In the third chapter I show how Zeno takes up and advances Parmenides’s criteria for philosophy by developing the genre of paradoxes. The paradoxes are based on Parmenides’s understanding of the principle of non-contradiction but allow Zeno to start with the position of his opponent in order to show how this position will yield inconsistencies. The mereological problems raised by the plurality paradoxes will be discussed, but the main focus will be on Zeno’s paradoxes of motion which seem to show motion to be self-contradictory. They confront natural philosophers with two kinds of problems, mereological ones and spatio-temporal ones. I argue against viewing the paradoxes as simply awaiting their solutions in modern mathematics to be solved. Instead, I interpret them as questioning the very consistency of the notions of time, space, and motion. In this way Zeno’s paradoxes will be shown to constitute one of the severest attacks on any project of conceptualizing motion. The paradoxes will therefore act as a touchstone for whether natural philosophy can develop in such a way as to meet Parmenides’s challenge.