The Princeton companion to mathematics.

*(English)*Zbl 1242.00016
Princeton, NJ: Princeton University Press (ISBN 978-0-691-11880-2/hbk; 978-1-400-83039-8/ebook). xx, 1034 p. (2008).

This is a wonderful book! To convince yourself of the correctness of this statement, it suffices to read Part I (called with much understatement “Introduction”) which explains on 76 pages many basic ideas and concepts of mathematics. Of course, most professional mathematicians know all this stuff, but you hardly find it anywhere written in such a clear and suggestive manner, with a minimum of formality and a maximum of information. The best way to get a detailed idea of the intention and the contents of the book is probably to resume the main editor’s own description in the Preface, with some additional remarks on contents and style.

“Part II is a collection of essays of a historical nature. Its aim is to explain how the distinctive style of modern mathematics came into being. What, broadly speaking, are the main differences between the way mathematicians think about their subject now and the way they thought about it 200 years ago (or more)? One is that there is a universally accepted standard for what counts as a proof. Closely related to this is the fact that mathematical analysis (calculus and its later extensions and developments) has been put on a rigorous footing. Other notable features are the extension of the concept of number, the abstract nature of algebra, and the fact that most modern geometers study non-Euclidean geometry rather than the more familiar triangles, circles, parallel lines, and the like.

“Part III consists of fairly short articles, each one dealing with an important mathematical concept that has not appeared in Part I. The intention is that this part of the book will be a very good place to look if there is a concept the reader does not know about but has often heard mentioned. If another mathematician, perhaps a colloquium speaker, assumes that you are familiar with a definition, and if you are too embarrassed to admit that in fact you are not, then you now have the alternative of looking these concepts up in this Companion.

“The articles in Part III would not be much use if all they gave was formal definitions: to understand a concept one wants to know what it means intuitively, why it is important, and why it was first introduced. Above all, if it is a fairly general concept, then one wants to know some good examples – ones that are not too simple and not too complicated. Indeed, it may well be that providing and discussing a well-chosen example is all that such an article needs to do, since a good example is much easier to understand than a general definition, and more experienced readers will be able to work out a general definition by abstracting the important properties from the example.

“Part III, called “Mathematical Concepts”, covers the following 99 topics: The axiom of choice; the axiom of determinacy; Bayesian analysis; Braid groups; buildings; Calabi-Yau manifolds; cardinals; categories; compactness and compactification; computational complexity classes; countable and uncountable sets; \(C^*\)-algebras; curvature; design; determinants; differential forms and integration; dimension; distributions; duality; dynamical systems and chaos; elliptic curves; the Euclidean alagorithm and continued fractions; the Euler and Navier-Stokes equations; the exponential and logarithmic functions; the fast Fourier transform; the Fourier transform; Fuchsian groups; function spaces; Galois groups; the Gamma function; generating functions; genus; graphs; Hamiltonians; the heat equation; Hilbert spaces; homology and cohomology; homotopy groups; the ideal class group; irrational and transcendental numbers; the Ising model; Jordan normal form; knot polynomials; \(K\)-theory; the Leech lattice; \(L\)-functions; Lie theory; linear and nonlinear waves and solitons; linear operators and their properties; local and global in number theory; the Mandelbrot set; manifolds; matroids; measures; metric spaces; models of set theory; modular arithmetic; moduli spaces; the monster group; normed spaces and Banach spaces; number fields; optimization and Lagrange multipliers; orbifolds; ordinals; the Peano axioms; permutation groups; phase transition; \(\pi\); probability distributions; projective space; quadratic forms; quantum computation; quantum groups; quaternions, octonions, and normed division algebras; representations; Ricci flow; Riemann surfaces; the Riemann zeta function; rings, ideals, and modules; schemes; the Schrödinger equation; the simplex algorithm; special functions; the spectrum; spherical harmonics; symplectic manifolds; tensor products; topological spaces; transforms; trigonometric functions; universal covers; variantional methods; varieties; vector bundles; von Neumann algebras; wavelets; the Zermelo-Fraenkel axioms.

