The Art of the Infinite: The Pleasures of Mathematics

by Robert M. Kaplan

In the fifth grade he realized to his amazement that the answer to 134 divided by 29 was . “What a tremendous labor-saving device!” he later wrote. “To me, ‘134 divided by 29’ meant a certain tedious chore, while was an object with no implicit work.

Finding the solution will show what multiplication “means”—and the intricacy of finding might make the pleasures of mathematics even more meaningful.

“The major part of every meaningful life is the solution of problems.”

“Mathematicians are like Frenchmen,” Goethe once said; “whatever you say to them they translate into their own language and forthwith it is something entirely different.”

Beauty is truth, truth beauty, and both are mathematics.

When Wittgenstein dismisses mathematics as nothing more than a string of tautologies, the mathematician answers: nothing less!

“Gentlemen,” said Benjamin Peirce to his students at Harvard University one day late in the nineteenth century, “that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore, we know it must be the truth.”

The way of mathematics is always to spiral up its widening tower to greater generality and higher simplicity.

What singles out mathematics is the way that its minute discriminations lead to ever greater generalities, as climbers reach their dazzling vistas by attending to the piton in the cleft.

Interlude The Infinite and the Unknown

How staggeringly far it has come in five thousand years—but for every answer found, a flurry of new questions arises. In the sequence of ratios of what we don’t yet know to what at any moment we do, the terms grow to infinity.

This has been the romance of imagination and the infinite. Like the beloved in tales as old as time, the infinite keeps escaping imagination’s stratagems, drawing it on through intrigues that must any moment surely untangle.

Hilbert’s remark that a mathematical problem should be clear and easy to understand, since complication is abhorrent; should be difficult enough to entice us but not completely inaccessible (“lest it mock our efforts”); and should be significant: “a guidepost on the tortuous path to hidden truths.”

This move is like a modulation in a late Beethoven quartet: inspired, outrageous, transforming. It trumps the original freedom of choice with a freedom of its own.

If you stare too long into an abyss, the abyss will stare back into you. —Nietzsche

To say that Cantor did infinitely more would be an understatement.

Such a clever way of pairing gives us the confidence now to think the unthinkable and face what Galileo shied away from: a more than infinity. For if you look at all the rational numbers Q, or even at just the positive rationals—the set of all these fractions—there are obviously more of them than there are natural numbers, since between any two fractions will lie another, until what was the space from one natural to the next will be crammed to bursting with them.

You will appreciate the exhilaration Cantor must have felt in winning such striking insights as this by going against the authority of the demigods Aristotle, Gauss, and his own contemporary Kronecker, who said that it was illegitimate to think of or deal with completed infinities. For them, as for almost all the world before Cantor, the infinite was potential.

You probably expect that so shattering a conclusion follows from a proof whose subtlety or abstruseness could never be contained in these pages—yet here it is, in a version Cantor came up with later: the most stunning work in the gallery of nineteenth-century art, and built once again on the strut of a slender diagonal.

Cantor had to show that there was no 1–1 correspondence between the sets and —not just that he couldn’t find one. The only logical approach to such negative statements was a proof by contradiction. His strategy would be to assume that a 1–1 correspondence had been made, and then to reduce this assumption, as they dismissively say, ad absurdum.

Now comes the diagonal stroke of genius. That first decimal place in the list’s first entry, a11, must, of course, be one of the digits from 0 to 9: for example, it is either 5, or not. Cantor asks us to create our own decimal number between 0 and 1 as follows. Like those on the list, it too will begin “0.”, but its first decimal place will be determined by what a11 is. If a11 is 5, our number will have a 6; but if a11 isn’t 5, we put a 5 in the first place of ours.

Notice, however, that it can’t possibly be the first entry on the list, since it differs from that entry at least in the first decimal place. It can’t be the second entry either, differing as it does from it in at least the second decimal place; nor the third, for the analogous reason, nor the fourth—nor the nth. It cannot, in fact, be anywhere on this list that was supposed to contain all of the reals in (0,1) because it differs from every entry on it in at least one decimal place—and that is the contradiction.

This proof—as simple and subtle as all great art—throws open the gates to what Hilbert called Cantor’s Paradise.

Now we can return to Galileo’s shorter and longer line-segments. The open interval (0, 1) has more points on it than all the counting numbers in the world, although there is no end of them. What about the longer segment (0, 2)? An astonishingly simple proof—another “Look!”—shows that this longer segment contains just as many points as the first: there is a 1–1 correspondence between them.

“How many” has nothing to do with “how long.”

