There’s a pretty thought experiment that’s sometimes attributed to Democritus though it’s actually due to a later popularizer of the atomic hypothesis1 and it goes like this: Suppose we use the world’s sharpest knife to cut a block of cheese in half, leaving two small blocks where before there was one large one. If cheese is made of atoms, some of the atoms end up in one half and the rest end up in the other. But if cheese is a continuous substance, and the block of cheese is analogous to a line segment in Euclidean geometry, then what happens to the points that are exactly lined up with the knife’s edge? Do they get duplicated? Do they vanish? Does the knife somehow slip to one side or the other of those knife’s-edge points? None of these options seems satisfying, but if we’re cutting something that’s truly continuous, what other options do we have?
In some ways the slipping-knife option, in which the symmetry of the block of cheese gets broken, seems the least satisfying to me. It suggests that you can never truly cut something into two equal pieces, not because of the imprecision of human instruments but because of something inherently strange about space – a strangeness that affects even the bare-bones one-dimensional kind of space that Euclid didn’t find interesting enough to say much about. Yet curiously, the third option in the parable, in which the knife must slip either to the left or to the right, received a kind of vindication a century and a half ago when German mathematician Richard Dedekind set out to think deeply about what real numbers are.
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