“Another use of Part III is to provide backup for Part IV, which is the heart of the book. Part IV consists of twenty-six articles, considerably longer than those of Part III, about different areas of mathematics which cover the following 26 branches: Algebraic numbers; analytic number theory; computational number theory; algebraic geometry; arithmetic geometry; algebraic topology; differential topology; moduli spaces; representation theory; geometric and combinatorial group theory; harmonic analysis; partial differential equations; general relativity and the Einstein equations; dynamics; operator algebras; mirror symmetry; vertex operator algebras; enumerative and algebraic combinatorics; extremal and probabilistic combinatorics; computational complexity; numerical analysis; set theory; logic and model theory; stochastic processes; high-dimensional geometry and its probabilistic analogues.

“A typical Part IV article explains some of the central ideas and important results of the area it treats, and does so as informally as possible, subject to the constraint that it should not be too vague to be informative. The original hope was for these articles to be “bedtime reading”, that is, clear and elementary enough that one could read and understand them without continually stopping to think. For that reason, the authors were chosen with two priorities in mind, of equal importance: expertise and expository skill. But mathematics is not an easy subject, and in the end the authors had to regard the complete accessibility they originally hoped for as an ideal that they would strive toward, even if it was not achieved in every last subsection of every article.

“Many Part IV articles contain excellent descriptions of mathematical concepts that would otherwise have had articles devoted to them in Part III. The authors originally planned to avoid duplication completely, and instead to include cross-references to these descriptions in Part III. However, this risked irritating the reader, so they decided on the following compromise. Where a concept is adequately explained elsewhere, Part III does not have a full article, but it does have a short description together with a cross-reference. This way, if the reader wants to look a concept up quickly, he can use Part III, and only if he needs more detail he will be forced to follow the cross-reference to another part of the book.

“Part V is a complement to Part III. Again, it consists of short articles on important mathematical topics, but now these topics are the theorems and open problems of mathematics rather than the basic objects and tools of study. As with the book as a whole, the choice of entries in Part V is necessarily far from comprehensive and has been made with a number of criteria in mind. The most obvious one is mathematical importance, but some entries were chosen because it is possible to discuss them in an entertaining and accessible way, others because they have some unusual feature (an example is the well-known “four-color theorem” which is discussed in V.12, though this might well have been included anyway), some because the authors of closely related Part IV articles felt that certain theorems should be discussed separately, and some because authors of several other articles wanted to assume them as background knowledge. As with Part III, some of the entries in Part V are not full articles but short accounts with cross-references to other articles.

“Part VI is another historical section, about famous mathematicians. It consists of short articles, and the aim of each article is to give very basic biographical information (such as nationality and date of birth) together with an explanation of why the mathematician in question is famous. Initially, we planned to include living mathematicians, but in the end we came to the conclusion that it would be almost impossible to make a satisfactory selection of mathematicians working today, so we decided to restrict ourselves to mathematicians who had died, and moreover to mathematicians who were principally known for work carried out before 1950. Later mathematicians do of course feature in the book, since they are mentioned in other articles. They do not have their own entries, but one can get some idea of their achievements by looking them up in the index.

“After six parts mainly about pure mathematics and its history, Part VII finally demonstrates the great external impact that mathematics has had, both practically and intellectually. It consists of longer articles, some written by mathematicians with interdisciplinary interests and others by experts from other disciplines who make considerable use of mathematics.

The final part of the book contains general reflections about the nature of mathematics and mathematical life. The articles in this part are on the whole more accessible than the longer articles earlier in the book, so even though Part VIII is the final part, some readers may wish to make it one of the first parts they look at.

“The order of the articles within the parts is alphabetical in Parts III and V and chronological in Part VI. The decision to organize the articles about mathematicians in order of their dates of birth was carefully considered, and the authors made it for several reasons: it would encourage the reader to get a sense of the history of the subject by reading the part right through rather than just looking at individual articles; it would make it much clearer which mathematicians were contemporaries or near contemporaries; and after the slight inconvenience of looking up a mathematician by guessing his date of birth relative to those of other mathematicians, the reader would learn something small but valuable.

