 The infinite has been of central concern to Western thought since the very first pre-Socratic fragment. It concerned the philosopher Anaximander (who flourished in the 6th century BCE), who identified the principle and origin of existing things as to apeiron. In Anaximander, the principle has both an ontological and an ethical significance. The Pythagoreans (6th century BCE) saw the infinite negatively and emphasized the lack of definiteness associated with it; they also gave it spatial connotations. Indeed, in the 5th century BCE the Pythagorean Archytas of Tarentum (see Huffman 2005, 540–550) gave the following argument for the spatial infinitude of the cosmos based on the contradiction that postulating a boundary to it would seem to entail. If the cosmos is bounded, then one could extend one’s hand or a stick beyond its boundary to find either empty space or matter. And this would be part of the world, which thus cannot be bounded on pain of contradiction. So the world is unbounded. Archytas identified this with the world being infinite. Kant similarly identified the unbounded and the infinite in his cosmological antinomy. In  Section 8  we will see that these notions should be distinguished, but a mathematically precise articulation of the distinction had to wait until the development of new conceptions of space in the 19th century.