulation. It marks the transition from empirically anchored to formal-deductive geometry. Its lucid axiomatic structure became pioneering for the mathematical thought of the 20th century. The 14th edition under review - corresponding to the 100-year anniversary - has been substantially supplemented and enriched, namely on one hand by contributions which compile the prehistory and evolution until the present, and on the other hand by documents and indices that complete the work and make it more accessible. The two most comprehensive contributions in the jubilee edition under review examine the evolution in the 20th century: on the one hand of the development of Hilbert's work

*Grundlagen der Geometrie*, taking into consideration the origin based on all editions that have appeared, and on the other hand the continuing development of the research topic 'Foundations of geometry' since Hilbert. This edition also boasts - for the first time - a long-expected bibliography. It comprises all of the 150 sources mentioned by Hilbert in the framework of its evolution and mentioned in all previous editions of the

*Grundlagen der Geometrie*. At the same time it serves as bibliography for all other contributions in the edition under review.

**1.6. Review by: Jan von Plato**.

*The Bulletin of Symbolic Logic*

**12**(3) (2006), 492-494.

This is a review of a 2004 edition of an English translation with title

*David Hilbert's Lectures on the Foundations of Geometry 1891-1902*.

Hilbert's book Grundlagen der Geometrie of 1899 has played a peculiar role in the development of the foundations of mathematics. It has been hailed as the dawn of a new era, as in Weyl's 1944 paper 'David Hilbert's mathematical work'. However, Hilbert's topic, the study of axiomatic geometry in the synthetic tradition of Euclid, has been an absolutely marginal field in 20th century mathematics. ... The importance of Hilbert's geometry is to be searched elsewhere than in its influence on research in geometry, so out-dated after a century that journals dedicated to geometry would not even publish such material anymore. Hilbert's role was rather, as Weyl emphasizes, to show the possibility of formalization of mathematical arguments to the extent that one can check their correctness without the need to think of their intuitive content. Thus, geometry gave a model for how to formalize mathematics and how to pose the central foundational questions, such as those of consistency, completeness, and decidability.

**2. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1912), by David Hilbert.**

**2.1. Review by: Thomas Haken Grönwall.**

*Bull. Amer. Math. Soc.*

**20**(1914), 326.

This book appears as volume 3 of a new Teubner collection entitled Fortschritte der mathematischen Wissenschaften in Monographien, under the general editorship of Professor Otto Blumenthal, and consists of a re-impression, with some insignificant changes, of the six papers on integral equations published by Hilbert in the

*Göttinger Nachrichten*from 1904 to 1910, with an additional chapter on the foundations of the kinetic theory of gases (also published in

*Mathematische Annalen*). These various papers not having been remoulded into an organic whole, the presentation of the theory is not a systematic one, but this defect is largely offset by an elaborate table of contents arranged by subjects and stating the main theorems in extenso. Since Hilbert's papers have already become classical and are familiar to all students of the subject, an enumeration of the section headings will suffice:

- General theory of linear integral equations;

- Application of the theory to linear differential equations;

- Application of the theory to problems in the theory of functions;

- Theory of functions of an infinite number of variables;

- A new exposition and extension of the theory of integral equations,

- Application of the theory to various problems in analysis, geometry, and theory of gases.

**3. Methoden der Mathematischen Physik Volume I (1924), by Richard Courant and David Hilbert.**

**3.1. Review by: Einar Hille.**

*Bull. Amer. Math. Soc.*

**31**(1925), 456-459

The present volume centres around one single physical problem, the oscillation problem, with its mathematical equivalents, the boundary value and expansion problems. These are the main problems. Incidentally the reader will pick up a good deal about methods which are applicable to other problems of mathematical physics, but he will have to supply the applications himself. ... A few words regarding the joint authorship should be appropriate. The book is obviously and avowedly written by Courant. It is true that most of the subject matter originated directly or indirectly with Hubert, whose spirit hovers over almost every page of the book. ... the book, in spite of its restricted scope, is rich in material and in points of view which are either novel or little known. The book - as most human work - is not perfect, but the imperfections are mostly on side-issues. It was obviously not meant as an opiate, but intended to stimulate interest, discussion and research in a field which still belongs to the richest in mathematical physics. We look forward to the appearance of Volume II with eager expectation.

