  It is remarkable that, while many of the responses to Zeno’s  paradoxes, and even some modern formulations of the paradoxes  themselves, depend on advanced mathematical techniques, Zeno’s  original arguments do not themselves appear to have involved any  particularly complicated mathematics. Several of the paradoxes  involve no specifically mathematical notions at all. The Achilles is  perhaps the best example since it employs only very ordinary notions, such as getting to where another has started from. The other extant  arguments for the most part deploy similarly prosaic notions: being  somewhere or being in a place, being in motion, moving past something else, getting halfway there, being of some size, having parts, being  one, being like, being the same, and so on. Where Zeno seems to have  leapt ahead of earlier thinkers is in deploying specifically  quantitative concepts, most notably quantitative concepts of limit  (peras) and the lack of limit (to apeiron). Earlier Greek thinkers had tended to speak of limitedness and unlimitedness  in ways suggesting a qualitative rather than a quantitative notion.  While one might suppose that Zeno’s turn to a more strictly  quantitative conception of limit and limitlessness could have been  inspired by his familiarity with Pythagorean philosophers and  mathematicians in Magna Graecia, we can in fact trace the philosophy  of limiters and unlimiteds only back as far as Philolaus, a  Pythagorean roughly contemporary with Socrates and thus a good deal  younger than Zeno.