You need to pick out faces in the crowding natural numbers to jog your imagination into glimpsing just how very big their infinity is.

Does it take the courage of Daedalus or the foolishness of Icarus to ask now: “Is there a greater infinity still than those of the naturals and the reals?” Does the asking imply a sort of imagination in whose presence ours shrivels to a dot? Or has abstraction somehow insulated the mind against the reality it calls up, so that the imagination we rightly praise is one of intuiting relationships and devising ways of rigorously proving that they hold?

Not if seeing has now replaced believing, making you know you are right.

“How many” has nothing to do with “how many dimensions.”

This revelation startled Cantor as much as it does us. He had found two sizes of infinity, and as anyone who indulges in counting expects, where there are two there must be many more.

You see why we said before that Cantor had done infinitely more than make sense of pairing numbers with their squares.

If you pause now to ask how he won his insights, the answer must surely involve a pioneer’s love of freedom more than comfort.

“Mathematics is freedom!” Cantor later proclaimed—and freedom is where “what if” meets “what then.”

The pattern is beautiful in itself and spoke to Cantor in terms of cardinality: what he saw was that the cardinality of any finite set is less than that of the set of its subsets.

That Cantor in fact came up with a proof makes you wonder again about how impersonal the works of mathematics are. If the theorem is out there, is its proof out there too? Could anyone have discovered it, is it part of our common heritage? Or is the proof and what it establishes now a part of our thinking the way the Mona Lisa is—but needn’t have been: necessary after the fact?

Once again Cantor approached his conjecture through a proof by contradiction—and once again he used his diagonal idea, but now etherealized to suit the unearthly remoteness of its subject.

There are infinitely many counting numbers. There are yet more reals. Now there are more subsets of the reals than reals themselves.

Have you fully appreciated what “and so on” must mean? There will be an 4, and an and an

For at the gates that Cantor flung Apart (and Hilbert later) Angelic fleas cavort in hosts Inordinately greater.

The first tremors that Cantor felt came from his colleagues. In France, Poincaré grew disgusted with set theory, which he thought pathological: he wrote that later generations would regard Cantor’s work as a disease from which they had recovered. Closer to home, Kronecker, the most powerful figure in the German mathematical establishment, had been opposed to Cantor’s ideas from the beginning, calling the work humbug and the man himself a charlatan and corrupter of youth.

An intellect focused on the infinite may overlook temporal indignities, but the psyche that intermediates between the intellect and the world cannot, and Cantor wrote that poverty and recrimination were the price he paid for his radical views.

Mathematicians usually enjoy generalizing their observations, but seeing plots everywhere brought Cantor suffering. He alienated some friends, discarded others and had his first serious breakdown when he was thirty-nine.

This Axiom of Choice asserts that given any collection of distinct, non-empty sets, if you need to (as Cantor did), you can always choose an element from each. It doesn’t tell you how to do this, but you can imagine a jackdaw plucking a shiny trinket from each of a possibly infinite row of boxes.

The problem with the Axiom of Choice is that it lets the Four Horsemen loose in the land: it allows an initiate, for example (by an ingenious train of reasoning), to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun. Not only are its results an affront to intuition, but by not requiring us to know how we do the choosing, it adds to the Formalist shift of mathematics sideways.

No longer do we construct objects cleated to their locales, but now rest content (like Hilbert and the Hungarian) with t...

It takes an inhuman force of character to make the beds while your house is falling down. At the same time that Cantor was trying desperately and unsuccessfully to prove the Continuum Hypothesis (if his theory couldn’t even locate the cardinality of the reals in the hierarchy of the alephs, what good was it?), he went about the beginner’s business of learning how to do arithmetic—but this time with transfinite cardinal numbers: the most radical extension of the franchise we have seen.

“We do not arbitrarily give laws to the intellect or to other things, but as faithful scribes we receive and copy them from the revealed voice of Nature.”

No wonder Hilbert so warmly defended Cantor’s work: “No one shall drive us from the Paradise that Cantor has created for us!”

But was Cantor a Formalist? He never presented his results in the formal context of stripped-down deductions from axioms. His mathematics isn’t about symbols that could mean this or that, but about what he saw as real ideas in the divine intellect, and corporeal objects in the world. Completed infinities were, for him, actual, not like the formless and merely potential apeiron of the Greeks. Hilbert purged mathematics of meaning; Cantor flooded his mathematics with metaphysics and theology.

Mathematics,” as Cantor had famously said, “is freedom!” But this motto is as ambiguous as it is bold, since there is freedom from as well as freedom of.