and other mathematical beasts
For a discipline that is supposed to yield certain knowledge, mathematics is awfully uncertain about what its objects are. The ancient Egyptians took a no-nonsense approach: Their geometry was about stretched ropes and plots of land near the Nile. But when the Greeks got hold of mathematics, they started loading it with metaphysical baggage. The Pythagoreans worshiped numbers as a divine gift and made them the foundation of reality. Plato took this a step further, enshrining mathematical forms like numbers and circles in a kind of eternal, heavenly realm, one that revealed itself to the intellect but not to the senses.
Scratch a working mathematician today and you're likely to find a Platonist underneath. The things described by mathematics--not just circles and numbers but more rococo entities like Galois fields and Gorenstein rings--are widely believed within the profession to exist objectively and eternally like Plato's ideas, transcending the world of space and time. René Thom, the French geometer who became famous in the 1970s as the father of catastrophe theory, has declared that "mathematicians should have the courage of their most profound convictions and thus affirm that mathematical forms indeed have an existence that is independent of the mind considering them." Mathematical abstractions are "like Mount Everest," says Oxford's Roger Penrose. They are "just there."
Other mathematicians are loath to swallow this metaphysical bolus. The distinguished University of Chicago algebraist Saunders MacLane, for one, dismisses the Platonist vision of mathematics as "a glorious illusion." British physicist David Deutsch, in his recent book The Fabric of Reality (Penguin), calls it a "rickety fantasy."
But what is the alternative? Consider, if you will, the Monster. This is an entity--a "group"--that mathematicians discovered in 1980, on their way to wrapping up a bit of unfinished business in abstract algebra. The concept of a group is one of the more abstract (and powerful) in higher mathematics. Speaking crudely, a group is a set of operations--operations on anything at all--that hang together in a nice algebraic way. In the case of the Monster, the operations are ways of shifting things around in higher-dimensional space--in 196,883 dimensions, that is.
The Monster is vast: The number of its elements can be written down roughly as 8 followed by 53 zeros. Where might the beast live, if not in Platonic heaven? Could it really be confined to the heads of a few mathematicians, like a collective neurosis? Could it lurk somewhere in the physical universe, like a black hole? Might it be just a fiction, like Prince Hamlet? A sociocultural construct, like the national debt? An unactualized possibility, like the fortune you could have made had you gone into software ten years ago?
Strange to say, all of these ontological hypotheses have their partisans today. While discord in the mathematical priesthood is nothing new--in the 1920s proponents of various alternatives to Platonism were persecuting one another with all the fury of early Christian heresiarchs--the debate over what mathematics really is has never been more bewildering.
Among those rendered irritable by all the philosophical confusion is Reuben Hersh, a retired mathematician whose book The Mathematical Experience (co-authored with Philip J. Davis), won the National Book Award in 1983. In his just-published What Is Mathematics, Really? (Oxford), Hersh has a great deal of fun saying boo to the goose that honks Platonism. "Forget foundations, forget immaterial, inhuman `reality,'" he exhorts. Mathematics is about "a world of ideas...created by human beings, existing in their shared consciousness." He calls this doctrine humanism. Moreover, he says, most of the thinkers who have shared his understanding of mathematics--Aristotle, d'Alembert, Polányi--have been "lefties" (good), whereas the Platonists and other foundationists have been "righties" (bad). Gottlob Frege died a Nazi, and Willard Quine is an avowed Republican!
One problem with Hersh's humanism: Ideas in the head, whether shared or unshared, do not possess the features we want to ascribe to mathematical objects. The Monster has 8 zillion zillion parts; no amount of Jamesian introspection will reveal a like multiplicity in one's idea of the Monster. Fine, Hersh concludes, then the Monster does not really exist--even though different mathematicians can reason about it and reach the same conclusions as to its attributes, which thereby receive "social validation."
Will mathematicians rally to Hersh's fashionable pragmatism? I hope not, for the deepest and best mathematics always seems to be done by Platonists. Even if Platonism is false, it is probably most effective for working mathematicians to behave as though it were true. Wait, doesn't that make it true from a pragmatic perspective?
It should be borne in mind, though, that too ardent a belief in mathematical Platonism can bring one to a sticky end. In the nineteenth century, Georg Cantor framed modern set theory, discovering in the bargain a never-ending hierarchy of infinite sets--a vision he believed was vouchsafed to him by God. He ended his days in an insane asylum. Arch-Platonist Kurt Gödel claimed to have an extrasensory perception of mathematical entities. He starved himself in 1978 in Princeton (where he was a member of the Institute for Advanced Study), under the paranoid delusion that people were putting poison in his food.
Platonic relationships, it seems, can be dangerous to your mental health.