**3.2. Review by: Hans Albrecht Bethe.**

*Science, New Series*

**119**(3080) (1954), 75-76.

This is from a review of the English translation with title Methods of Mathematical Physics Volume I (1953).

This English language version of the well-known "Courant-Hilbert" will be very much welcomed, especially by younger physicists in all fields; the older generation will, of course, already have on their shelves a very familiar and much-used copy of the former German edition. The original book, Methoden der Mathematischen Physik was written in a period when the interests of mathematicians and physicists were clearly diverging and it presented in a clear and systematic way the results of much mathematical research which was very relevant to the needs of physicists. It succeeded admirably in its purpose of making physicists aware of the essential unity of the mathematical methods which they were employing in widely differing fields. It also provided a much-needed source book for many mathematical developments which were to be found only in the mathematical journals and which might otherwise have been overlooked by physicists. ... The influence of this book in the development of physical theory has been significant.

**3.3. Review by: Joaquin Basilio Diaz.**

*Mathematical Reviews*, MR0065391

**(16, 426a)**.

This is from a review of the English translation with title

*Methods of Mathematical Physics*Volume I (1953).

This book is a welcome translation of the second edition of the well known "Methoden der mathematischen Physik" (1931). The text covers the following subjects: linear transformations and quadratic forms, development of arbitrary functions in series of orthogonal functions, linear integral equations, calculus of variations, eigenvalue and vibration problems, application of variational calculus to eigenvalue problems, and special functions, as in the German original.

**3.4. Review by: T A A B.**

*The Mathematical Gazette*

**39**(328) (1955), 175.

This is from a review of the English translation with title

*Methods of Mathematical Physics*Volume I (1953).

The mathematical event of 1924 was, possibly, the publication of the first edition of Volume I of " Courant-Hilbert ". Some mathematical physicists may have opened their eyes widely at the emphasis on what we now call linear analysis and on the connection of eigenvalues with variational principles; but the prescience of the authors was fully justified by succeeding developments, so that now "Courant-Hilbert " is not a survey preliminary to exploitation but a map of an organised and settled domain. Its value is thus changed in type but not diminished, and an English translation is very welcome to those who have not the gift of tongues. Courant himself has prepared the translation and has made some additions and improvements, but the book is substantially the equivalent of the second (1931) German edition.

**4. Grundzüge der theoretischen Logik (1928), by David Hilbert and Wilhelm Ackermann.**

**4.1. Review by: Henry George Forder.**

*The Mathematical Gazette*

**14**(197) (1928), 273-274.

The critical work on the foundations of Analysis during last century has led to investigations in fundamental logic during the last decades. The first germs of the modern movement are found in Boole, whose work was continued by C S Peirce and Schroder. Partly independent of these researches, we have the various editions of Peano's Formulario, wherein Peano and his collaborators succeeded in putting a number of mathematical theories into logical symbolic form; this work was based on a most profound analysis of reasoning and its aim was to express all mathematics in a completely symbolic way without the use of any ordinary language. Meanwhile the disturbing contradictions that had arisen in the theory of sets by a too naive use of the notion of class had necessitated a closer scrutiny of logical principles, and this was carried out in Whitehead and Russell's Principia Mathematica, with the help of a development of Peano's symbolism. Hilbert and his followers also use Peano's ideas, but with an aim and method different from that of previous investigators, and suggested by the axiomatic treatment of other mathematical theories. ... The book under review is an introduction to a promised larger work dealing more fully with Hilbert's theory. It treats of the calculus of propositions and of propositional functions, the foundations of mathematics being kept in view throughout. Towards the end of the book the well-known paradoxes are thoroughly examined, the doctrine of types described, its influence on the theory of the Dedekind section shown, its imperfections pointed out, and a remedy briefly sketched. For a full treatment of the latter and its application to the foundations of mathematics we must await the larger work. Hilbert's papers on this subject are well known, but being mostly in lecture form, they are sometimes rather sketchy, and this first instalment of a full and authoritative exposition makes us hope the sequel will soon be published.

