# Reviews of Paul Halmos' books

We list some of Paul Halmos' books and give brief extracts from some reviews and some extracts from Prefaces of these books. We have not included

For extracts from reviews of

*Naive set theory*(1960), perhaps his most famous book, since we have devoted a separate page to reviews of this text.For extracts from reviews of

*Naive set theory*see THIS LINK.**Click on a link below to go to that book**- Finite dimensional vector spaces (1942)

- Finite dimensional vector spaces, 2nd Edition (1958)

- Measure theory (1950)

- Introduction to Hilbert space and theory of spectral multiplicity (1951)

- Lectures on ergodic theory (1956)

- Entropy in ergodic theory (1959)

- Algebraic logic (1962)

- Lectures on Boolean algebras (1963)

- A Hilbert space problem book (1967)

- A Hilbert space problem book (2nd edition) (1982)

- Bounded integral operators on $L^{2}$ spaces (1978) with Viakalathur Shankar Sunder

- I Want to Be a Mathematician: An Automathography (1985)

- I Want to Be a Mathematician: An Automathography (2005) [AMA Reprint]

- I have a photographic memory (1987)

- Problems for Mathematicians Young and Old (1991)

- Linear Algebra Problem Book (1995)

- Logic as algebra (1998) with Steven Givant

**1. Finite dimensional vector spaces (1942), by Paul R Halmos.**

**1.1. Review by: E R Lorch.**

*Mathematical Reviews*MR0033869

**(11,504d).**

A purpose of this study is to present n-dimensional transformation theory from the abstract point of view. Normally, due to the existence of a finite basis, elementary matrix theory has an aspect not at all suggesting that the infinite dimensional theory of operators is a natural extension of it. The author exploits as completely as possible the methods and notions of the infinite in his presentation of the finite case; such a program has long been needed.

**1.2. Review by: Mark Kac.**

*Bull. Amer. Math. Soc.*

**49**(5) (1943), 349-350.

In this book the author presents the topics covered usually in an introductory course in algebra (matrices, linear equations, linear transformations, and so on) from the point of view of a modern analyst interested in general vector spaces. The ever-growing interest in Hilbert and more general linear spaces makes the appearance of the book very timely, especially since it furnishes an excellent introduction to the subject certainly within the grasp of a first-year graduate student or even a good senior or junior. The topics are treated in such a manner as to make future generalizations look both natural and suggestive. This sometimes is done at the expense of the shortness of exposition. Some theorems, as the author himself confesses, could be proved in fewer lines. He prefers, however, longer proofs that admit a generalization to shorter ones that do not.

**2. Finite dimensional vector spaces, 2nd Edition (1958), by Paul R Halmos.**

**2.1. From the Preface.**

My purpose in this book is to treat linear transformations on finite- dimensional vector spaces by the methods of more general theories. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications, and to do so in a language that gives away the trade secrets and tells the student what is in the back of the minds of people proving theorems about integral equations and Hilbert spaces. The reader does not, however, have to share my prejudiced motivation. Except for an occasional reference to undergraduate mathematics the book is self-contained and may be read by anyone who is trying to get a feeling for the linear problems usually discussed in courses on matrix theory or "higher" algebra. The algebraic, coordinate-free methods do not lose power and elegance by specialization to a finite number of dimensions, and they are, in my belief, as elementary as the classical coordinatized treatment. I originally intended this book to contain a theorem if and only if an infinite-dimensional generalization of it already exists. The tempting easiness of some essentially finite-dimensional notions and results was, however, irresistible, and in the final result my initial intentions are just barely visible. They are most clearly seen in the emphasis, throughout, on generalizable methods instead of sharpest possible results. The reader may sometimes see some obvious way of shortening the proofs I give. In such cases the chances are that the infinite-dimensional analogue of the shorter proof is either much longer or else non-existent.

**2.2. Review by: J L B Cooper.**

*The Mathematical Gazette*

**44**(348) (1960), 142-143.

The purpose of this book is to give an account of the theory of linear operators and manifolds in finite-dimensional spaces suitable as an introduction to the theory of operators in Hilbert Space. It adopts an approach based on operator theory, makes little use of matrices and determinants, though these are mentioned, and gives some indication of the use of transcendental methods, such as Zorn's lemma. After an account of operators in general finite spaces, including a rather sketchy account of the Jordan canonical form, the scalar product spaces and the decomposition of hermitian and normal operators in these are treated with great thoroughness. Finally convergence of operators, with an elementary ergodic theorem, are studied, and a brief account of Hilbert Space follows. The book is written clearly and carefully, and has numerous examples well chosen to illustrate its point of view. It can be recommended strongly for the student of its subject.

