Robert and Ellen Kaplan are the founders of The Math Circle, a school open to anyone of any age that teaches the enjoyment of mathematics. Their book, The Art of the Infinite: The Pleasures of Mathematics, uses the concept of infinity as a touchstone for understanding mathematical thinking. In the excerpt below, the authors take a moment to explain why the search for infinity is so essential.
I had removed the black earth’s boundary stones:
Once she had been enslaved and now was free.
This is what Solon, the great Athenian lawgiver, wrote some twenty-five hundred years ago. Taking boundaries away, however, can lead from fusion to confusion and so to chaos. We know where we are when our thoughts, like our words, are sharply defined.
The Greeks had a word for the infinite and it was apeiron (òΠεlpov), which literally meant “without boundary” and translates equally well into “indefinite”. Why should they, why should we, so concern ourselves with the endless, when it may only amount to the vague? Anaximander, who lived a hundred and fifty years before Socrates, recognized the foolishness of claiming that one element or another – earth, air, fire, water – was the source of everything else. Rather, he said, the source is the apeiron– as if distinction rose out of indistinction, the way it does in so many creation myths. We think this way still, seeing speciation on a grand scale evolving from the unspecified, and minutely differentiated tissues from stem cells.
The infinite disguised as the indefinite is our only begetter. But in this same guise it is how we imagine the world truly to be: made up ultimately not of separate objects, molecules, atoms, electrons, or quanta, but, past the ever more granular, to be as partless as the ocean, where our little prisms of selves spray up and soon enough submerge. Just as we picture continuity in the material world by rocks between boulders, stones between rocks, pebbles between stones, and sand to fill in the crevices, so we fractions in the spaces between integers – and for fractions “ever smaller” means denominators becoming infinitely large. If the heavens are full; if everything flows; if time is a river: then not only how we began but how we go on is drenched in that ambiguous apeiron.
“Tell me if ever anything was finished,” da Vinci scribbled again and again over his late drawing of tumbling chaos. He tried to give some form to this chaos by representing it as cascades and waves and whirl pools, since their immensity was at least shaped by comprehensible forces. Our hope is to find some structure to the infinite, behind what might be only superficial indefinition: regularities governing infinite ensembles; powers, dominions, and thrones among its blurred degrees.
Mathematics is the art of the infinite because whatever it focuses on with its infinite means discloses limitless depth, structure, and extent.