“In the other parts, some attempt has been made to arrange the articles thematically. This applies in particular to Part IV, where the ordering attempts to follow two basic principles: first, that articles about closely related branches of mathematics should be close to each other in the book; and second, that if it makes obvious sense to read article A before article B, then article A should come before article B in the book. This is easier said than done, since some branches are hard to classify: for instance, should arithmetic geometry count as algebra, geometry, or number theory? A case could be made for any of the three and it is artificial to decide on just one. So the ordering in Part IV should not be taken as a classification scheme, but just as the best linear ordering we could think of.

“As for the order of the parts themselves, the aim has been to make it the most natural one from a pedagogical point of view and to give the book some sense of direction. Parts I and II are obviously introductory, in different ways. Part III comes before Part IV because in order to understand an area of mathematics one tends to start by grappling with new definitions. But Part IV comes before Part V because in order to appreciate a theorem it is a good idea to know how it fits into an area of mathematics. Part VI is placed after Parts III-V because one can better appreciate the contribution of a famous mathematician after knowing some mathematics. Part VII is near the end for a similar reason: to understand the influence of mathematics, one should understand mathematics first. And the reflections of Part VIII are a sort of epilogue, and therefore an appropriate way for the book to sign off.”

From this detailed description by the editor it should be clear what the intention of the book is, and how this is realized. Summarizing, it is not exaggerated to say that the outcome is really unique. The book offers an overwhelming wealth of ideas, results, methods, and applications of virtually all important branches of mathematical sciences, and it is written in an extremely clear and suggestive way. Even when some of the articles are tough going, by the very nature of the topics they cover, they discuss what they discuss in a clearer and less formal way than a typical textbook, throughout with remarkable success. In fact, the contributors have achieved this by looking at illuminating examples, which they sometimes follow with more general theory and sometimes leave to speak for themselves. This is probably the most important and commendable characteristic of the Companion.

As a result, this impressive work is, to the reviewer’s best knowledge and experience, completely different from, and much better than, any encyclopedic work on mathematics and mathematicians published before. In fact, it is a wonderful book!

“Part II is a collection of essays of a historical nature. Its aim is to explain how the distinctive style of modern mathematics came into being. What, broadly speaking, are the main differences between the way mathematicians think about their subject now and the way they thought about it 200 years ago (or more)? One is that there is a universally accepted standard for what counts as a proof. Closely related to this is the fact that mathematical analysis (calculus and its later extensions and developments) has been put on a rigorous footing. Other notable features are the extension of the concept of number, the abstract nature of algebra, and the fact that most modern geometers study non-Euclidean geometry rather than the more familiar triangles, circles, parallel lines, and the like.

“Part III consists of fairly short articles, each one dealing with an important mathematical concept that has not appeared in Part I. The intention is that this part of the book will be a very good place to look if there is a concept the reader does not know about but has often heard mentioned. If another mathematician, perhaps a colloquium speaker, assumes that you are familiar with a definition, and if you are too embarrassed to admit that in fact you are not, then you now have the alternative of looking these concepts up in this Companion.

“The articles in Part III would not be much use if all they gave was formal definitions: to understand a concept one wants to know what it means intuitively, why it is important, and why it was first introduced. Above all, if it is a fairly general concept, then one wants to know some good examples – ones that are not too simple and not too complicated. Indeed, it may well be that providing and discussing a well-chosen example is all that such an article needs to do, since a good example is much easier to understand than a general definition, and more experienced readers will be able to work out a general definition by abstracting the important properties from the example.

