**8. Achilles paradox**

One of the most famous paradoxes of philosophy is the Zeno's Paradox about Achilles and the tortoise. The paradox goes like this:

It states that Achilles will never catch up with a hulking turtle if she begins her movement before him.

For several centuries this paradox was a "horror" of philosophers and theoretical scientists. And right now the explanations that are offered to ordinary people are very vague.

Meanwhile, it seems to me that everything is easily explained if we analyze the discreteness of time that Zeno offers and that which his interlocutors understood.

Let's try to figure out the details of Zeno's reasoning. His main position: Achilles will not catch up with the turtle, because in the interval of time for which he will reach her position, she will go further. Zeno suggests that the interlocutors consider the whole movement as a sequence of states and intervals at those points where the turtle has already visited. These intervals will be shorter and shorter until they become infinitely small. However, Zeno did not apply such concepts. Actually, turning to the concepts of infinitesimal ones, he remains himself and leaves his interlocutors in terms of finite time intervals … and space too. And he comes and leads the rest of the participants in the conversation to a clear contradiction. He tells them: "I show you my time and space, which I interrupt at any time when I want and as many times as I want. You too can tear your time, in which Achilles easily catches up with a turtle, on as many sites as you like. So our times are the same, but in mine Achilles is forever behind the turtle. So in yours, he will not catch up with her."

All ordinary people understand the discreteness of time as the same and fall into the "trap." In their head, time is unbroken, infinite, and the "time of Zeno" is only that time in which the tortoise is ahead of Achilles, and its division into an infinite number of sections—intervals. He "equates" it with the time of the interlocutor infinite time. Zeno says "ALWAYS," but it is in his time, and he evens out times with the number of intervals. But the intervals for Zeno and his interlocutors are different. *Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems… DOI: http://dx.doi.org/10.5772/intechopen.91467*

And this is the trick of Zeno, because the number of intervals is not a length of time especially if the intervals are infinitesimal. The paradox turns into Sophism. We do not know knowingly did it Zeno ... Hardly. Otherwise, he would have created a theory of infinitesimal quantities 2000 years earlier than Descartes and Leibniz, who created higher mathematics in which discreteness and, especially, its infinitesimal values play a fundamental role. Judging by Zeno's other aporias—for example, "On the Arrow," he felt a "discrepancy" between ordinary discrete thinking, based on observations and practical experience and continuity, which scientists spoke about in his time. And he showed this problem in every way in the Achilles paradox—irresponsibly changing the discreteness of time.
