A literary-mathematical puzzle

In 1968, Stanislaw Lem, the great Polish science-fiction writer, published a book entitled Głos Pana. It was translated into English and published under the title His Master’s Voice in 1983. It tells the story of how a message from an alien intelligence is detected accidentally in neutrino radiation reaching earth, and the attempts of a small group of scholars and scientists to decode the message (largely unsuccessful).

The book purports to be a manuscript by Professor Peter E. Hogarth, the leading mathematician of his time, left unfinished at the time of his death. Hogarth makes no pretence of describing the attempt objectively; he is interested in the contrast between the cosmic nature of the message and the very human features of the scholars working on it.

I think that Hogarth is one of the best-realised mathematicians in literature. His views of himself, his colleagues, and his subject are never clichés. For example, he takes the view (shared by many mathematicians) that mathematical ideas are pre-linguistic, and we sometimes have a hard struggle putting them into words. But he says,

Mathematics never reveals man to the degree, never expresses him in the way, that any other field of human endeavour does: the extent of the negation of man’s corporeal self that mathematics achieves cannot be compared with anything.

The paragraph that follows is speculative mathematical philosophy of a high order.

He also says more down-to-earth things that most mathematicians would agree with:

It sounds odd, perhaps, that I intended to learn through lecturing, but this had happened to me more than once before. My thinking always goes best when a link forms between me and an active and critical audience. Also, one can sit and read esoteric works, but for lectures it is imperative to prepare oneself, and this I did, so I cannot say who profited more from them, I or my students.

or:

The creative potential, the capacity to solve problems, changes in a man in ebbs and flows, and over this he has little control. I had learned to apply a kind of test. I would read my own articles, those I considered the best. If I noticed in them lapses, gaps, if I saw that the thing could have been done better, my experiment was successful. If, however, I found myself reading with admiration, that meant I was in trouble.

The book is also curiously self-referential. A couple of the characters are very dismissive of the science-fiction genre:

One day I found [Dr Rappaport] amid large packages from which spilled attractive, glossy paperbacks with mythical covers. He had tried to use, as a “generator of ideas”—for we were running out of them—those works of fantastic literature, that popular genre (especially in the States), called, by a persistent misconception, “science fiction”. He had not read such books before; he was annoyed—indignant, even—expecting variety, finding monotony. “They have everything except fantasy”, he said. Indeed, a mistake. The authors of these pseudo-scientific fairy tales supply the public with what it wants: truisms, clichés, stereotypes, all sufficiently costumed and made “wonderful” so that the reader may sink into a safe state of surprise and at the same time not be jostled out of his philosophy of life.

One small incident concerns me here. When Hogarth arrives at the HMV project, one of the people he finds there is Tihamer Dill, the son of Hogarth’s former teacher. He relates how he showed his early work to Dill Senior, only to be rejected in favour of his fellow student Myers. The subject of Dill’s and Myers’ work was combinatorics:

Myers followed in [Dill’s] footsteps, and I have to admit that he was not bad at combinatorial analysis—a branch, however, that even then I considered to be dried up.

So Hogarth convinced himself that Dill had a personal antipathy towards him:

I do not think I ever finished any larger paper in all my younger work without imagining Dill’s eyes on the manuscript. What effort it cost me to prove that the Dill variable combinatorics was only a rough approximation of an ergodic theorem! Not before or since, I daresay, did I polish a thing so carefully; and it is even possible that the whole concept of groups later called Hogarth groups came out of that quiet, constant passion with which I plowed Dill’s axioms under.

At the time he wrote this, combinatorics was not a recognised branch of mathematics. 1968 was the year I began my doctoral studies in finite group theory. I very much enjoyed using arguments of a graph-theoretic and combinatorial nature, but didn’t realise at the time that these constituted a separate subject (indeed, even now I am not sure that they do). How did Lem know about “combinatorial analysis” then? It was possible that, in Poland, he was closer to Hungary, the centre of the upsurge in combinatorics.

In 1936, Erdős and Turán conjectured that, if a set A of natural numbers has positive upper density (this means that there is a positive ε so that it happens for infinitely many n that a fraction at least ε of the first n natural numbers belongs to A), then A contains arbitrarily long arithmetic progressions. This was a partial quantification of a theorem of van der Waerden, one of the “founding documents of Ramsey theory”, which says that, if the set of natural numbers is partitioned into finitely many parts, then at least one of these parts contains arbitrarily long arithmetic progressions. (Clearly, in a partition into k parts, some part has upper density at least 1/k.)

In a major work in 1956, Klaus Roth proved that such a set must contain 3-term arithmetic progressions. His proof was analytic. Then, in 1975, Endre Szemerédi proved the general conjecture. His proof was a tour de force and introduced combinatorial methods, most notably a special case of what is now called the Szemerédi Regularity Lemma, an indispensible tool in modern graph theory, and closely connected with the theory of graph limits I mentioned in a recent posting.

But a bigger shock came two years later when Hillel Furstenberg gave a new proof, using ergodic theory!

There is no suggestion that the emotional relationship between Furstenberg and Szemerédi is anything like that between Hogarth and Dill in Lem’s novel. Moreover, Furstenberg did not “plow under” Szemerédi’s Regularity Lemma. In fact, it is known now that the two approaches have a lot in common; the combinatorial methods can be translated into ergodic theory and vice versa. There is no doubt at all that each field has nourished the other with new techniques.

But I am left with the question: Did Lem anticipate, to some extent, Furstenberg’s proof of Szemerédi’s theorem?

There is a possible solution which I cannot rule out, and on which I would be grateful for information. I do not have access to the original Polish version of Lem’s novel. It is possible that he updated it with hindsight (the translation was later than Furstenberg’s proof, though the original novel was earlier). The translation is credited to Michael Kandel, but is copyright Stanislaw Lem, so it is reasonable to assume that the author and translator both had a hand in it.

Advertisements

About Peter Cameron

I count all the things that need to be counted.
This entry was posted in books, history and tagged , , , , . Bookmark the permalink.

One Response to A literary-mathematical puzzle

  1. Pingback: Weekly Picks « Mathblogging.org — the Blog

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s