Infinity: Novelist's Math, Physicist's Drama

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.

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