Numbers are central to science. They underlie what Galileo Galilei and Isaac Newton called the primary properties of things, the properties that can be measured (John Locke listed these as number, motion and rest, size, figure, and impenetrability). These underlie secondary properties (like colors and musical harmonies and discords), which in turn underlie the tertiary properties, like beauty, which make life worth living.

The centrality of numbers to science indirectly confers on them philosophical significance, but they have also played a direct role in metaphysics. Plato's theory of universals begins from the problem of the One over Many. Behind the superficial diversity of things in the world, it is often the case that there is one thing that many numerically distinct individuals share in common. For instance, when one doubles the length of the string on a lyre or the length of a column of air in a...