"The Greek search for essences had led the Pythagoreans to picture the universe as a multitude of mathematical points completely subject to the laws of number—a sort of arithmetic geometry... The rival Eleatic philosophy of Parmenides upheld the essential "oneness" of the universe and the impossibility of analyzing it in terms of the "many." Zeno of Elea sought dialectically to defend his master's doctrine by demolishing the Pythagorean association of multiplicity with number and magnitude. ...The paradoxes, as one sees now, involve such notions as infinite sequence, limit, and continuity, concepts for which Zeno nor any of the ancients gave precise definition. ...their influence was profound. The Greeks banned from their mathematics any thought of an arithmetic continuum or of an algebraic variable, ideas which might have led to analytic geometry; and they refused to place any confidence in infinite processes, the methods which would have led to calculus. Whereas the Pythagoreans had envisioned a union of arithmetic and geometry, Greek mathematicians after Zeno saw only the mutual incompatibility of the two fields."