**4.2. Review by: Snz.**

*Annalen der Philosophie und philosophischen Kritik*

**7**(1928), 157.

From the Foreword by Hilbert: This book deals with theoretical logic (also called mathematical logic, logic calculus or algebra of logic). - By using and supplementing ... material, W Ackermann ... has produced its present structure and its definite overall presentation. ... Theoretical logic ... is an application of formal methods of mathematics to the field of logic. It addresses logic as a similar formula based language in which it has long been customary to express mathematical relationships. In mathematics it would already be regarded a utopia, if one wanted to use only ordinary language in the construction of a mathematical discipline.

**4.3. Review by: Max Black.**

*The Mathematical Gazette*

**23**(255) (1939), 334.

This is from a review of the 2nd edition of 1938.

This is the second edition of the well-known introduction to symbolic logic which first appeared ten years ago. Few changes have been made in the treatment, which is still conspicuous for rigour and clarity. ... The changes made have increased the value of what was already a first-rate book indispensable to the serious student of mathematical logic.

**4.4. Review by: E N.**

*The Journal of Philosophy*

**35**(14) (1938), 390-391.

This is from a review of the 2nd edition of 1938.

The many-mansioned discipline known as "symbolic logic" has for a long time ceased to be a simple affair, and introductions to it vary according to the special interests to which they cater. The chief emphasis of an introductory work may thus fall upon symbolic logic as a calculus devised for solving problems not capable of being handled by traditional formal logic; upon modern symbolism as an instrument for distinguishing and exhibiting abstract logical forms; upon the reducibility of mathematics to general logic, and hence upon providing material preparatory to reading Principia Mathematica; or upon the syntactical and semantic studies of recent years, and upon the import of these studies for the issues in the foundations of mathematics and in the general philosophy of the formal sciences. This second edition of a well-known book, although it touches upon all the aspects of the subject indicated, accentuates even more than the original version the syntactical researches relevant to Hilbert's Beweistheorie. With one important qualification to be mentioned below, it is an excellent introduction into this phase of the subject, perhaps the best one available. The exposition is uniformly clear and amply illustrated with fully worked-out examples. The reader is rapidly taught to understand the problems and some of the techniques in meta-logical researches, and is carefully prepared to read more advanced treatises.

**4.5. Review by: Willard Van Quine.**

*The Journal of Symbolic Logic*

**3**(2) (1938), 83-84.

This is from a review of the 2nd edition of 1938.

As those familiar with the first edition will recall, the four chapters correspond to four increasingly comprehensive levels of logic. Chapter 1 deals with the propositional calculus; postulates and rules of inference are presented, and their completeness is proved ... Chapter 2 supplements the propositional calculus, in effect, with the Boolean algebra of one-place predicates ... In Chapter 3 a deductive system is presented which involves propositional variables, individual variables, predicate variables of one and many places, the propositional connectives, and predication and quantification with respect to individuals. ... In Chapter 4 logic receives its full generality: predication and quantification are al- lowed not only with respect to individuals but also with respect to predicates and propositions. The paradoxes are presented, and Russell's simple theory of types is adopted for their avoidance.

**4.6. Review by: Alonzo Church.**

*The Journal of Symbolic Logic*

**15**(1) (1950), 59.

This is from a review of the 3rd edition of 1949.

The changes which have been made as compared to the second edition are described in the preface ...

**4.7. Review by: Reuben Louis Goodstein.**

*The Mathematical Gazette*

**34**(308) (1950), 147-148.

This is from a review of the 3rd edition of 1949.

Though it is generally regarded as the youngest of the mathematical disciplines, symbolic logic was conceived two and a half centuries ago by the same mind which fathered the differential and integral calculus, for it was Leibniz who first formulated the idea of a mathematical calculus by which the truth or falsehood of a proposition might be evaluated with the same facility as the answer to an addition sum. ... The Hilbert-Ackermann Grundzüge der Theoretische Logik was written in 1927 as an introduction to David Hilbert and Paul Bernays' projected two-volume work on the foundations of mathematics, but it soon proved to be of independent interest and value, and the second edition (1937) anticipated the appearance of volume 2 of the Hilbert-Bernays Grundlagen der Mathematik by nearly two years.

