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 the development of mathematics. They are described in Aristotle's great work 'Physics' and are presented as four arguments in the form of paradoxes, stated below :
1. The Racecourse or Dichotomy Paradox -
There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end. In order to traverse a line segment it's necessary to reach the halfway point, but this requires first reaching the quarter - way point, which first requires reaching the eighth - way point, and so on without end. Hence motion can never begin.
This problem isn't alleviated by the well - known infinite sum ½ + ¼ + 1/8 ... = 1 because Zeno is effectively insisting that the sum be tackled in the reverse direction. What is the first term in such a series ?
( See David Darling : The Universal Book of Mathematics, 2004. )
2. Achilles and the Tortoise -
This is perhaps the most famous of the Zeno's paradoxes.
The slower when running will never be overtaken by the quicker; for that which is persuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. Thus, Achilles, however fast he runs, will never catch the plodding Tortoise who started first. And yet, of course, in the 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 the 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 the same speed in opposite directions. Then A takes one time, t, to traverse the width of B, but half the time, t/2, to traverse the width of C. But these are the same length, 1. So A takes both t and t/2 to traverse the 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 the 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 the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable."
( See David Darling : The Universal Book of Mathematics, 2004 )
The NSTP Theoretical Resolution of the First Two Paradoxes
Zeno's paradoxes, except the last two, are not a matter of language or symbolic theories (e.g. set theory) or equations. They are deep rooted in profound concepts, whose appropriate analysis and synthesis shall resolve the paradoxes.
The first two of Zeno's paradoxes are out of the misbelief that space exists in the ontological sense, i.e. as a reality, out there. In fact, space is a virtual reality, a form/kind of illusion. Consequently (spatial) motion is also a form of illusion ( to non - spatial observer/s ). Thus reality is not constrained by spatial infinities as whatever that is seen as happening in space is a mere illusion, with no resemblance to reality. And illusion could be of any logically possible kind. In other words, the thoughts modulating / creating / responsible for the spatial illusion do not have to bother whether the mover has to first reach half of the distance and so on, or the faster has to first reach the point where the slower started or has infinitely many gaps to traverse, etc. The only thing is that they, the thoughts, produce some dynamic spatial pattern ( actually / physically represented in the form of appropriate non - spatial states of consciousness ), as if a mover moving or the faster overtaking the slower. That's it.
[ In analogy with today's desktop computers a software programmer / graphic designer do not at all have to worry with Zeno's first two arguments / paradoxes. All s/he has to do is to write a program in order to create / generate an appropriate dynamic / changing pattern on the computer monitor screen. The same is true with the whole universe, whose non - spatial mechanism is stated in the NSTP ( Non - Spatial Thinking Process ) theory. ]
Resolution of the Third Paradox
The first proposition / assumption in the third paradox is false.
Resolution of the Fourth Paradox
In the fourth argument / paradox there is no consideration of speeds ( or the concept of speed ). As A and C are travelling in the opposite directions their speeds add up. And as time taken = distance covered / speed, double speed makes the mover cover the distance in half time.
And even if this solution has any flaw/s then ultimately there is the NSTP theory, with its idea of spatial illusion, to resolve the paradox.
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