To Infinity and Beyond

BUT IF THE AIR IS PURER AT HIGHER ALTITUDES, it is also thinner. As Cantor ascended into this rarefied realm, he began to suffer a series of mental breakdowns. In the touching biography, The Mystery of the Aleph, Amir Aczel recounts how Cantor was in and out of sanitariums, landing with increasing frequency in his university's nervenklinik. It is still a matter of debate whether it was the mathematics that drove him mad or the madness that drove him to pursue the math. Certainly he had plenty of colleagues who thought the very concept of transfinite numbers appalling. Leopold Kronecker, the most powerful mathematician in Germany at the time (and Germany was then the leading mathematical nation), repeatedly tried to prevent the publication of Cantor's papers, condemning him to a life of provincial isolation.

Cantor took this rejection as a sign of conspiracy. As with John Nash, the great brain began to wander. He started to concern himself with Theosophy, Freemasonry and the Rosicrucians; he attempted to prove that Shakespeare's plays had been written by Francis Bacon. The nervenklinik became a second home. "When you look long into an abyss," Nietzsche tells us, "the abyss also looks into you." Few men of any stripe have stared as long as Cantor.

During his life, Cantor remained a marginal force, but since his death his stature as a mathematician has, like his beloved transfinites, only continued to swell. Today he is enshrined in the pantheon of mathematical giants. What is more, the study of transfinite numbers has itself exceeded all limits. Harvard mathematician Robert Kaplan is co-author of a forthcoming book, The Art of the Infinite, with his linguist wife, Ellen. Kaplan notes that today the transfinites are the center of an active research program which has led "to the discovery of numbers that dwarf even Cantor's remotest dreams." These peculiar "numbers," it must be said, bear little relationship to the common conception of that term; we are here in a realm of such arcane abstraction that some mathematicians themselves continue to question the validity of these entities.

The classes of infinities now under study sound indeed like refugees from Lewis Carroll's Wonderland remade by the Monty Python team: There is the First Inaccessible Cardinal and, after it, the Hyperinaccessibles. Then comes the first Mahlo Cardinal, which, as the Kaplans wryly remark, "sounds more like a medieval grandee than any kind of number." There are Indescribable Cardinals as well as Huge, Supercompact and Rowbottom Cardinals, to say nothing of the Extendible and Ineffable Cardinals. Last but not least are the Inexpressible Cardinals. "Devising them isn't only a game of one-upmanship on a gigantic scale," the Kaplans write, "but a serious attempt to prove important theories which are unprovable."

From Cantor's work inevitably followed the revolutionary theories of Kurt Goedel, which have changed forever our understanding of the foundation of rational thought. As Goedel demonstrated, every logical system inevitably contains paradoxes, propositions that can be proved both true and false. At the start of the 19th century, John Playfair noted the ineluctably creative dimension of modern science, an activity whose primary virtue is not necessarily empirical. By following the lead of rational conjecture, Playfair mused, we become aware "how much further reason may sometimes go than the imagination may dare to follow."