The paradoxes, a.k.a. aporias, of Zeno of Elea are built upon a paralogism that portrays the movement as if it is a thing divisible in the reality proper, in the same manner as it is divisible in the imagination. However, the course of an object that moves itself in the reality proper is indivisible, because it doesn’t presents solutions of continuity. If the trajectory of an object is interrupted in the real world, when it recommences its movement then it is not the same old movement, but a new one.
By this Zeno supposed, or pretended to suppose, that an arrow that was threw towards a target could have its movement intersected on its half, and, in the following, the remaining of its movement intersected on its subsequent half, and on this manner continually, ad infinitum, without the movement changing its nature. Notwithstanding, a continued movement occurring in the real space does not admit to be divided in this arbitrary manner because it's an indivisible unity.
Shortly, what Zeno was intending that us should believe was that the arrow came through till half of its way towards the target, and then stopped. In the following, it recommenced its movement going till half of the remaining course, and stopped again. And, so on, the arrow proceeded in that bizarre manner, never reaching to the target. It's a true thing that if the arrow proceeded in this strange way, stopping and recommencing its trajectory at every half of its course, it will never reach to the target. However, its succeeding movements will not be the same movement, and they will be smaller every time, till they'll turn themselves imperceptibles to the human eyes and, in the following, to the common microscopes; and, at last, absolutely imperceptibles. Space sectioned and measured in this manner is infinitely divisible.
Thus, the crafty sophistry of Zeno conducts us to intercept and divide the movement of the arrow in every subsequent half of its journey, but only in our imagination, convincing us that this is the same continuous movement that happens in reality proper, and never allowing us to consider that this is not the same movement in its entireness. Then, he manages us to transfer the manipulated conclusion of this experiment to the field of reality. And we don’t realize plainly this because we are not actually thinking; but he is, in fact, Zeno that is conducting our thoughts. Sophistry functions because we permit that someone, craftily, manipulate our thoughts, thinking for us.
The same is the case with the well-known paradox called "Achilles and the tortoise". Here, the a priori supposition is that departing the tortoise on the very moment that Achilles do the same thing, but from a distance away in front of him, nearer to the place marked as the arrival place (let's assume that the tortoise's advantage is of fifty meters), Achilles traverses half of the distance (five and twenty meters) whereas the tortoise goes forward only a little. In the case of Achilles making a halt in this very instant and recommencing their run afterwards, Achilles, with certainty, would have never reached to the tortoise. Yet, in very little time these different movements would have turned themselves imperceptibles, because the real space can be infinitely sectioned. But, this is not the case that happens with our competitors, because they never stop to a lunch. Thus, Achilles does not traverses half of the way, or a third of the way or any other arbitrary division of this, but, incessantly, surpassed all these points.
Taking this reasoning unto its extreme course we could imagine that the tortoise remains still, not moving himself an inch forward. Nevertheless, if Achilles runs toward it only halfway down the course and make a stop, recommencing to run and stop in the same manner, repeatedly, he would never reach to it. Not even in a billion years, although the total inactivity of the animal.
Only if the protagonists of Zeno's paradoxes should have performed continually interrupted real trajectories (not in imagined space), in their runs towards their targets, they will have never reach to them. Notwithstanding, because in the reality proper they never stop, Achilles reaches to the tortoise and wins the race, in the same manner that the arrow goes unstopped to its target.