**2.3. Review by: Albert Wilansky.**

*Amer. Math. Monthly*

**66**(6) (1959), 528-529.

Foreign mathematicians are warned not to search in dictionaries for zeroish, zeroness, askable, .... In style, Professor Halmos follows G H Hardy in the role of a "missionary preaching to the cannibals."

**3. Measure theory (1950), by Paul R Halmos.**

**3.1. From the Preface.**

My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of reference for the more advanced mathematician. I have tried to keep to a minimum the amount of new and unusual terminology and notation. In the few places where my nomenclature differs from that in the existing literature of measure theory, I was motivated by an attempt to harmonize with the usage of other parts of mathematics. There are, for instance, sound algebraic reasons (or using the terms "lattice" and "ring" for certain classes of sets - reasons which are more cogent than the similarities that caused Hausdorff to use "ring" and "field." It might appear inconsistent that, in the text, many elementary notions are treated in great detail, while, in the exercises, some quite refined and profound matters (topological spaces, transfinite numbers, Banach spaces, etc.) are assumed to be known. The material is arranged, however, so that when a beginning student comes to an exercise which uses terms not defined in this book he may simply omit it without loss of continuity. The more advanced reader, on the other hand, might be pleased at the interplay between measure theory and other parts of mathematics which it is the purpose of such exercises to exhibit.

**3.2. Review by: J L B Cooper.**

*The Mathematical Gazette*

**35**(312) (1951), 142.

[The book] gives a comprehensive account of those aspects of the theory of measure and integration which are important in general measure spaces and in topological spaces and groups. ... The exposition throughout is masterly. Proofs are clear, precise and elegant. Care has been taken to avoid the indigestion which accompanies the reading of abstract theories, by discussions in each section aimed at giving the reader an intuitive grasp of the subject and an idea where he is going, and by well-chosen sets of examples. These last should not be overlooked by the reader: they contain not only helpful illustrative matter on the main text, but also accounts of further developments of the subject.

**3.3. Review by: Harry M Gehman.**

*Mathematics Magazine*

**26**(3) (1953), 173-174.

[The book] is intended for use as a textbook in graduate courses in Measure Theory. It should also prove useful for reference purposes with its collection of theorems, its well chosen problems, and its penetrating comments on the subjects which it touches. ... Unlike some textbooks, the exercises in Halmos' book are not trivial. They contain well chosen special cases to illustrate the theory, alternate methods of proofs of theorems, and additional definitions and theorems to extend the material of the text.

**3.4. Review by: S Kakutani.**

*Mathematical Reviews*MR0033869

**(11,504d).**

This book presents a unified theory of the general theory of measure and is intended to serve both as a text book for students and as a reference book for advanced mathematicians. This book is written in a very clear style and will make an excellent text book for those graduate students who are already familiar with the theory of Lebesgue integration in a Euclidean space. But the treatment of the subject is rather abstract so that the book is perhaps not to be recommended for beginners. On the other hand, this book contains many new results obtained in the last ten years, and will be a very useful source of reference for research mathematicians.

**3.5. Review by: J C Oxtoby.**

*Bull. Amer. Math. Soc.*

**59**(1) (1953), 89-91.

In this book Professor Halmos presents an account of the modern theory of measure and integration in the generality required for the study of measure in groups. Thus finiteness conditions are imposed only where necessary, and algebraic and topological aspects are appropriately stressed. Although written primarily for the student, the many novel ideas in the book and its store of interesting examples and counter examples have already made it an indispensable reference for the specialist. The clarity of expression and the sprightly style which are characteristic of the author make the exposition a pleasure to follow. ... It seems likely that this book will come to be recognized as one of the few really good text books at its level. It can hardly fail to exert a stimulating influence on the development of measure theory.

**4. Introduction to Hilbert space and theory of spectral multiplicity (1951), by Paul R Halmos.**

**4.1. Review by: J L B Cooper.**

*The Mathematical Gazette*

**36**(317) (1952), 218-219.

The main purpose of this book is to make available in English the theory of the unitary invariants of normal operators in Hilbert Space, and the last of its three chapters is devoted to this subject. The first two contain an excellently set out account of Hilbert Space and of the bounded operators, projectors and bounded hermitian and normal operators on it. The development of the theory of these last operators is made to rest on measure theory: as a result, and by making full use of the latest work on the subject, the author gives a treatment in which the geometry of the space plays a large part. ... The book can be recommended as an introduction to Hilbert Space theory, for a certain class of readers. Its merits are its accuracy, its concise and lively style and its concern to give the reader an idea of the general direction of the arguments. However, it is clearly written for readers with a training in the abstract modes of mathematical reasoning and, particularly, in measure theory. It would not be very suitable for readers interested in physical applications of the theory, partly because these applications involve only separable spaces, but mainly because no mention is made of unbounded operators and consequently of differential operators ... These minor points should not obscure the fact that Professor Halmos has written a stimulating and useful book

**4.2. Review by: B Sz-Nagy.**

*Mathematical Reviews*MR0045309

**(13,563a)**.

The main purpose of this book is to present the so-called multiplicity theory and the theory of unitary equivalence, for arbitrary spectral measures, in separable or not separable Hilbert space. This is developed in chapter III; the preceding parts serve as an introduction.

