Volume 11, No. 8—November 2001  
Table of contents for this issue  
 
HYPOTHESES

A Matter Of Choice
Can a mathematical idea have political import?
By Jim Holt

THE LITERARY THEORIST JULIA KRISTEVA has written that "the functioning of poetic meaning obeys the principles designated by the axiom of choice." Alan Sokal, in his famous send-up of postmodern theory, "Transgressing the Boundaries," suggested that the axiom of choice had something to do with the abortion-rights movement and the feminist agenda for equality. Later, in the book Fashionable Nonsense, Sokal and his co-author, Jean Bricmont, correctively declared that the axiom of choice "has no political implications whatsoever."

What is this much-invoked thing called the axiom of choice? Is it really devoid of political significance, as Sokal and Bricmont claim? Or could it turn out to pack an ideological punch beyond the imagination of even the most wild-eyed Left Bank postmodern theorist?

To understand what the axiom of choice is, start with this homely example, apparently thought up by Bertrand Russell. Suppose you have an infinite number of pairs of shoes and you want to pick out one shoe from each pair. There is an obvious rule for doing this: Take the left shoe from each pair (or use the right-shoe rule—it doesn't matter). Now suppose you have an infinite number of pairs of socks and you want to select one sock from each pair. Since socks in a pair, unlike shoes, are identical, there is no rule for defining a set that consists of precisely one sock from each pair. The choice for each pair would have to be arbitrary; and since there are infinitely many pairs, that means an infinite number of arbitrary choices. Here is where the axiom of choice comes to the rescue. It allows one to assume the existence of such a "choice set," even though there is no rule for constructing it.

The axiom of choice caused fierce controversy when it was first framed by the German logician Ernst Zermelo in 1904. Anti-choicers scoffed at the idea of a mathematical object existing by fiat. Pro-choicers countered by insisting that the axiom was self-evidently true and that it was crucial for mathematics and for science itself. Zermelo pointed out that many vocal anti-choicers had unwittingly been using the axiom of choice in their work before he explicitly formulated it.

Today, mathematicians are overwhelmingly pro-choice. Without the axiom of choice, much of modern mathematics would simply not exist. Algebraists need it to prove that every vector space has a basis; topologists need it to prove that the product of two compact spaces is compact; set theorists need it to prove that if two infinite numbers aren't equal, one must be bigger than the other. The axiom crops up in a variety of exotically named forms, including the Hausdorff maximal principle and Zorn's lemma. (The latter, as I recall, was borrowed as the moniker for a downtown New York jazz group in the 1980s.)

The axiom of choice is useful, no doubt about it. But is it true? In a sense, this question is unanswerable. In 1938, Kurt Gödel showed that if the axiom of choice is added to set theory, no inconsistency can possibly result. In 1963, Paul Cohen of Stanford proved the converse: If the negation of the axiom of choice is added to set theory, no inconsistency can possibly result. The upshot is that mathematicians are free either to accept or to reject this axiom without fear of falling into absurdity. So, as long as it makes for richer mathematics—and, according to Julia Kristeva, richer poetics—why not be a pro-choicer?

But wait. Subversion lurks. Let us go back to the year 1924. The scene is the Scottish Café, in the city of Lvov (then in Poland, now in Ukraine). Among the logicians and mathematicians who haunt this bohemian spot are Stefan Banach and Alfred Tarski. Together, using the axiom of choice, they come up with a theorem that is literally incredible: It is possible to take a solid sphere, dissect it into a finite number of pieces, and then, without stretching or bending those pieces in any way, reassemble them to form two solid spheres each of which is the same size as the original. Equivalently, it is possible to take a solid sphere the size of an orange, dissect it into a finite number of pieces, and reassemble them to form a solid sphere the size of the sun.

The Banach-Tarski paradox, as this theorem came to be called, certainly appears dangerous. It is a sort of mathematical miracle of the loaves and fishes, one that threatens to abolish scarcity, that linchpin of bourgeois economics, and usher in a postcapitalist utopia rather like the one envisaged by Marx. (Just think of what it would do to the gold market.) And it all hangs on the axiom of choice.

Fortunately (or unfortunately, depending on your ideological predilections), no one could slice up physical matter deftly enough to make good on this theorem, not even the guys behind the lox counter at Zabar's. Matter is composed of discontinuous atoms, so it can't be split into pieces of arbitrarily fine detail, as required by the Banach-Tarski dissection. The late physicist Richard Feynman recalled making just that objection when mathematicians at Princeton bet him that an orange could be re-pieced together into something as big as the sun: "No, you said an orange, so I assumed you meant a real orange," he protested.

The axiom of choice has no role to play in the material dialectic after all. It leaves the world as it is. Oh well. There's always poetics.

JIM HOLT

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