Infinity: Novelist's Math, Physicist's Drama

March 7, 2004 -- Numbers and narratives, statistics and stories. From Rudy Rucker's Spaceland to Apostolos Doxiadis' Uncle Petros and the Goldbach Conjecture, from plays such as Copenhagen, Proof, and Arcadia to many non-standard mathematical expositions, the evidence is building.

There has always been some interplay between mathematics and literature, but the border areas between them appear to be growing. Increasingly, fiction seems to come with a mathematical flavor, mathematical exposition with a narrative verve.

Two recent works on the mathematical notion of infinity illustrate this phenomenon. One is by novelist David Foster Wallace, the author of the exuberant 1,088-page novel Infinite Jest, among other works of fiction. His new book Everything and More: A Compact History of Infinity sketches the history of humanity's attempts to understand infinity. It begins with the Greeks and ends with modern logicians, Georg Cantor in particular. In between are accounts of the attempts by many mathematicians to get a handle on the discombobulating notions of the infinitely big and the infinitesimally small.

The other work on the topic is a play entitled Infinities by English physicist and cosmologist John Barrow. So far performed only in Europe, the play dramatically explores various counterintuitive aspects of infinity, from a scenario devoted to Jose Luis Borges' parable of the Library of Babel to one about the implications of mathematician David Hilbert's Hotel Infinity.

Hotel Infinity

To get a feel for the latter, imagine a scenario in which you arrive at a hotel, hot, sweaty and impatient. Your mood is not improved when the clerk tells you that they have no record of your reservation and that the hotel is full. "There is nothing I can do, I'm afraid," he intones officiously. If you're in an argumentative frame of mind and know some set theory, you might in an equally officious tone inform the clerk that the problem is not that the hotel is full, but rather that it is both full and finite.

You can explain that if the hotel were full but infinite (the above-mentioned Hilbert's Hotel Infinity), there would be something he could do. He could tell the guest in room 1 to move into room 2; the original party in room 2 he could move into room 3, the previous occupant of room 3 he could move into room 4, and so on. In general, the hotel could move the guest in room N into room (N + 1). This action would deprive no party of a room yet would vacate room 1 into which you could now move.

Infinite hotels clearly have strange logical properties, and they don't stop there. Infinite hotels that are full can find room not only for one extra guest, but for infinitely many extra guests. Assume the infinitely many guests show up at Hotel Infinity demanding a room. The clerk explains that the hotel is full, but one of the extra guests suggests the following way to accomodate the newcomers. Move the guest in room 1 into room 2 and the guest in room 2 into room 4. Move the guest in room 3 into room 6, and, in general, the guest in room N into room 2N. All the old guests are now in even-numbered rooms, and the infinitely many new guests can be moved into the odd-numbered rooms.

In a sense this property of infinity has been known since Galileo, who pointed out that there are just as many even numbers as there are whole numbers. Likewise there are just as many whole numbers as there are multiples of 17. The following pairing suggests why this is true: 1 - 17, 2 - 34, 3 - 51, 4 - 68, 5 - 85, 6 - 102, and so on.

Where physicist Barrow's play relies on a sort of abstract drama to get its ideas across, novelist Wallace's book has a more conventional format and covers more mathematical ground. Among many mind-bending oddities, it discusses different orders of infinity; in a quite precise sense the set of all decimal numbers is "more infinite" than the set of all fractions, which is "no more infinite" than the set of all whole numbers. Wallace also discusses Georg Cantor's unprovable Continuum Hypothesis, which deals with these various orders of infinity.

Bridging the Two Cultures

The task Wallace has chosen is heavy going, but he brings to it a refreshingly conversational style as well as a reasonably authoritative command of mathematics. Because the language is smart and inventive, the book provides enough enjoyment to induce the mathematically unsophisticated reader to slog through the many difficult patches along the way.

And this is part of the value of the confluence of these two works and of others like them. Although it's extremely unlikely that a novelist will prove a new theorem and only slightly less improbable that a mathematician will write a great novel, these attempts to span the chasm between the so-called two cultures should be applauded. Mathematical exposition is too important to be left only to mathematicians, and the wide variety of literary forms available should not be off-limits to mathematicians and physicists.

Maybe one day Heartbreak Hotel will be just down the creative block from Hotel Infinity, but right now it's time to check out.

Professor of mathematics at Temple University and winner of the 2003 American Association for the Advancement of Science award for the promotion of public understanding of science, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Plays the Stock Market. His Who’s Counting? column on appears the first weekend of every month.