**4.8. Review by: Reuben Louis Goodstein.**

*The Mathematical Gazette*

**35**(314) (1951), 293-294.

This is from a review of the English translation with title Principles of Mathematical Logic (1950).

This translation of Hilbert and Ackermann's Grundziige der Theoretischen Logik has been made from the second German edition (1938) ... The translation is noteworthy both for its fidelity and vitality. It is not in any sense a literary translation, for there are words, phrases and constructions which have kept sufficiently close to the original German to offend a critical ear, but it is clear, consistent and unambiguous, and shows every sign of careful thought and meticulous care in its preparation.

**4.9. Review by: Geoffrey Thomas Kneebone.**

*Philosophy*

**27**(103) (1952), 375-376.

This is from a review of the English translation with title

*Principles of Mathematical Logic*(1950).

This well-known book is described by its American editor as "a classic text in the field of mathematical logic," and most justly so. It was first published, in German, in I928, and reissued in a second edition in I938. We are now offered an English version, well translated and printed, and worthy of the original. Hilbert and Ackermann's Principles has a very special place in the literature of mathematical logic because it treats, with unsurpassed excellence of presentation, of just those parts of symbolic logic which are clear of the realm of controversy and which are therefore of permanent value to mathematicians and philosophers. ... It is therefore most welcome that this little book, in which these logical systems are made so easy to understand, should now be available in English.

**4.10. Review by: William Hunter McCrea.**

*The British Journal for the Philosophy of Science*

**2**(8) (1952), 332-333.

This is from a review of the English translation with title

*Principles of Mathematical Logic*(1950).

This is a translation of the very well-known Grundzüge der theoretischen Logik of Hilbert and Ackermann, one of the classic texts of mathematical logic. The translators, the editor and the publishers deserve to be thanked for making it available. It is made from the second German edition (1938) and it should be mentioned that a third German edition has been published this year (1951). ... For over twenty years, this has deservedly been a standard work on the subject. The prospect of its remaining such for many years to come has been enhanced by its now being made available in the present excellent translation to a still wider class of readers.

**4.11. Review by: G Zubieta R.**

*The Journal of Symbolic Logic*

**16**(1) (1951), 52-53.

This is from a review of the English translation with title

*Principles of Mathematical Logic*(1950).

This translation differs from the original book in two respects: (1) a terminology more familiar to the readers of American textbooks is used; and (2) some errors appearing in the original are amended, following the indications of criticisms which have pointed out those mistakes.

**4.12. Review by: F H Fischer.**

*The Journal of Symbolic Logic*

**25**(2) (1960), 158.

This is from a review of the 4th edition of 1959.

The fourth edition of this well-known classic differs notably from the preceding one.

**4.13. Review by: Richard Milton Martin.**

*Mathematical Reviews*, MR0104563

**(21 #3316)**.

This is from a review of the 4th edition of 1959.

In the latest edition of this famous text, several substantial changes and improvements are made. In the first place, the method of truth-tables is elaborated more fully. The axiom-systems given throughout are based on those of Gentzen. A new section on intuitionistic sentential logic is added, as well as one on Ackermann's "strenge Implikation". In the chapter on the lower predicate calculus a new section on singular descriptions is added in addition to some material on many-sorted theories.

**5. Anschauliche Geometrie (1932), by David Hilbert and Stefan Cohn-Vossen.**

**5.1. Review by: Herbert Westren Turnbull**.

*The Mathematical Gazette*

**17**(225) (1933), 277-280.

This remarkable book ... comes as a grateful gesture from Hilbert, who is one of the most eminent of mathematicians, and from his collaborator who has borne the brunt of editing the subject-matter right worthily. Here, at a ripe stage in a career of outstanding achievement, a prince of analysts and logicians has been content to lay aside all intricacy and yet deal with high matters of geometry, often in the most elementary way, and always with a direct appeal to the reader's intuition. The emphasis in geometry, just as in any other part of mathematics, may be laid upon the logic of the argument, and proceed from abstraction to abstraction: or, again, it may be laid upon the quality of the subject-matter itself. In the latter case the mathematician has become an artist and seeks only to present his facts with vividness and charm.

**5.2. Review by: T A A B.**

The Mathematical Gazette 36 (317) (1952), 231-232.