“Part III, called “Mathematical Concepts”, covers the following 99 topics: The axiom of choice; the axiom of determinacy; Bayesian analysis; Braid groups; buildings; Calabi-Yau manifolds; cardinals; categories; compactness and compactification; computational complexity classes; countable and uncountable sets; \(C^*\)-algebras; curvature; design; determinants; differential forms and integration; dimension; distributions; duality; dynamical systems and chaos; elliptic curves; the Euclidean alagorithm and continued fractions; the Euler and Navier-Stokes equations; the exponential and logarithmic functions; the fast Fourier transform; the Fourier transform; Fuchsian groups; function spaces; Galois groups; the Gamma function; generating functions; genus; graphs; Hamiltonians; the heat equation; Hilbert spaces; homology and cohomology; homotopy groups; the ideal class group; irrational and transcendental numbers; the Ising model; Jordan normal form; knot polynomials; \(K\)-theory; the Leech lattice; \(L\)-functions; Lie theory; linear and nonlinear waves and solitons; linear operators and their properties; local and global in number theory; the Mandelbrot set; manifolds; matroids; measures; metric spaces; models of set theory; modular arithmetic; moduli spaces; the monster group; normed spaces and Banach spaces; number fields; optimization and Lagrange multipliers; orbifolds; ordinals; the Peano axioms; permutation groups; phase transition; \(\pi\); probability distributions; projective space; quadratic forms; quantum computation; quantum groups; quaternions, octonions, and normed division algebras; representations; Ricci flow; Riemann surfaces; the Riemann zeta function; rings, ideals, and modules; schemes; the Schrödinger equation; the simplex algorithm; special functions; the spectrum; spherical harmonics; symplectic manifolds; tensor products; topological spaces; transforms; trigonometric functions; universal covers; variantional methods; varieties; vector bundles; von Neumann algebras; wavelets; the Zermelo-Fraenkel axioms.

“Another use of Part III is to provide backup for Part IV, which is the heart of the book. Part IV consists of twenty-six articles, considerably longer than those of Part III, about different areas of mathematics which cover the following 26 branches: Algebraic numbers; analytic number theory; computational number theory; algebraic geometry; arithmetic geometry; algebraic topology; differential topology; moduli spaces; representation theory; geometric and combinatorial group theory; harmonic analysis; partial differential equations; general relativity and the Einstein equations; dynamics; operator algebras; mirror symmetry; vertex operator algebras; enumerative and algebraic combinatorics; extremal and probabilistic combinatorics; computational complexity; numerical analysis; set theory; logic and model theory; stochastic processes; high-dimensional geometry and its probabilistic analogues.

“A typical Part IV article explains some of the central ideas and important results of the area it treats, and does so as informally as possible, subject to the constraint that it should not be too vague to be informative. The original hope was for these articles to be “bedtime reading”, that is, clear and elementary enough that one could read and understand them without continually stopping to think. For that reason, the authors were chosen with two priorities in mind, of equal importance: expertise and expository skill. But mathematics is not an easy subject, and in the end the authors had to regard the complete accessibility they originally hoped for as an ideal that they would strive toward, even if it was not achieved in every last subsection of every article.

“Many Part IV articles contain excellent descriptions of mathematical concepts that would otherwise have had articles devoted to them in Part III. The authors originally planned to avoid duplication completely, and instead to include cross-references to these descriptions in Part III. However, this risked irritating the reader, so they decided on the following compromise. Where a concept is adequately explained elsewhere, Part III does not have a full article, but it does have a short description together with a cross-reference. This way, if the reader wants to look a concept up quickly, he can use Part III, and only if he needs more detail he will be forced to follow the cross-reference to another part of the book.

“Part V is a complement to Part III. Again, it consists of short articles on important mathematical topics, but now these topics are the theorems and open problems of mathematics rather than the basic objects and tools of study. As with the book as a whole, the choice of entries in Part V is necessarily far from comprehensive and has been made with a number of criteria in mind. The most obvious one is mathematical importance, but some entries were chosen because it is possible to discuss them in an entertaining and accessible way, others because they have some unusual feature (an example is the well-known “four-color theorem” which is discussed in V.12, though this might well have been included anyway), some because the authors of closely related Part IV articles felt that certain theorems should be discussed separately, and some because authors of several other articles wanted to assume them as background knowledge. As with Part III, some of the entries in Part V are not full articles but short accounts with cross-references to other articles.

