Monday, December 06, 2010

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.

9 comments:

Narad said...

Concerning Ramanujan: it is not entirely certain that he really did believe that Namagiri whispered theorems to him in his dreams. G.H. Hardy, for example, gives a rather different account of Ramanujan's religious beliefs. See, for example, the account beginning at the bottom of p. 3 (Lecture I) of the collection "Ramanujan: twelve lectures on subjects suggested by his life and work".

vinayakpathak said...

I don't understand how a good mathematician can ever believe in God. For being a good mathematician you need to be excellent at logical reasoning; to be a believer in God, you need to be a mish-mash of logical fallacies. I can easily see a believer in God, when asked to prove something, first assumes it's true and then "proves" it. I guess if a person is both, then he probably is a believer in a very vague, pantheistic kind of way. And I am pretty sure that if it turns out that having such a belief really does have positive effects on your mathematical abilities, amphetamines have much better effects and they don't even screw up with your belief system.

Jeffrey Shallit said...

Vinayak:

People are very good at compartmentalizing their beliefs, so it doesn't surprise me at all that excellent mathematicians can be theists. And it's a matter of record that they can.

I know a good mathematician who believes in all sorts of silly "alternative" health practices.

Joel Reyes Noche said...

Excellent mathematicians regularly believe things without proof, in particular, axioms and definitions. I would say that this involves some "faith."

Takis Konstantopoulos said...

Joel Reyes Noche said...
Excellent mathematicians regularly believe things without proof, in particular, axioms and definitions. I would say that this involves some "faith."


Forget the adjective "excellent". It's unnecessary.

I'm afraid you confuse the words "faith/believe" and "mathematics/science". If you had worked in the latter you would have quickly realized that neither axioms nor definitions are freebies in mathematics. A lot of serious work goes into them, and this is far from "faith" or "belief".

However, to an untrained person it appears that proofs is the only thing mathematics is about. But it just ain't so.

Joel Reyes Noche said...

@Takis,
I agree with what you're saying, hence the scare quotes around the word "faith." In retrospect, I guess I should also have put scare quotes around the word "believe." Thanks for the clarification.

Mike from Ottawa said...

"It could have happened another way; but it did not." But if this all they were saying, then so what?"

Maybe someone from your U's history department can explain it to you, since the same could be said of most, if not all, history.

Applies as well to large swathes of paleontology and geology as well, or any science where contingency is an element.

Jeffrey Shallit said...

Mike from Ottawa:

Actually, I've published a paper in a history journal, so I think I have some idea what is done in academic history. But thanks for your advice.

Anonymous said...

To end my posting spree on this blog, a personal opinion on mathematics and its practitioners and a pointer to a good book.
The art of mathematics requires honest provable reasoning. Proof of correct reasoning does not require consensus (Galileo) or the misbegotten efforts of certain parties re: Poincare Conjecture. Regarding mathematical taxonomy in general reminds me of the tower of Babel and a few choice quotes from Paracelcus regarding the medical practicioners of his era .. as does a sojourn through the Sapir-Whorf hypothesis. As Einstein said, make it as simple as possible but no simpler. Mathematics may need another Dyson to synthesize equivalent statements from diverse forms of arbitrary expressions. Even canned software like Maple & Mathematica do the same things differently. At least in FORM you can roll your own.
To close, for those who have that special place on their desk to bang their head or who experience jaw-dropping stupidity on a semi-regular basis take a look at "Systemantics" by John Gall. The book may help your perspective on certain `goings-on`. I've had my 2nd edition copy since the early '90's.
Happy holidays everyone and may one of you out there win a Noble or a consolation ig Noble!