The brilliant intuitive and concrete approach to geometry provided by Hilbert and his collaborator Cohn-Vossen needs little recommendation. Six chapters (Simple curves and surfaces, regular systems of points, configurations, differential geometry, kinematics, topology) serve to lead the average reader to a number of vantage points from which large domains can be surveyed. The authors have Clifford's gift of carrying the reader in a few short strides from the elements to the very heart of a geometrical topic, and their knack of exhibiting an underlying unity connecting and elucidating apparently dis-similar regions, a knack which the novice cannot too early begin to cultivate, is in the best Klein tradition. ... No better stimulus for the young geometer could be found.

**5.3. Review by: William Munger Boothby**.

*The Mathematics Teacher*

**47**(2) (1954), 126.

This is from a review of the English translation with the title

*Geometry and the Imagination*(1952).

This is a translation of the famous 'Anschauliche Geometrie' of Hilbert and Cohn-Vossen, and it represents a most welcome addition to the semi-popular books on mathematics in English. The purpose of the book is to present, in as intuitive and pictorially clear a fashion as possible, without cumbersome analytical accompaniment, several well-chosen, illustrative topics and problems from diverse fields of geometry.

**5.4. Review by: Cyrus Colton MacDuffee.**

*Science, New Series*

**116**(3023) (1952), 643-644.

This is from a review of the English translation with the title

*Geometry and the Imagination*(1952).

David Hilbert was a very great mathematician whose research extended into almost every field of mathematics. Furthermore, he was a great teacher and expositor, with a genius for presenting basic ideas uncluttered by details. His insight penetrated far beyond the obvious and brought to light relations previously unobserved. With a self-confidence supported by his pre-eminent position as a mathematician, he did not hesitate to devote attention to mathematics of the most elementary sort, such as arithmetic and plane geometry, and he was able to endow these humble topics with a dignity and depth unsuspected by more superficial observers.

**5.5. Review by: Edwin Arthur Maxwell.**

*The Mathematical Gazette*

**37**(322) (1953), 295.

This is from a review of the English translation with the title

*Geometry and the Imagination*(1952).

A glance down the index (twenty-five columns of it) reveals the breadth of range:- Annulus; Atomic structure; Automorphic functions; Bubble, soap; Caustic curve; Colour problem; Density of packing, of circles; Four-dimensional space; Gears, hyperboloidal; Graphite; Lattices; Mapping; "Monkey saddle"; Table salt; Zinc. These are but a few of the topics brought before the geometer's view. The title invokes the imagination, and the text must surely capture it.

**6. Grundlagen der Mathematik. Vol. I (1934), by David Hilbert and Paul Bernays.**

**6.1. Review by: Henry George Forder.**

*The Mathematical Gazette*

**18**(231) (1934), 338-340.

This is the long-awaited account of Hilbert's metamathematics, or, as it is now called, the Beweistheorie. The publication has been delayed to some extent by the necessity of taking into account the recent results of Gödel, and as a consequence the book is issued in two parts, the first and longest of which is the subject of the present review. Hilbert's method ... partly consists in transferring the formalist view, now usual in geometry, to logic and arithmetic. ... Anyone who reads this book will hope that he will live long enough to see the sequel. It is easily the most important contribution to the question since the Principia. The subject is not yet a garden, but it is no longer a wilderness.

**6.2. Review by: Rudolf Carnap.**

*The Journal of Unified Science (Erkenntnis)*

**8**(1/3) (1939), 184-187.

In discussions on the foundations of mathematics in the last decades, three main opinions have been proposed: the reasoned conception of Frege and Russell the so-called logicism, the doctrine of intuitionism expressed by Brouwer and Weyl, and that of Hilbert and his colleagues, especially Bernays, who developed the concept of formalism.

**6.3. Review by: Geoffrey Thomas Kneebone.**

*The Journal of Symbolic Logic*

**35**(2) (1970), 321-323.

This is from a review of the 2nd edition of 1968.