**4.3. Review by: E R Lorch.**

*Bull. Amer. Math. Soc.*

**58**(3) (1952), 412-415.

There is little doubt that the author of this book enjoyed himself thoroughly during its preparation. Reading the result afforded this reviewer considerable pleasure. In one hundred and nine well-packed pages one finds an exposition which is always fresh, proofs which are sophisticated, and a choice of subject matter which is certainly timely. Some of the vineyard workers will say that P R Halmos has become addicted to the delights of writing expository tracts. Judging from recent results one can only wish him continued indulgence in this attractive vice. The present work may confidently be recommended. However, beginners in the field should be cautioned before they rush off to secure a copy. Unless one is equipped and in training, one should not attempt the expedition. One must not be misled by the title. For this introduction to Hilbert space, one has to be an expert in measure theory. As a matter of fact it is best to have read the author's book on measure theory or its equivalent. ... Most pages exhibit a zest for play as well as work which is refreshing. Indeed, at times one may have a vague apprehension that the author is preparing a prank or baiting a trap; however it seldom turns out to be more than a friendly tweak given with a wink. Such an intimate style, in the present desert of works written with an unexceptionable scientific detachment, is warmly welcome. It is certainly a facet to the general success enjoyed by Halmos, previous books.

**5. Lectures on ergodic theory (1956), by Paul R Halmos.**

**5.1. Review by: Yael N Dowker.**

*Mathematical Reviews*MR0097489

**(20 #3958)**.

This book is the first work on ergodic theory in book form since E Hopf's 'Ergodentheorie' appeared on 1937. Its contents are based on a course of lectures given by the author at the University of Chicago in 1956. The first of these facts makes the book very welcome; more so since the book is written in the pleasant, relaxed and clear style usually associated with the author. The material is organised very well and painlessly presented. A number of remarks, ranging from the serious to the whimsical, add insight and enjoyment to the reading of the book.

**5.2. Review by: Yael N Dowker.**

*Bull. Amer. Math. Soc.*

**65**(4) (1959), 253-254.

The author, in the apology (preface) to the book, asks the reader to regard these notes as "designed to rekindle" interest in the subject. From this point of view and considering the excellent and effortless style of the book it is doubly regretful that the material discussed is so restricted in time and person. There is almost no indication of work done during the last decade, and the reviewer cannot but be disappointed that the reader is left unaware of the recent sparks of interest found by workers in such branches of ergodic theory as, for instance, those related to probability theory, number theory, abstract ergodic theory, dynamical systems in general and geodesic flows in particular.

**6. Entropy in ergodic theory (1959), by Paul R Halmos.**

**6.1. From the Preface.**

Shannon's theory of information appeared on the mathematical scene in 1948; in 1958 Kolmogorov applied the new subject to solve some relatively old problems of ergodic theory. Neither the general theory nor its special application is as well known among mathematicians as they both deserve to be; the reason, probably, is faulty communication. Most extant expositions of information theory are designed to make the subject palatable to non-mathematicians, with the result that they are full of words like "source" and "alphabet". Such words are presumed to be an aid to intuition; for the serious student, however, who is anxious to get at the root of the matter, they are more likely to be confusing than helpful. As for the recent ergodic application of the theory, the communication trouble there is that so far the work of Kolmogorov and his school exists in Doklady abstracts only, in Russian only. The purpose of these notes is to present a stop-gap exposition of some of the general theory and some of its applications. While a few of the proofs may appear slightly different from the corresponding ones in the literature, no claim is made for the novelty of the results. As a prerequisite, some familiarity with the ideas of the general, theory of measure is assumed; Halmos's

*Measure theory*(1950) is an adequate reference.

