Zeno of Elea's (b.490 BC) arguments against motion precipitated a crisis in Greek thought. All of these, concerning motion, have had a profound influence on development of mathematics. They are described in Aristotle's great work 'Physics' and are presented as four arguments in form of paradoxes, stated below :
1. The Racecourse or Dichotomy Paradox -
There is no motion because that which is moved must arrive at middle of its course before it arrives at end. In order to traverse a line segment it's necessary to reach halfway point, but this requires first reaching quarter - way point, which first requires reaching eighth - way point, and so on without end. Hence motion can never begin.
This problem isn't alleviated by well - known infinite sum ½ + ¼ + 1/8 ... = 1 because Zeno is effectively insisting that sum be tackled in reverse direction. What is first term in such a series ?
( See David Darling : The Universal Book of Mathematics, 2004. )
2. Achilles and Tortoise -
This is perhaps most famous of Zeno's paradoxes.
The slower when running will never be overtaken by quicker; for that which is persuing must first reach point from which that which is fleeing started, so that slower must necessarily always be some distance ahead. Thus, Achilles, however fast he runs, will never catch plodding Tortoise who started first. And yet, of course, in real world, faster things do overtake slower ones.
3. The Arrow -
An arrow cannot move at a place at which it is not. But neither can it move at a place at which it is. But a flying arrow is always at place at which it is. That is, at any instant it is at rest. But if at no instant is it moving, then it is always at rest.
4. The Moving Blocks or Stadium -
Suppose three equal blocks, A , B, C, of width 1, with A and C moving past B at same speed in opposite directions. Then A takes one time, t, to traverse width of B, but half time, t/2, to traverse width of C. But these are same length, 1. So A takes both t and t/2 to traverse distance 1.
( See Simon Blackburn : Dictionary of Philosophy, 1996 )
The German set theorist Adolf Frankel ( 1891 - 1965 ) is one of many modern mathematicians ( Bertrand Russell is another ) who have pointed out that 2,000 years of attempted explanations have not cleared away mysteries of Zeno's Paradoxes : "Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as theory of convergent series or theory of sets. In end, however, difficulties inherent in his arguments have always come back with a vengeance, for human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable."