In a recent edition of his monthly column in The Believer, the author Nick Hornby wrote that, if you wanted to read something that will impress other people, "I would suggest that you can't beat a collection of letters by an author — and if that author is a poet, then so much the better." He goes on to justify this idea by saying that "the implication is clear: you know the poet's work inside out… and you now need something else, something that might help to shed some light on some of the more obscure couplets." I took Hornby's advice to heart, and spent a recent cross-country plane flight reading the collection *John Von Neumann: Selected Letters*, recently co-published by the London Mathematical Society and the American Mathematical Society. I hate to be the bearer of bad news, but if my fellow travelers felt any admiration or awe at the superior intellect that my choice of reading material was supposed to indicate then they sure hid it well. I, on the other hand, walked away from the book with an increased admiration for von Neumann and his work.

If there is any mathematician who needs no introduction, it would probably be von Neumann, an idea reinforced by the fact that he was honored by the US Postal Service with a stamp in 2005. Von Neumann is probably best known either for his work in what would now be considered computer science — where his name would show up on a short list of people who contributed to the birth of the digital computer — or his work as one of the founders of game theory. In addition to those areas of applied mathematics, von Neumann did quite a bit of work in the pure realms of mathematics, where he made significant contributions to the area of operator algebras in general and its applications to quantum mechanics more specifically. The breadth of his work is easily matched by its depth, and in his foreword to the book under review Peter Lax makes a compelling case that von Neumann "should be thought of as a triple Nobel laureate or, possibly, a 3 1/2-fold winner."

Miklós Rédei put together this collection, consisting of selected correspondence that von Neumann wrote during his lifetime. Rédei writes in his introduction that he chose to include those letters in the archives that would "contribute to our understanding of John von Neumann as a scientist — broadly interpreted — and as a public figure" and not to include most of the available letters that were exclusively of a personal nature. That said, many of the letters do give glimpses of his personal life, and some of the best moments in the collection include descriptions of his family vacations and his travels. Another insight into his personal life is given in a very touching introduction to the volume by Marina von Neumann Whitman, the only child of John von Neumann, who writes about her father's legacy and his thoughts about it. The first section of the volume under review consists of the three introductions I have mentioned — by Lax, Rédei, and Whitman — and then a series of photographs from von Neumann's life

The second section of the book consists of a series of short essays introducing von Neumann to a reader who might not be familiar with his story. The first of these is biographical in nature, telling the story of von Neumann from his birth in Budapest in 1903 through his studies in Germany and moving to the United States in 1930 up to his death from cancer (likely contracted during his work on the atomic bomb) in 1958. The next sections give very brief introductions to his work in logic, operator algebras, unbounded operators, quantum mechanics, quantum logic, ergodic theory, computer science and game theory. These essays are concise and well-written, and give as much introduction and background to these areas as one could hope for considering that they total to about 30 pages. While it would be lying to say that they are written for a layperson, they are written well enough as to give a mathematician working in another field a basic understanding of the problems that von Neumann worked on.

After these introductions, the real fun begins and there are over 200 pages of letters that von Neumann wrote to a wide range of mathematicians. The recipients read like a who's who of mathematicians (and scientists more generally) from the 1940s and 1950s — Dirac, Gödel, Kaplansky, Kloosterman, Oppenheimer, Schrödinger, and Veblen are just a handful of the characters who one finds in the pages of this book. Rédei has done an excellent job in annotating the letters, with footnotes describing which articles and books the letters refer to as well as discussing what the original language of the letters were and where the originals were found.

But the real star of the book is the letters themselves, which cover a wide range of topics. There are letters to Gottschalk and Rademacher from the University of Pennsylvania as well as to Bush from Harvard addressing the future of those institutions in general and the mathematics programs in particular. Another series of letters deals with the fact that von Neumann came up with an independent proof of what we call Gödel's second incompleteness theorem, but decided not to publish it when he realized that Gödel himself had a proof. There are letters to the physicist Louis Tuckerman about game theory in general and, in particular, some board games in which one can prove that a player has a winning strategy without giving any kind of construction as to what that strategy might be. And, perhaps most touching, there is a letter to a Swiss woman politely telling her that the physics paper that her son wrote and that she forwarded to von Neumann was not worth publishing or further pursuing.

It is clear from these letters that von Neumann was a visionary — "I would like to mention that the importance of accelerating approximating and computing mathematics by factors like 10,000 lies not only in that one might thereby do in 10,000 times less time problems which one is now doing…but rather in that one will be able to handle problems which are considered completely unassailable at present" he writes to Lewis Strauss of the Navy — and also that he is a very engaging writer.

When compiling a collection of letters, one of the questions an editor faces is how to organize them. Rédei made the decision to organize these letters by recipient, and while this approach seems like the right decision in many cases, there are places where the collection would have benefitted from being organized chronoligically — in particular, when multiple letters to different recipients address the same topics around the same time. Most of the examples of this are mathematical in nature, as von Neumann writes multiple people about the problem he is working on. However, several examples are more personal in nature, and I found one pair of letters particularly amusing: von Neumann wrote to both Paul Dodd, the dean of the University of California, and James Killian, the president of MIT, on February 24, 1956. Both letters are negotiating the details of jobs that von Neumann would like to start at the beginning of 1957 at each of the two schools. It is especially interesting to note that even in this era before cut-and-paste, large paragraphs of these letters are completely identical, something which is likely to be obscured for readers who read them over 100 pages apart.

Many of the letters are political in nature or at least touch on political issues. In the 1930s, von Neumann was very concerned about the situation that would become World War II, and when he wrote to mathematicians in Europe he would often ask about their impression of the political situation with the Nazis, And, in 1935, he wrote to Blaschke that "Although not a German, I am very much indebted to German science and especially to many representatives of mathematics and physics in every part of Germany…nevertheless, I cannot reconcile it with my conscience to remain a member of the German Mathematical Society any longer." As the war situation progressed, von Neumann's involvement grew and eventually led to his work on the atomic bomb, something that shows up in a handful of the letters. One partucularly interesting letter is one written to Saunders Mac Lane in 1948 trying to convince him to work with the Army Committee on Mathematics — "Each one of us has his hesitations in this field, as you do, and I certainly had mine. After all is said and done, I came to the conclusion that this work is well worth doing, and in fact, that we have, as mathematicians, a certain moral responsibility."

I could go on for much longer excerpting passages that I find particularly interesting, as the many pieces of the inflight magazine I tore off for bookmarks would attest, but instead I recommend that you check out a copy of the book for yourself. These letters are put together in a very nice volume, and give quite a bit of insight into one of the great mathematical minds of all time, even if his name might not be as impressive to strangers as Pablo Neruda or Emily Dickinson.

Darren Glass is assistant professor of mathematics at Gettysburg College. He can be reached at [email protected].