**7. Algebraic logic (1962), by Paul R Halmos.**

**7.1. From the Introduction.**

It has often happened that a theory designed originally as a tool for the study of a physical problem came subsequently to have purely mathematical interest. When that happens, the theory is usually generalized way beyond the point needed for applications, the generalizations make contact with other theories (frequently in completely unexpected directions) and the subject becomes established as a new part of pure mathematics. The part of pure mathematics so created does not (and need not) pretend to solve the physical problem from which it arises; it must stand or fall on its own merits. Physics is not the only external source of mathematical theories; other disciplines (such as economics and biology) can play a similar role. A recent (and possibly somewhat surprising) addition to the collection of mathematical catalysts is formal logic; the branch of pure mathematics that it has precipitated will here be called

*algebraic logic.*Algebraic logic starts from certain special logical considerations, abstracts from them, places them into a general algebraic context, and,

*via*the generalization, makes contact with other branches of mathematics (such as topology and functional analysis). It cannot be overemphasized that algebraic logic is more algebra than logic. Algebraic logic does not claim to solve any of the vexing foundation problems that sometimes occupy logicians. All that is claimed for it is that it is a part of pure mathematics in which the concepts that constitute the skeleton of modern symbolic logic can be discussed in algebraic language. The discussion serves to illuminate and clarify those concepts and to indicate their connection with ordinary mathematics. Whether the subject as a whole will come to be considered sufficiently interesting and sufficiently deep to occupy a place among pure mathematical theories remains to be seen.

**7.2. Review by: Donald Monk.**

*Amer. Math. Monthly*

**71**(6) (1964), 708-709.

By algebraic logic the author means that branch of general algebra which deals with algebraic structures mirroring in some sense certain formal logics. ... As Halmos indicates, one of the main problems in algebraic logic is to state and prove algebraically various important theorems of logic. ... Halmos' book is highly recommended as an introduction for those who wish to study logic from a purely algebraic point of view.

**7.3. Review by: Leon Henkin.**

*Science, New Series*

**138**(3543) (1962), 886-887.

Hopefully, the fragmentation of science caused by increasing specialization will be counterbalanced by the development of new modes of unification. In the contemporary development of mathematics one such unifying influence is the massive intrusion of algebraic concepts and methods into all mathematical fields. Mathematical logic has been intertwined with algebra from its beginnings, through Boole's discovery that simple laws of logic can be expressed symbolically as algebraic equations. But only in very recent years has the algebraic viewpoint in logic been systematized to the point where an almost complete account of logic can be given in algebraic terms. Alfred Tarski pioneered this enterprise through successive exploration of Boolean, relation, and cylindric algebras. The most detailed contributions are found in the papers of Paul Halmos on polyadic Boolean algebras. In this volume Halmos collects his papers (unchanged) and adds a brief preface, a bibliography, and an index.

**7.4. Review by: The Editors.**

*Mathematical Reviews*MR0131961

**(24 #A1808)**.

The book consists of a collection of 10 papers of the author, together with a bibliography of 13 supplementary items, on algebraic logic.

**8. Lectures on Boolean algebras (1963), by Paul R Halmos.**

**8.1. From the Preface.**

In 1959 I lectured on Boolean algebras at the University of Chicago. A mimeographed version of the notes on which the lectures were based circulated for about two years; this volume contains those notes, corrected and revised. Most of the corrections were suggested by Peter Crawley. To judge by his detailed and precise suggestions, he must have read every word, checked every reference, and weighed every argument, and I am very grateful to him for his help.

**8.2. Review by: A D Wallace.**

*Science, New Series*

**144**(3618) (1964), 531-532.

The literary style is a mature version of the author's earlier presentations and there is a trace of current Gallic hauteur as well as some Bourbakian pontification. ... But this is an entirely minor and subjective opinion, and the book is most highly recommended as a generally elegant and perceptive introduction to the basic facts concerning Boolean algebra available to the very well-trained senior and to the average second-year graduate student.

**8.3. Review by: R Sikorski.**

*Mathematical Reviews*MR0167440

**(29 #4713)**.

This is a very good introduction to the theory of Boolean algebras. It contains a discussion of all fundamental notions like subalgebras, homomorphisms, ideals, filters, complete algebras, √-complete algebras, etc. ... The book contains almost no references to the literature.

**8.4. Review by: R S Pierce.**

*The Journal of Symbolic Logic*

**31**(2) (1966), 253-254.

The theory of Boolean algebras is one of the most attractive parts of mathematics. On the one hand, Boolean algebras arise naturally in such diverse fields as logic, measure theory, topology, and ring theory, so that the study of these objects is motivated by important applications. At the same time, the theory which has been developed constitutes one of the most elegant parts of modern algebra. Finally, the subject still poses many challenging questions, some of which have considerable importance. A graduate student who wishes to study Boolean algebras will find three excellent books to smooth his way ... For an introduction, the book by Halmos is probably the best of these monographs. It offers a quick route to the most attractive parts of the theory ... Like all of the books by Professor Halmos, Lectures on Boolean algebras is written with admirable style and clarity. This work is a welcome addition to the literature of mathematics.

