The Modern Atomists

Abstract

In our story of the infinite we have looked mainly at the infinitely large, perhaps because it has received so much attention since Cantor’s pioneering work in the 1880s, perhaps also because there seems to be something about the infinitely large that captures the imagination in a way that the infinitely small cannot. This bias, however, is hardly justified. In the history of mathematics the infinitely small has played a role at least as important as its counterpart on the other extreme of the scale. If nothing else, it lies at the root of the notion of continuity, an idea that goes back all the way to the Greeks, whose philosophers heatedly debated the possibility of endless division. And much later, disguised as the infinitesimal, it would become the cornerstone around which the calculus was developed.¹ In any event, from a purely mathematical point of view the distinction between “large” and “small” is not really as fundamental as it may seem, since we can always use the function y = 1/x (or its two-dimensional equivalent, the transformation of inversion) to change the one into the other.

Where the telescope ends, the microscope begins.

Who is to say of the two, which has the grander view?

—Victor Hugo (1802–1885), Les Misérables