This is the first volume of the long-awaited second edition of Bernays's masterly presentation of the main results in proof theory obtained by Hilbert and his collaborators before 1934. The modifications made in the new edition do not affect the general character of the book, since they are for the most part minor additions or amendments, or clarifications of points of detail. ... The style of Grundlagen der Mathematik, and the spirit in which it is written, are very different from what is now usual in systematic expositions of logic and proof theory, and the book has to a very high degree the virtues of a more reflective and less specialized age than that of today. Nowadays the non-formal background of metamathematics is normally taken for granted, and authors proceed in a wholly abstract manner, getting down to technicalities from the outset. ... Although the book has now been in existence for thirty-five years, it is still very much alive, and the logical, philosophical, and mathematical background that is to be gained from a careful reading of it remains altogether unique.

**7. Methoden der Mathematischen Physik Volume II (1937), by Richard Courant and David Hilbert.**

**7.1. Review by: Edward Thomas Copson**.

*The Mathematical Gazette*

**22**(250) (1938), 302-306.

The first volume of "Courant-Hilbert" was published in 1924, and proved so successful that a second edition was brought out in 1931. In the first edition the authors promised a second volume dealing with the existence theorems of the equations of mathematical physics from the calculus of variations viewpoint. Publication had to be delayed owing to the unsatisfactory state of the theory in 1924, a state since rectified mainly by the work of Professor Courant and his pupils. The promise of fourteen years ago has been brilliantly fulfilled with the publication of the work now under review. The underlying idea throughout the book is to emphasise general methods, and not to present a dull succession of theorems. Yet a very wide range indeed is covered, and that in no superficial manner.

**7.2. Review by: Garrett Birkhoff.**

Science, New Series 99 (2573) (1944), 322.

The two volumes by Courant and Hilbert are already widely known among mathematicians and physicists for their clarity, rigor and breadth of view. They constitute an outstanding source of material on expansion methods and partial differential equations. American mathematical physics will be benefited both during the war and after by having them available at a greatly reduced price.

**7.3. Review by: Joaquin Basilio Diaz.**

*SIAM Review*

**6**(4) (1964), 463-466.

This is a review of the English translation under the title Methods of Mathematical Physics, Volume II (1962).

The two volumes of Courant and Hilbert's "Methoden der mathematischen Physik" have been regarded, since their appearance, as standard source books for applied mathematicians. And this is the second volume of the English version, contributing to "breaking through the language barrier", so to speak. The preface, by Professor Courant, explains the genesis of the book; this English version is said to have been in preparation ever since the appearance during the last war (1943) of the Interscience Publishers reprint of volume II of the German edition, under license of the United States Government. It also explains the dedication of the book to Kurt Otto Friedrichs as "a natural acknowledgement of a lasting scientific and personal friendship".

**7.4. Review by: Menahem Max Schiffer.**

*Science, New Series*

**137**(3527) (1962), 334.

This is a review of the English translation under the title Methods of Mathematical Physics, Volume II (1962).

The second volume of the mathematical classic, the Courant-Hilbert Methoden der Mathematischen Physik appeared in 1937, and it is still an indispensible handbook for anyone who has to deal with partial differential equations. It has served as a basis for countless courses in applied mathematics and advanced mathematical physics; it has stimulated and strongly influenced mathematical research during the past quarter century. The present volume is the long-expected English translation and, at the same time, a very much enlarged and revised edition of the original book, which covers the subject matter found in the first six chapters of the German edition. The seventh chapter, which deals with existence proofs for elliptical methods, has been omitted in this translation, and will form the nucleus of a projected third volume of Courant-Hilbert.

**7.5. Review by: Michael James Lighthill.**

*Mathematical Reviews*, MR0140802

**(25 #4216)**.

Volume I, by the authors, was published in 1953. This completely rewritten version of the famous classic on partial differential equations has been greatly influenced by major applied mathematical researches of the 1950's. In fact, modern analytical knowledge of the subject that is valuable in applications is covered to a truly extraordinary extent in this volume. It may be read independently of Volume I, while a shorter Volume III is also promised, concerned with existence proofs (particularly those for elliptic equations) and with the construction of solutions by methods such as finite differences. ... Compared with its first edition, the present volume both contains a much greater volume of information and succeeds in rendering it more easily intelligible, improvements made possible, above all, by the vast volume of research conducted since the appearance of the first edition and based directly or indirectly upon its influence and that of its author.