**9. A Hilbert space problem book (1967), by Paul R Halmos.**

**9.1. Review by: C R Putnam.**

*Mathematical Reviews*MR0208368

**(34 #8178).**

The book is divided into three parts: problems, hints and solutions. According to the instructions, if one is unable to solve a problem, even with the hint, he should, at least temporarily, grant it (if it is a statement) and proceed to another. He should be prepared, however, to use the as yet unproved statement in the hope of thereby getting some clues as to its solution. If he does solve a problem he should look at the hint and the solution anyway for perhaps some other variations on the theme. The rules are simple and the advantages of following them to the conscientious and diligent reader are surely obvious and incalculable. Nevertheless, there is the ever lurking temptation to combine each problem-hint-solution into a unified whole, to be read together and at one time. Although this procedure admittedly thwarts, at least partially, the aim of the book, it is certainly not without its own rewards. Thus, a researcher thereby may quickly find a compact presentation of a particular issue, often including some history, references, and enlightening side remarks.

**10. A Hilbert space problem book (2nd edition) (1982), by Paul R Halmos.**

**10.1. Review by: Peter Fillmore and Nigel Higson.**

*Amer. Math. Monthly*

**91**(9) (1984), 592-594.

A recurring theme is the perturbation of an operator by a compact operator, beginning with the Fredholm alternative and a 1909 result of Weyl on the invariance of the limit points of the spectrum under such perturbations. ... Because of the diversity of topics that it includes, single operator theory is ideally suited to a book of the nature of Halmos's (we can imagine a Homotopy Theory Problem Book being less successful). The second edition contains 250 problems as against 199 in the first. It remains an excellent, compact survey of single operator theory, as well as a valuable source for Hilbert space techniques. The form, content, and style of the second edition deserve the same high praise as the first.

**10.2. Review by: Philip Maher.**

*The Mathematical Gazette*

**73**(465) (1989), 259-260.

It might seem somewhat bizarre to be reviewing here a book that was first published in 1967 and one, moreover, whose subject matter is at the upper end of accessibility for the Gazette readership. Yet Halmos's 'A Hilbert space problem book' has never been reviewed in these pages before; the second edition (under review) differs quite a bit from the first; and, of course, anything by Halmos, however technical, promises to be lucid and entertaining. In fact, this book - with its well-nigh unparalleled form - is an object lesson in the communication of mathematics. For it does not follow that boring and so-often unilluminating format of definition, lemma, theorem, proof (a format which seems virtually de-rigeur for the exposition of advanced mathematics). Rather, this book consists of problems; some 250 of them (and their solutions) which cover much of the theory of (single) linear operators on Hilbert space. Of course, I am aware that other books have appeared that approach parts of mathematics in a problem-orientated (as distinct from, so to speak, a subject-orientated) way: one thinks of Burns's 'A pathway to number theory' for example. But Halmos's books is the first to do this at an advanced (rather than at an elementary) level.

**11. Bounded integral operators on $L^{2}$ spaces (1978), by Paul Richard Halmos and Viakalathur Shankar Sunder.**

**11.1. Review by: W A J Luxemburg.**

*Mathematical Reviews*MR0517709

**(80g:47036)**.

The book contains a very valuable amount of information on integral operators. The theory is richly illustrated by examples and counterexamples. Many open problems are discussed. A large number of them have been solved recently by Schep and W Schachermayer, and the solutions will appear soon in print. Also, the history of the subject is well presented. The style of the book is lucid, lively and entertaining, as we have come to expect from the senior author.

**11.2. Review by: Adriaan Zaanen.**

*Bull. Amer. Math. Soc. (N.S.)*

**1**(6) (1979), 953-960.

The Halmos-Sunder book is written in a lively style, as was to be expected. It is a mine of information for any analyst interested in operators, in particular operators on L2

*spaces. The theory is illustrated by examples as well as by counterexamples; open problems are mentioned and sometimes analyzed. A list of bibliographical notes gives information about the history of the subject. The preface ends with the remark that the book contains only a part of a large subject, with only one of several approaches, and with explicit mention of only a few of the many challenging problems that are still open. I agree, but I hope that nevertheless it will be clear from my comments in this review that I believe the present contribution to operator theory by Halmos and Sunder is a valuable one.*

