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Great Mathematics Books of the Twentieth Century: A Personal Journal

Lizhen Ji
International Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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If ever a book deserved to be described as a “labor of love,” it is this one. Cliché as it is, the expression captures several important things about Lizhen Ji’s project. First, it clearly required an enormous amount of work to collect all the information in these 667+60 pages. If the subtitle “A Personal Journey” means that Ji has actually read all the books he lists, then all I can say is that I am awed and humbled. Second, the phrase recognizes that Ji must have really wanted to do this, and in particular that he loves mathematics books. But third, the phrase implies that the way we know that love was required for this particular labor is that there is something eccentric about the whole project. Love, after all, famously renders people blind to the faults of the beloved.

Ji’s love song to mathematics books has two components whose pages are numbered separately. The first component is a tribute to books from before the twentieth century, mostly by way of images of their title pages. The second and much larger component is a list of good mathematics books from the past century (generously understood). This is like a personal “basic library list” with extensive annotations. Let’s look at each of these in turn.

Ji includes about 60 pages of photographs of title pages from famous older books, many in full color. A bunch of these appear in a kind of insert at the beginning of the book, but others appear at the beginning of the relevant section of the text. Most of the photographs were taken by Ji himself, plumbing the riches of the University of Michigan’s library collection.

It is interesting to look at these images, but caution is needed with their descriptions. Ji is clearly not much interested in bibliographical niceties, and sometimes is just inaccurate. A picture of a printed edition of Archimedes’ works is just labeled “A book by Archimedes.” (It is in fact the Basel 1544 edition of the Greek text with Latin translation, the first European printed edition of Archimedes.) On the same page is a book captioned “Archimedes’ collected works” which is in fact a 1675 book by Isaac Barrow, Archimedis Opera: Apollonii Pergaei Conicorum Libri IIII. Theodosii Sphaerica: Methodo Nova Illustrata, & Succincte Demonstrata. As Wikipedia says this is “an edition with numerous comments of the first four books of the On Conic Sections of Apollonius of Perga, and of the extant works of Archimedes and Theodosius of Bithynia.” So yes, the works of Archimedes are part of what is in the book, but there is more to say than that!

The pictures are included mostly for the reader’s pleasure, and as a hat tip towards older authors, so perhaps the eccentricities don’t matter. Still, it is a bit weird to see Euler’s algebra textbook represented by its English edition, Riemann’s collected works in their French edition, and Kronecker represented by a page from a book in French (by whom?) containing a “Mémoire sur les Systèmes Modulaires de Kronecker.”

Ninety percent of the book consists of the list of books, organized by topic. The organization can be idiosyncratic, but the choice of books is usually what one would expect. It’s easiest to discuss an area with which I’m very familiar, so let’s illustrate by looking at section 7.1, from the chapter on “Number Theory”:

7.1 Number Theory
                7.1.1 Hardy and Wright, An Introduction to the Theory of Numbers
                7.1.2 Books on basic number theory
                7.1.3 Davenport, The Higher Arithmetic
7.1.4 Hasse, Number Theory
                7.1.5 Borevich and Shafarevich, Number Theory
                7.1.6 Khinchin, Three Pearls of Number Theory
                7.1.7 Artin, Galois Theory

This is followed by sections 7.2 “algebraic number theory,” 7.3 “analytic number theory,” 7.4 “transcendental number theory,” 7.5 “arithmetic algebraic geometry,” and 7.6 “modular forms and automorphic representations.” Just looking at section 7.1, some things stand out. First, these are all fairly sophisticated books; they do not include any introductory textbooks. Second, while section 7.1.2 is called “books on basic number theory,” in fact all of these books contain some kind of “basic” number theory, though most of them actually go well beyond that. (Perhaps what is meant in 7.1.2 is other books on basic number theory.) Davenport is probably the most accessible of these books; the most sophisticated might be either Hasse or Borevich-Shafarevich (both do algebraic number theory, for example). Artin’s book is not about number theory at all, but since there is no section on Galois Theory, it’s perhaps not unreasonable to place it here. Some folks might wonder about Serre’s A Course in Arithmetic. For mysterious reasons, it appears under “arithmetic algebraic geometry.”

(For curious readers, here are the books mentioned in section 7.1.2: L. K. Hua, Introduction to Number Theory; M. Rosen, Number Theory in Function Fields; R. Lidl and H. Niederreiter, Finite Fields; K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory; N. Koblitz, p-adic Number and p-adic Analysis; M. Schroeder, Number Theory in Science and Communication.)

For comparison, here are the elementary number theory books the MAA lists as BLL***, i.e., considers “essential for undergraduate libraries”: Rademacher’s Lectures on Elementary Number Theory, Weil’s Number Theory for Beginners, Niven-Zuckerman-Montgomery’s An Introduction to the Theory of Numbers, and Hardy-Wright. I suspect Ji would consider Weil’s little book beneath notice, and think of Niven-Zuckerman-Montgomery as too much of a textbook. Of course, part of the difference is that the MAA targets its list quite specifically to undergraduate libraries, while Ji is definitely focused on much more advanced books. (He is not always consistent about this, however. Fraleigh’s abstract algebra textbook is on the same page as Birkhoff’s Lattice Theory.)

Ji doesn’t give us just a list, of course. Each book gets at least a half page of description. Most of these are structured similarly: a short paragraph by Ji describing the book, followed by one or more quotations from other commenters. These come from many sources of varying reliability: MathSciNet (usually good, but representing an initial reaction to a new publication), reviews from the Bulletin of the AMS (typically more considered), reviews from (anyone’s guess), and descriptions by the author or translator of the book (interested parties). The authors of the reviews are rarely mentioned, so the typical reader will find it hard to know what to make of them.

(Full disclosure: yes, it bugs me that there are no quotes from MAA Reviews, but on the other hand most of the books we have reviewed are not from the twentieth century. The fact that none of my own books is included is less annoying, since I doubt that any of them are “great mathematics books.” And no, the one review I wrote for the Bulletin is not quoted.)

The annotations are mostly a lost opportunity. I would have been very interested to hear what Ji thinks of the books he has included, but his personal assessment is rarely heard. When he does say something, it is usually both brief and bland: “this is a classic”, “this is an important book written by a master”, “this is a self-contained introduction”. As one would expect, the author has more to say (but not a lot more) about books that are closer to his own area of expertise.

The least useful sections are those on expository books and on the history of mathematics. In both cases this is because Ji writes very much as a mathematician. His choice of expository books is almost entirely a function of how much “real mathematics” the book contains. Most of the books he chooses are good, but his assessment of who will enjoy them can be way off. Ji’s discussion of history, unfortunately, comes very close to the stereotype of “when it comes to history, he’s a very good mathematician.”

So we go back to the “labor of love” thing. Will readers find this book useful? One possible audience will be librarians looking of advice on acquisitions. The obvious competition here is the MAA’s list of library recommendations, but there are also books such as Using the Mathematics Literature, a collection of bibliographic surveys produced by experts and put together by a librarian. The disappointing thing about Ji’s recommendations is how “safe” they are: all the “usual suspects” are here, and readers will not find comments indicating that a book is actually not nearly as good as people think. Nor will they find many surprising or newer titles that have not yet become established “classics.”

Another potential audience would be folks like me, who like to know what others think about books I like (or dislike). But they will mostly be disappointed here: Ji’s voice is heard too rarely and is drowned out by the nameless voices from MathSciNet and Too bad: there’s a 200-page book here that would be very interesting and useful.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He is the editor of MAA Reviews. As anyone who has visited his house or his office will confirm, he really likes books.