Counting things was the beginning of mathematics. The integers came first: 1, 2, 3, …. Later the need to measure straight lines and flat areas led to the study of geometry in ancient Egypt and India. But mathematics stumbled when it came to curves, spheres, continuous quantities and smooth changes. Early mathematics could not grasp our more fluid world, could not bring its changes and subtleties to life. Mathematics had to learn about change and infinity. It had to enter the labyrinth of the continuum.

Zeno's famous paradoxes may seem to be merely teasing riddles or bewildering games, but they are much, much more than that. They provoked the first great debates over infinity in the European tradition. Two thousand years later, students were still immersed in study of the paradoxes, and one of them, the Englishman Isaac Newton, grew up to create a new kind of mathematics of change: the infinitesimal calculus. Today, the jets we fly in, the bridges we cross and the devices that play our music were all designed using Newton's calculus.

Since space, motion and time are often thought of as continuous and infinite, Zeno's paradoxes were also the first deep enquiry into their structures. Philosophers, however, have tended to study Zeno's paradoxes of motion as if they were primarily about space, motion and time. Plato portrayed Zeno as Parmenides' younger lover, and historians have tended to agree that Zeno's paradoxes were an indirect defence of his friend's strange philosophy.