**12. I Want to Be a Mathematician: An Automathography (1985), by Paul R Halmos.**

**12.1. Review by: John Baylis.**

*The Mathematical Gazette*

**70**(453) (1986), 253-255.

P R Halmos has a well-deserved reputation as an expositor of mathematics. One of his favourite techniques is to excavate for the simplicity underlying layers of complexity, to extract it and to display it in a strikingly illuminating way. He also of course has made important contributions to mathematics per se, but on his own self-assessment his research achievements rank only fourth, behind writing, editing and teaching. These abilities as teacher of and doer of mathematics, combined with mild but significant eccentricity has made him a 'name' in mathematics, a name about whom the mathematical public will be sufficiently curious to guarantee interest in his autobiography. ... this is a fascinating addition to recent mathematical culture by one of its makers. The main message I absorbed from it was a set of conditions required for success in mathematics: talent, yes; single-mindedness, almost as obvious; sense of humour, essential when the going gets tough; and love, yes that is the right word - you must love mathematics, and that means all the ingredients, passion, pain and loyalty.

**12.2. Review by: Henry Helson.**

*Mathematical Reviews*MR0789980

**(86m:01059)**.

This autobiography is a frank, personal, witty commentary on mathematicians and mathematics by one of the most influential, and observant, mathematicians of our time. It is much more about the profession of mathematics than about the personal life of its author. ... He makes two main points. The first is the importance of being literate. The ability to speak and write effectively, preferably in more than one language, is essential to effectiveness in all professional activity. And second, a real professional must work in all aspects of the job: research of course, but also teaching in several formats, exposition at several levels, refereeing and editing, departmental chores, and participation in meetings and conferences. His standard is high.

**12.3. Review by: John A Dossey.**

*The Mathematics Teacher*

**79**(6) (1986), 481-482.

This autobiographical sketch details the professional aspects of the career of the distinguished American mathematician Paul Halmos. It gives the reader a delightfully candid view of the evolution of his career of research, teaching, travel, editing, and service from his secondary school days to the present. ... The book is exciting, witty, and well worth the time invested in its study. It communicates what it means to be a mathematician.

**12.4. Review by: Gian-Carlo Rota.**

*Amer. Math. Monthly*

**94**(7) (1987), 700-702.

Every mathematician will rank other mathematicians in linear order according to their past accomplishments, while he rates himself on the promise of his future publications. Unlike most mathematicians, Halmos has taken the unusual step of printing the results of his lifelong ratings. By and large, he is fair to everyone he includes in his lists (from first rate (Hilbert) to fifth rate (almost everybody else)), except towards himself, to whom he is merciless (even in the choice of a title to the book: "I want to be a mathematician," as if there were any question in anybody's mind as to his professional qualifications). ... The leading thread of his exposition, what makes his narration entertaining (rather than just interesting), is mathematical gossip, which is freely allowed to unfold in accordance to its mysterious logic. The reader will be thankful for being spared the nauseating personal details that make most biographies into painful reading experiences ... Whatever does not relate to the world of mathematics is ruthlessly and justly left out (we hardly even learn whether he has a wife and kids).

**13. I Want to Be a Mathematician: An Automathography (2005), by Paul R Halmos. [American Mathematical Association Reprint]**

**13.1. Review by: Fernando Q Gouvea.**

http://www.maa.org/node/105650

Instead, the book is about his life as a mathematician among mathematicians. It shows us a little about how Halmos thinks about mathematics, about what interested and motivated him, and about how he interacted with others. It includes a lot of what might, somewhat uncharitably, be described as "gossip": stories and anecdotes about mathematics, mathematics departments, and mathematicians. In my experience, mathematicians love this sort of thing. Those of my colleagues who have read this book have enjoyed it. My students have liked it much less, partly because they aren't that interested in the world of mathematics, partly because they feel "turned off" by what they describe as Halmos's "arrogance." I think what bothers them is Halmos's bluntness about what counts and what does not count as significant in mathematics. That Halmos's harshness is mostly directed at his own work doesn't change my students' assessment. If all this famous guy can say, after trying for fifty years, is "I want to be a mathematician," they argue, then we students have no chance at all.

**14. I have a photographic memory (1987), by Paul R Halmos.**

**14.1. From the Preface.**

Would you look at some of the snapshots I have taken in the last 40-odd years? If you put a penny into a piggy bank every day for 45 years, you'll end up with something over 16,000 pennies. I have been a snapshot addict for more than 45 years, and I have averaged one snapshot a day. Over a third of the pictures so accumulated have to do with the mathematical world: they are pictures of mathematicians, their spouses, their brothers and sisters and other relatives, their offices, their dogs, and their carillon towers. The pictures were taken at the universities where I worked, and the places where I was a visitor (for a day or for a year), and, as you will see, many of them were taken over food and drink. That's rather natural, if you think about it. It is not easy, and often just not possible, to snap mathematicians at work (in a professional conversation, thinking, lecturing, reading) - it is much easier to catch them at tea or at dinner or in a bar. In any event, the result of my hobby was a collection of approximately 6000 "mathematical" pictures, and when it occurred to me to share them with the world I faced an extremely difficult problem of choice. ... The people included are not necessarily the greatest mathematicians or the best known. If I think a picture is striking, or interesting, or informative, or nostalgic, then it is here, even if the theorems its subject has proved are of less mathematical depth than those of a colleague whose office is two doors down the hall.

