Review of The Man Who Loved Only Numbers

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When Paul Erdős died in 1996 Paul Hoffman had known him for about 10 years and interviewed him a number of times. Hoffman had also known and interviewed many of Erdős’ friends and associates. So it’s fair to say that Hoffman had a lot more knowledge of the subject of his biography than most biographers (unless they’re family or close friends) ever do. The biography is indeed quite good, and provides a clear and informative portrait of the very unique, appealing, and colorful individual that Erdős was.

However, what defined Paul Erdős even more than his idiosyncrasies and his humanity was that he was quintessentially a mathematician – a singularly talented and prolific one. Another biography of Erdős, by PhD physicist Bruce Schechter (My Brain is Open: The Mathematical Journeys of Paul Erdos), was published at almost the same time as Hoffman’s. (I have reviewed that as well, and I’ll try not to duplicate much from that other review.) Hoffman freely concedes that he is not a mathematician. While Schechter isn’t a professional mathematician, he is more able to give a complete and accurate account of Erdős’ work than Hoffman.

The picture of Erdős that comes across in both biographies is much the same, since both books are based on a lot of the same source material. Hoffman’s book has the advantage that the source of all Erdős quotes (except those Hoffman got directly) are documented in footnotes. But because Hoffman also can report many anecdotes from his interviews with Erdős himself and with his associates, this biography offers a somewhat clearer picture of the subject as a person, apart from his work.

One anecdote is especially revealing. On an occasion in the late 1960s Erdős (who was about 55 at the time) was staying with an old friend from Hungary in southern California, very near the beach. One day Erdős went out to walk on an esplanade above the beach. Only ten minutes later his hosts received a phone call, from someone who lived close to the esplanade about five blocks away, to report that Erdős turned up on their doorstep saying he was lost and needed help finding his way back to his friend’s place. So Erdős, in spite of his prodigious memory for details of his mathematics and his mathematical collaborators, couldn’t even remember how to retrace his steps. The natural explanation is that he was so absorbed in his mathematical thoughts that no vestige of his short walk had registered in his memory.

As other anecdotes made clear, Erdős was fully capable of recalling details of mathematical conversations he’d engaged in years before, and he could also keep track of more or less simultaneous conversations he carried on with several different mathematicians at the same meeting. He could also recall details of perhaps thousands of technical papers he’d read decades before. It seems reasonable to conclude that his ability to concentrate and to recall mathematical detail had a great deal to do with his singular power as a mathematician – as exemplified by his ability either to solve quickly new mathematical problems or at least to judge accurately their level of difficulty almost effortlessly.

In contrast to this clear portrait of Erdős as a person, Hoffman’s lack of mathematical background means he must rely on the testimony of others he interviewed to describe Erdős’ mathematics. The result is a somewhat less satisfactory account. One example is what Hoffman calls “friendly numbers”. He says this means a pair of numbers (a and b) where the sum of proper divisors of a is equal to b and the sum of proper divisors of b is equal to a. This is actually the definition of “amicable numbers”. The example given is the pair 220 and 284, which was known to Pythagoras. Those are indeed “amicable numbers”. The accepted definition of “friendly numbers” is numbers that have the same ratio between themselves and their own sum of divisors. By this definition, 220 and 284 aren’t “friendly”.

An even more serious problem is the discussion of Bernhard Riemann’s non-Euclidean geometry. Hoffman writes “He [Riemann] builds a seemingly ridiculous assumption that it’s not possible to draw two lines parallel to each other. His non-Euclidean geometry replaces Euclid’s plane with a bizarre abstraction called curved space.” It’s not actually bizarre at all, since the surface of any sphere is one example. Straight lines on the surface of a sphere are (parts of) “great circles” (which are by definition the largest circles that can be drawn on a sphere, like the Earth’s equator). Great circles are never “parallel”, since they always intersect.

Apart from these and a few other mathematical glitches, Hoffman’s book is almost free of trivial proofreading errors. But there’s one glaring exception (at least in the paperback edition). Sixteen pages of pictures are included in the middle of the book, and given appropriate page numbers. However, the footnotes at the end of the book are all keyed to page numbers, and they haven’t been corrected to account for the picture pages, so that all references to pages after the pictures are off by 16 – a very annoying problem if one wants to actually check these references.

In spite of these problems, Hoffman’s book provides a fine portrait, based on personal experiences, of Erdős the man.

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