He is always getting closer, but never makes up the lost ground completely. Intuitively at least, it is clear that the human runner can quickly outdistance the tortoise (Papa-Grimaldi, 1996), but Zeno's paradox seems to defy this capability. In this regard, Cohen asks, "Achilles will certainly, with his celebrated speed, soon get very close behind the tortoise -- but why can't he, logically speaking at least, ever overtake the reptilian competitor?" (2002, p. 38). A similar paradox is presented by Zeno in "The Arrow," discussed further below.

Paradox of the Arrow. According to Dowden, this paradox concerns "a moving arrow [that] must occupy a space equal to itself at any moment. That is, at any moment it is at the place where it is. But places do not move. So, if at each moment, the arrow is occupying a space equal to itself, then the arrow is not moving at that moment because it has no time in which to move; it is simply there at the place" (2010, para. 3). This paradox suggests that anything that is moving is not actually moving, but is rather suspended in a static state from moment to moment. In this regard, Cohen asks, "What can we say about the arrow in flight? At each instant, Zeno says, it is motionless, since it would have time to move, that is, to occupy at least two successive positions, only if it were accorded at least two instants. At any given moment, it is thus at rest at a given point. Motionless at each point of its path, it is motionless the entire time that it is moving" (2002,...

38).
Conclusion

The research showed that Zeno was an ancient Greek philosopher who predated Aristotle and Plato and who propounded a series of four paradoxes, the Achilles, the Dichotomy, the Arrow and the Moving Rows, all of which concern motion and all of which have challenged mathematicians for more than two thousand years. Of these four paradoxes, the Achilles and the Arrow were shown to be the most well-known, and both of these involved similar extensions of Zeno's logic as it applied to moving objects. As stated, Zeno's paradoxes do appear to defy easy solutions, but they ignore the harsh realities of motion wherein a single stride by Achilles will overtake the tortoise irrespective of how infinitesimally it moves and the arrow in fact remains in motion from moment to moment irrespective of its status at any given point in time. Therefore, to assert that an object in motion is "motionless" is not a paradox, but is rather an oxymoron.