**14.2. Review by: Arthur M Hobbs.**

*Mathematical Reviews*MR0934204

**(89f:01067).**

This is a book of considerable merit in several different ways. Beginning with the most frivolous use, it is very nearly the perfect book for a mathematician's coffee table (it only lacks colour). Less trivially, with 604 photographs of (mostly) mathematicians and two photographs of the old and new buildings at Oberwolfach, each picture accompanied by an informative caption, the book is pleasant reading for any mathematician, for many mathematician's spouses, and for any other person interested in mathematics as it is lived. ... as a historical document, this book will be valuable in at least two ways. It gives a fascinating cross-section of mathematical life in the mid-twentieth century, and it provides considerable insight into the personality and interests of Professor Halmos himself.

**15. Problems for Mathematicians Young and Old (1991), by Paul R Halmos.**

**15.1. Review by: Stan Wagon.**

*Amer. Math. Monthly*

**99**(9) (1992), 888-890.

What makes a problem interesting? Its statement should be simple, not requiring excessive explanations, and the solution should be readily understandable by the intended audience. [In] Halmos's book ... his problems ... meet these criteria admirably. ... Halmos has won several writing awards, and the reader won't be disappointed in the prose with which he wraps the problems ...The book lives up to its title, which promises problems for both young and old. But is it the young or the old who are more likely to be impressed by the pretty and elegant elementary problems? I'm not sure. In any event, for the experienced mathematician or beginning graduate student seeking meatier fare the book contains several chapters with problems for the more mature reader.

**15.2. Review by: Lionel Garrison.**

*The Mathematics Teacher*

**85**(7) (1992), 592.

Buy this book. In fact, buy several. Give them to your students and colleagues, and save one for when your first copy wears out. ... [this book], impels the reader to get out a pencil and start doing mathematics. Paul Halmos, one of the pre eminent mathematicians and teachers of our time, here shares his personal treasury of favourite problems. Perhaps a third of these 165 problems are at least comprehensible to able high school students who have studied calculus and probability and who enjoy being challenged. The remainder generally assume the standard undergraduate pro gram in analysis, topology, and abstract algebra.

**15.3. Review by: G A Heuer.**

*Mathematical Reviews*MR1143283

**(92j:00009).**

The 165 problems in this charming book are classified roughly by the mathematical subdiscipline into which they most naturally fit. ... While a good many of the problems are catchy enough to prick the interest of most readers without further help, most are preceded by the kind of skilful discussion for which the author is so well known, to make the problem more appealing and to help make clear why this is exactly the natural question to pose in this context. The title is apt: there is much here for the veteran professional mathematician; there is also a good bit for the budding mathematician who is still a high school pupil.

**16. Linear Algebra Problem Book (1995), by Paul R Halmos.**

**16.1. Review by: Nick Lord.**

*The Mathematical Gazette*

**81**(490) (1997), 168-170.

... it is the quality of Halmos' commentary that made [the book] such a treat to review and which means that I can recommend it without hesitation to all lecturers (who think they can teach linear algebra) and all sufficiently mature students (who think they have learnt it). His quiet emphasis on details, his penetrating insights into what makes an approach to a proof plausible, or the mode of construction of a counter-example clear, and his 'heart-on-sleeve' approach to problem-posing, in explaining why he cast the problems in the form he did, are matchless: you feel you are in the company of a master expositor, striding together through the world of linear algebra while he points out the flora, fauna, topography and the way ahead. [It is] a book bristling with lovely touches ...

**16.2. Review by: Robert Messer.**

*Amer. Math. Monthly*

**105**(6) (1998), 577-579.

For students going on in mathematics, linear algebra serves as a transition to upper-level mathematics courses. In addition to learning the subject matter of linear algebra itself, these students must be fortified with a degree of mathematical maturity in working with axioms and definitions, basic proof techniques, and mathematical terminology and notation. These issues cannot be left to chance; they must be addressed explicitly to prepare students for courses such as abstract algebra and real analysis. As a textbook for a linear algebra course, Paul Halmos's 'Linear Algebra Problem Book' satisfies these criteria. ... The conversational style of writing in this book occasionally lapses into annoying chattiness. A definition can be guessable and an answer conjecturable. A corollary can be unsurprising or minute but enchanting. Within three sentences the zero linear functional has two symbols and goes from most trivial to most important and ends up uninteresting.

