How restless were the ancient Greek philosophers in their quest to comprehend existence. And they set such tasks that not only philosophers think about to this day. Zeno's aporias are one such paradox. The most famous are the aporia "Achilles and the Turtle" and "Arrow".
Aporia "Achilles and the Turtle" boils down to the fact that no matter how fast Achilles ran, he will never catch up with the turtle, even if it moves ten times slower, being at some distance from Achilles. Zeno argues as follows: while Achilles reaches the place where the turtle began to move, the latter will crawl away some distance further. When Achilles overcomes this distance, the tortoise will crawl away even further, although not by much. But in this way, no matter how much Achilles reaches the previous location of the turtle, it will still be ahead. And even if the distances and time will be very tiny, this process will drag on indefinitely, and as a result, Achilles will never catch up with the turtle.
It seems to be nonsense: how can Achilles never catch up with the turtle? But from the point of view of philosophy and other sciences, the task is set logically correctly. As it turned out, the point is in the idea of space, time and infinity. If space and time are continuous, and infinity physically exists, then Achilles, it seems, should not catch up with the turtle. But, nevertheless, it is catching up.
Even in Ancient Greece, attempts were made to resolve Zeno's paradoxes. Aristotle, although he considered space-time indivisible, but, like the atomists, who already believed space-time to be discrete, nevertheless curtailed the possibility of infinitely fragmentation of time. Therefore, it is believed that Aristotle could not explain how a finite period of time is composed of infinite parts.
Kant and Hegel noted dialectics, that is, the contradiction of movement, indicated in the aporias, in the last century drew attention to the discrepancy between the mathematical model and the physical reality of movement (in particular, Hilbert). Cauchy, using the concept of the limit and convergence of mathematical series, and Robinson - hyperreal numbers (both of which are from the field of higher mathematics), proved that even in infinity Achilles will still catch up with the turtle, but it is clear that this higher mathematics is not very close to physical reality … It could also be noted that, according to Einstein's theory of relativity, Achilles and a turtle moving at different speeds will have different time flows, although the difference will be negligible, but in a dispute with infinities, any finite difference always "wins". Quantum mechanics, with its discreteness and uncertainties,only added to the difficulties in understanding and resolving the aporias.
Although it is worth adding that in the case of Achilles and the tortoise, one aspect should be taken into account: if Achilles in any way fixed, that is, for any infinitely small period of time, he would stop every time he got to the previous place of the tortoise, and even if he had a speed that is millions of times higher than the speed of a turtle, then he would not have caught up with this turtle in reality. But, to our delight, Achilles runs without bothering and calmly overtakes the turtle. Unfortunately, this fact does not remove all questions on the structure of space-time and infinity.
The situation with the "Strela" aporia is even more confusing. Considering the flight of an arrow, Zeno notes that at any moment of time the arrow, occupying a certain space, rests in it. That is, there is no actual movement. In this aporia, the problems of understanding space-time and infinity again appear, but the problem of the illusion of motion is added to them. If there were books in Ancient Greece, you would think that Zeno, like our students, indulged in such simple fun. Where there is page numbering in the books, children draw little people on different pages in different poses, and then, quickly flipping through these pages, you can see that the little man is performing an intricate "dance". You can also take a piece of film and make sure that every frame in it is static, but when watching a movie, all these characters move for some reason. From this, if you wish, you can conclude that we all live in the "Matrix", but this is not a matter of proving the illusion of movement (the illusion of which is very painfully removed if you stand on the path of a flying arrow), but in faith, because even in the cosmogony of Islam, there is an opinion that God instantly destroys and restores the universe again, as on a film.
Thus, as it was said, these restless ancient Greeks set tasks that are still being solved and solved, and allow a deeper understanding of themselves and the surrounding reality.
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