“Part VI is another historical section, about famous mathematicians. It consists of short articles, and the aim of each article is to give very basic biographical information (such as nationality and date of birth) together with an explanation of why the mathematician in question is famous. Initially, we planned to include living mathematicians, but in the end we came to the conclusion that it would be almost impossible to make a satisfactory selection of mathematicians working today, so we decided to restrict ourselves to mathematicians who had died, and moreover to mathematicians who were principally known for work carried out before 1950. Later mathematicians do of course feature in the book, since they are mentioned in other articles. They do not have their own entries, but one can get some idea of their achievements by looking them up in the index.

“After six parts mainly about pure mathematics and its history, Part VII finally demonstrates the great external impact that mathematics has had, both practically and intellectually. It consists of longer articles, some written by mathematicians with interdisciplinary interests and others by experts from other disciplines who make considerable use of mathematics.

The final part of the book contains general reflections about the nature of mathematics and mathematical life. The articles in this part are on the whole more accessible than the longer articles earlier in the book, so even though Part VIII is the final part, some readers may wish to make it one of the first parts they look at.

“The order of the articles within the parts is alphabetical in Parts III and V and chronological in Part VI. The decision to organize the articles about mathematicians in order of their dates of birth was carefully considered, and the authors made it for several reasons: it would encourage the reader to get a sense of the history of the subject by reading the part right through rather than just looking at individual articles; it would make it much clearer which mathematicians were contemporaries or near contemporaries; and after the slight inconvenience of looking up a mathematician by guessing his date of birth relative to those of other mathematicians, the reader would learn something small but valuable.

“In the other parts, some attempt has been made to arrange the articles thematically. This applies in particular to Part IV, where the ordering attempts to follow two basic principles: first, that articles about closely related branches of mathematics should be close to each other in the book; and second, that if it makes obvious sense to read article A before article B, then article A should come before article B in the book. This is easier said than done, since some branches are hard to classify: for instance, should arithmetic geometry count as algebra, geometry, or number theory? A case could be made for any of the three and it is artificial to decide on just one. So the ordering in Part IV should not be taken as a classification scheme, but just as the best linear ordering we could think of.

“As for the order of the parts themselves, the aim has been to make it the most natural one from a pedagogical point of view and to give the book some sense of direction. Parts I and II are obviously introductory, in different ways. Part III comes before Part IV because in order to understand an area of mathematics one tends to start by grappling with new definitions. But Part IV comes before Part V because in order to appreciate a theorem it is a good idea to know how it fits into an area of mathematics. Part VI is placed after Parts III-V because one can better appreciate the contribution of a famous mathematician after knowing some mathematics. Part VII is near the end for a similar reason: to understand the influence of mathematics, one should understand mathematics first. And the reflections of Part VIII are a sort of epilogue, and therefore an appropriate way for the book to sign off.”

From this detailed description by the editor it should be clear what the intention of the book is, and how this is realized. Summarizing, it is not exaggerated to say that the outcome is really unique. The book offers an overwhelming wealth of ideas, results, methods, and applications of virtually all important branches of mathematical sciences, and it is written in an extremely clear and suggestive way. Even when some of the articles are tough going, by the very nature of the topics they cover, they discuss what they discuss in a clearer and less formal way than a typical textbook, throughout with remarkable success. In fact, the contributors have achieved this by looking at illuminating examples, which they sometimes follow with more general theory and sometimes leave to speak for themselves. This is probably the most important and commendable characteristic of the Companion.

As a result, this impressive work is, to the reviewer’s best knowledge and experience, completely different from, and much better than, any encyclopedic work on mathematics and mathematicians published before. In fact, it is a wonderful book!

Reviewer: Jürgen Appell (Würzburg)