Naming Infinity

Loren Graham, an American historian of science, and Jean-Michel Kantor, a French mathematician, have collaborated on a recent book, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity (Harvard University Press, 2009).

Graham and Kantor's thesis is that a Russian religious cult, "name worshipping", was partly responsible for advances in set theory made by Russian mathematicians in the early 1900's, whereas the French "rationalistic, Cartesian" approach prevented similar breakthroughs. Indeed, on p. 189 they claim "under the influence of their [Borel, Baire, Lebesgue's] ultra-rationalistic traditions, they lost their nerve". (However, this supposed "loss of nerve" is not supported by any documentation.)

I found it an interesting book, but one that did not successfully support its thesis. After all, placing the responsibility for an historical event on some other single event or belief is highly problematic: consider that there are still people who debate the conclusion that slavery was a primary cause of the American civil war. Why one group solved a mathematical problem, while another failed, is not always so simple to determine. It can be a matter of pure intellectual ability, or a desire to focus on one area rather than other, or something entirely random, like a conversation in a café. Graham and Kantor hedge a bit on p. 100, where they write, "when we emphasize the importance of Name Worshipping to men like Luzin, Egorov, and Florensky, we are not claiming a unique or necessary relationship. We are simply saying that in the cases of these thinkers, a religious heresy being talked about at the time when creative work was being done in set theory played a role in their conceptions. It could have happened another way; but it did not." But if this all they were saying, then so what?

Srinivasan Ramanujan believed that his family's deity, Namagiri, whispered equations in his ear. But it doesn't necessarily follow that without a belief in Namagiri, he would not have created the mathematics he did. Maybe he would have been an even better mathematician had he not been imbued with his Hindu superstitions.

Similarly, it's not clear to me that Russian mathematicians like Egorov, Luzin, Florensky, and others could not have been just as successful -- or even more so -- had they not subscribed to their bizarre Christian cult. It even seems that their cult could induce ridiculous statements like the one the quote from Bely: "When I name an object with a word, I thereby assert its existence." That will certainly be news to unicorns and the set of all sets. Similarly, Luzin wrote that he "consider[ed] the totality of all natural numbers objectively existing" (italics in original), which raises the question, what precisely does it mean for a number to "exist"? This point is not really discussed by the authors. But it seems to me that, for example, that the number ℵ₀ "exists" in exactly the same way that the number 4 "exists".

The book is generally well-presented, although I found a couple of things to quibble about. On page 58, for example, the authors claim that "explicit namings of even one of them [normal numbers in base 10] have been very difficult to obtain". But this is not true. For example, Champernowne's number .123456789101112 ..., which consists of the decimal expansions of the integers concatenated in increasing order, has a very simple proof of normality found by Pillai in 1939. Many, many other examples are known, such as the result of Davenport and Erdős in 1952 that .f(1)f(2)f(3)... is normal if f is any polynomial taking positive integer values at the positive integers. And I was also annoyed by the misspelling "Riemanian surfaces" on p. 113 --- "Riemann" has two n's, not one --- and the failure to put the proper accents on Wacław Sierpiński on p. 118.

Nevertheless, this is a little book worth reading, even if its main thesis is poorly supported. For me, the best part of the book was the history of Soviet mathematics in Chapters 7 and 8. Although I knew the names of most of the Russian mathematicians discussed, I had not explicitly realized the extent of mathematical talent in Moscow in the 1920's. And the mistreatment of Egorov, Luzin, and Florensky under Communism, and the bravery of people like Chebotaryov who stood up for them and suffered as a result, is a cautionary tale worth knowing about.