**16.3. Review by: Jaroslav Zemánek.**

*Mathematical Reviews*MR1310775

**(96e:15001).**

Understanding simple things such as basic linear algebra does not seem to be an easy task. Indeed, the author offers original insights illuminating the essence of the associative and distributive laws, and the underlying algebraic structures (groups, fields, vector spaces). The core of the book is, of course, the study of linear transformations on finite-dimensional spaces. The problems are intended for the beginner, but some of them may challenge even an expert. ... Needless to say, more emphasis on the history of the subject would be attractive in a book of this type. In brief, the reviewer regrets that the author chose not to go deeper into the subject; he shows a few trees, but makes little attempt to see the forest.

**17. Logic as algebra (1998), by Paul Halmos and Steven Givant.**

**17.1. Review by: Stephen D Comer.**

*The Journal of Symbolic Logic*

**63**(4) (1998), 1604.

This slim book provides an introduction to logic with the goal, as suggested by the title, of demonstrating that logic can be profitably understood from an algebraic viewpoint. Instead of making the point by providing the reader an extensive treatment of algebraic logic, the book takes a modest concrete approach by limiting its consideration to that part of logic most familiar to a general audience, namely the propositional calculus and the monadic predicate calculus. The presentation reflects Halmos's pedagogical style. Topics are introduced first by logical considerations, then the ideas are abstracted, and finally they are placed in an appropriate general algebraic context. The book reads like an essay. This is not to say the presentation is not rigorous. It is. Details are presented when necessary to convey the ideas, but without overwhelming readers. The presentation is crisp and lucid yet informal. It is as if the principles of logic are being explained over a cup of coffee. The book is directed towards a general (mathematically literate) audience with an interest in modern logic. Nevertheless, the prerequisite of "a working knowledge of the basic mathematical notions that are studied in a first course in abstract algebra" should be taken seriously. It is not a textbook (in the usual sense) even though it is based on notes from a course in logic by Paul Halmos.

**17.3. Review by: Graham Hoare.**

*The Mathematical Gazette*

**84**(499) (2000), 172-173.

[The book] is intended 'to show that logic can (and perhaps should) be viewed from an algebraic perspective .... Moreover, the connection between the principal theorems of the subject and well-known theorems in algebra become clearer.' Readers anticipating arguments based on truth tables or diagrams of switching circuits will be disappointed. In compensation they will be entertained by a rich array of algebraic concepts such as prime and maximal ideals, filters, homomorphisms, equivalence classes, kernels, quotient algebras and duality, all in the service of logic. As the authors state, 'propositional logic and monadic predicate calculus - predicate logic with a single quantifier, are the principal topics treated'. ... the whole will serve as a neat, succinct, introduction to logic particularly for readers very much at home with algebraic concepts.

**17.4. Review by: Myra R Lipman.**

*The Mathematics Teacher*

**92**(4) (1999), 371.

The authors of this book have targeted a wide-ranging audience; however, the book cannot be all things to all people. ... Because of the scope and depth of material, this book would be most useful as a classroom reference or supplement. If the book included many more examples and some exercises, it would be outstanding and certainly more helpful to students. Viewing logic from an algebraic perspective is an intriguing concept, and the authors succeed in giving a concise overview of relatively complex material in an intelligible manner.

**17.5. Review by: Natasha Dobrinen.**

*The Bulletin of Symbolic Logic*

**16**(2) (2010), 281-282.

This is an excellent and much-needed comprehensive undergraduate textbook on Boolean algebras. It contains a complete and thorough introduction to the fundamental theory of Boolean algebras. Aimed at undergraduate mathematics students, the book is, in the first authors words, "a substantially revised version of Paul Halmos'

*Lectures on Boolean algebras*." It certainly achieves its stated goal of "steering a middle course between the elementary arithmetic aspects of the subject" and "the deeper mathematical aspects of the theory" of Boolean algebras.

**17.6. Review by: Marcel Guillaume.**

*Mathematical Reviews*MR1612588

**(99m:03001).**

... this booklet is a gem, whose reading the reviewer ... highly recommends, fundamentally because it gives a better designed, direct and concise overview of logic, going beyond the topics explicitly treated, which are in fact reduced just to the minimum needed in order to explain and introduce the key notions, and to explain and prove fundamental theorems in the special cases considered. As to the form, the style is vivid and clear, using simple words, and free of long and complex technicalities. The text is rich in brief comments explaining the ideas behind the reasoning and calculations, and frequently refers to simple examples and to the basic notions of universal algebra.

Last Updated August 2016