**8. Grundlagen der Mathematik. Vol. II (1939), by David Hilbert and Paul Bernays.**

**8.1. Review by: Max Black.**

*Mind*

**49**(194) (1940), 239-248.

The philosopher who assumes that the title of Grundlagen der Mathematik describes a modern version of Principia Mathematica, written from the formalist standpoint, will be disappointed. The character of the formalist programme (as further explained below) demands that philosophy should be treated with the frigid politeness accorded to a gate-crasher. The generous scale of the treatise is accounted for by the desire to achieve accuracy in the detailed discussion of technical methods, often published here for the first time, and always involving notions whose exact definition requires meticulous attention to detail. Thus although the primary object is to present proofs of consistency (Widerspruchsfreiheit) of mathematical systems, no such proof appears in the first volume, whose 468 pages are occupied by preliminary considerations, such as the possibility of transforming formulae in the propositional calculus into a standard form, the extent to which logical quantifiers can be eliminated from an axiom system, the exact specification of the rules for the use of recursive definitions, and proofs of categoricity (Entscheidungsbarkeit) and completeness (Vollständigkeit) for simple systems. The new volume, where we reach proofs of consistency for the first time, is by far the more interesting.

**8.2. Review by Stephen Cole Kleene**.

*The Journal of Symbolic Logic*

**5**(1) (1940), 16-20.

Proof-theory or metamathematics is the investigation of formal logico-mathematical systems by "finite" methods.

**8.3. Review by: Henry George Forder.**

*The Mathematical Gazette*

**24**(260) (1940), 225-227.

... the subjects of Hilbert's investigations are demonstrations, supposed to be set out in full detail in the symbolism of mathematical logic; in particular, he discusses arithmetical demonstrations, set out in this way without any word of the ordinary language, and moving forwards by definite rules, which correspond to the laws of logic. He views these demonstrations from outside as a pure play of symbols and discusses the properties of such assemblages of symbols. In this work mathematics becomes, as it were, self-conscious, turning its weapons on itself. The goal which Hilbert had first in mind was a proof of the self-consistency of analysis ...

**8.4. Review by: Peter Gabriel Bergmann.**

*The American Mathematical Monthly*

**71**(3) (1964), 338.

This book was published as Methods of Mathematical Physics, Vol. II: Partial Differential Equations (1962).

This work is not a translation of the German book by the same name that appeared first in 1937 and which was reprinted in this county [USA] in 1943. Rather, it represents a new edition, prepared by Professor Courant during the past two decades, and available in English as the original language. ... the "Courant-Hilbert" is not primarily a textbook to be studied cover-to-cover, nor is it a handbook in which to look up a forgotten numerical constant; a very detailed table of contents, a good index, and an extensive bibliography, which includes both research papers and conference proceedings, as well as a few monographs, help the reader to discover any particular item within the context in which he might study it and understand the underlying grand design. All in all, it is most fortunate that this work is now available and that it incorporates the results of many papers of very recent date.

**8.5. Review by: Geoffrey Thomas Kneebone.**

*The Journal of Symbolic Logic*

**39**(2) (1974), 357.

This is from a review of the 2nd edition of 1970.

This new edition of the second volume of Grundlagen der Mathematik rounds off in a most satisfactory way the definitive account of the contributions of Hilbert's school to our understanding of the mathematics of formalized arithmetic. As Bernays himself acknowledges, the work can no longer claim to give a comprehensive survey of present knowledge in the whole field of foundations of mathematics; but it nevertheless does full justice to a large and very important body of research, [It consists of] Bernays's masterly account of the foundational research that was inspired by Hilbert.

**9. Hilbert's invariant theory papers (1978), by David Hilbert.**

**9.1. Review by: Claudio Procesi.**

*Mathematical Reviews*, MR0512034

**(80e:01035)**.

The four papers included in this translation are the most important written by Hilbert on invariant theory. These papers are of particular interest for two reasons: they represent one of the crucial steps of the birth of modern commutative algebra, as well as a very close link with the vast research program on the algebra of binary forms. Since the whole mathematical community is now rediscovering the power and richness of the mathematics of the last century, it is of particular importance to understand again an aspect of algebra, invariant theory, which has been unjustly neglected although its deep influence on today's mathematics is extremely great (but hardly observed). One should point out that these papers contain ideas which have not yet been fully developed and could even now be the basis of rather interesting research.

**10. Natur und mathematisches Erkennen (1992), by David Hilbert.**

**10.1. Review by:**

**Skuli Sigurdsson.**

*Isis*

**84**(3) (1993), 600-602.

David Hilbert (1862-1943) dominated the mathematical scene at the turn of the century. He was the main defender of the status quo in the Grundlagenkrise that culminated shortly after World War I. The problems he studied and the exacting standards he set are constitutive of modern mathematics. Hilbert was also deeply committed to physics and meta-physics-this was a hallmark of the Gottingen Denkstil - and he studied a variety of problems in mathematical physics from the early 1900s onward. Although his research on the foundations of physics received mixed reviews from his contemporaries, he was nominated for the Nobel Prize in physics every year from 1929 to 1933. ... The lectures were intended to be a preparation for a theory of knowledge. The first three question and refute customary views concerning mathematics: the role of intuition and experience, the role of presuppositions, and the noninfallibility (Nichtuntrüglichkeit) of mathematical conclusions are themes. The next group of lectures correct common views concerning physics: here the main issues are the formation of physical concepts, the laws of physics and eternal laws of nature, and the relationship between theory and experiment. The final group of lectures address philosophical questions: physical lawfulness and causality, natural events and probability, and the role of ideal structures. The lectures form, on the one hand, a sequel to Hilbert's address to the International Congress of Mathematicians in Paris in 1900. There he had taken stock of the field and put his stamp of approval on promising avenues for further research, including the axiomatization of physics. In the succeeding two de-cades, frequently with the help of his assistants, Hilbert applied his recently developed theory of integral equations to physical problems (e.g., the kinetic theory of gases) and discovered, simultaneously with Albert Einstein, the field equations of the general theory of relativity. Hilbert's lectures are thus partly a report on his progress in physics in the intervening years.

**10.2. Review by: Ivor Grattan-Guinness.**

*Mathematical Reviews*, MR1316394

**(95j:00003)**.

The manuscript upon which this text is based is a typescript which had been prepared at the time by Hilbert's assistant Paul Bernays and includes Hilbert's own annotations. His lectures were divided into three parts, covering intuition and assumptions in mathematics and its proofs; development of physical theories and laws, and their relationship to experiment; and physical lawfulness and causality, with related questions including probability theory.

**11. Theory of algebraic invariants (1993), by David Hilbert.**

**11.1. Review by: Luchezar L Avramov.**

*Mathematical Reviews*, MR1266168

**(97j:01049).**

B Sturmfels, the editor, writes, "Around the turn of the century, the University of Göttingen was a Mecca for mathematicians and students from around the world, including the United States. The visitors took back with them a large number of handwritten lecture notes. The present notes of Hilbert's 1897 course on invariant theory comprise 527 handwritten pages, taken by Hilbert's student Sophus Marxsen." ... The course lives up to the promise made at the very beginning: "The necessary prerequisite for an understanding of the following is a knowledge of differentiation and of the basis theorems from the theory of determinants." ... Two things make reading this book a particularly enjoyable experience. This is probably the closest one could get today to actually attending a course by Hilbert. And, as Sturmfels notes, "It is this bridge from nineteenth-century mathematics into twentieth-century mathematics which makes these course notes so special and distinguishes them from other treatments of invariant theory."

**12. The theory of algebraic number fields (1998), by David Hilbert.**

**12.1. Review by: Tauno Metsänkylä.**

*Mathematical Reviews*, MR1646901

**(99j:01027).**

The present edition will surely be welcomed by the vast majority of mathematicians preferring English as their scientific language. What makes this book so important is of course its weighty impact on the number theory of our century. Yet the book also contains some material of immediate interest to present-day students and researchers. In the introduction to the English edition, Lemmermeyer and Schappacher discuss the reception and criticism of the "Zahlbericht" by the mathematical community over the years. They also compare in a stimulating way Hilbert's treatment of Kummer theory with Kummer's original approach, and comment on some details of the text, notably those representing Hilbert's own contribution to the subject.