Zeno Malae - the mathematics of the author of the Achilles and the Tortoise paradox

Zeno Malae is considered one of the first mathematicians

What is mathematics? This question sounds a bit rhetorical since we all know that it is basically the same database of formulas and equations that are supposed to describe something in general, all kinds of logical connections between one thing and another. Surprisingly, this is not the way in which this field of numbers was treated in ancient times, but rather they saw it as a kind of ad hoc help; If an Egyptian from the Pharaonic period wanted to know how much water would fit in a square-like water trough, he would calculate the length of the two diagonals in the tank and multiply them by each other, the result was the desired amount of water. After several attempts he realized that it was not an exact result but a pretty good estimate and this made up his mind. This was the custom until the golden age of the Greeks - mathematics was computational, the kind that could help the average person on a daily basis in his business, household affairs, and the like. The question of the validity of these calculations was not on the agenda. When the Greeks came, he brought with him the most important foundation of mathematics known as proof, i.e. a clear and defined order of operations that will always lead to a certain result that can be expected. This is actually the beginning of mathematics as we know it today and it grew out of the ideas of great and deep philosophers who were given an "earthly" explanation in the form of mathematical equations. The connection between philosophy and mathematics was so strong that it is difficult to find a philosopher who was not a mathematician or a mathematician who was not a philosopher. One of the most important of them is the subject of the following article.

What we know about Zeno we got mainly from the records of Plato and a number of other writers, but this information is not very much. Zeno was born in the city of Alaea (now Italy) in 490 BC and already in his youth he began to engage in the world of philosophy. Zenon the philosopher was deeply influenced by his close friend who was also an important philosopher named Parmenides - the main figure in the theory of monism, which is based on the principle that all things that exist in the world are actually one eternal reality called "Is". From this principle it follows that since everything is one thing - there can be no states of non-existence - because everything exists, or movement - because there is nowhere to move if the whole world and the details in it are a single thing. One of the famous books he wrote about it contained 40 paradoxes on the matter of continuity, 4 of which had a decisive influence on the entire world of mathematics for a long time. The main themes of the book, as I mentioned, were about the idea of ​​the uniformity of the individuals existing in the world (ie there is no multiplicity) and the immobility.

One example of the way he attacked the idea that there is more than one thing in this world is that if we assume we take a certain size and divide it then surely it can be divided again an infinite number of times and we can never reach a final point where it can be said that it is a part by itself and does not belong to that initial part , Zeno also claimed that a situation in which no thing has a definite size is never possible. The explanation for the last claim was given by the philosopher Simplicius on behalf of Zeno by saying that if we add something without size to another factor or subtract this same something from it, then that factor will neither increase nor decrease; Since there was no change in this factor, then what we actually added or subtracted was nothing, and from this it follows that a factor without magnitude cannot exist.

The other thing he dealt with was of course the immobility and the decisive importance given to these paradoxes by Aristotle when he combined 4 of them (which we mentioned above) in his important book - Physics. These were the paradoxes known to many of us as dichotomy, Achilles and the tortoise, the arrow and the stadium. Here are three of them:

The claim of the first paradox was that there can be no movement of any body because in order for that body to reach the end point of the trajectory it must first reach the middle point of the trajectory. This is of course a philosophical claim and its translation into the language of mathematics is what is now called a series; Numerically, we can say that if we have a line with a length of 1 cm then in order to get from 0 to 1 we must first go by 0.5 cm, but also in order to get to 0.5 cm we must first get to 0.25 cm which is the half way of the half way and so on, if we continue this to infinity then actually the movement will never have meaning because we will never be able to reach the end. In mathematical formulation it looks like this – 1/2 + 1/4 + 1/8 +…. = 1, this equation has no solution because there will always be a small part that will be missing to complete to a sum of 1 and therefore logically a movement represented by a transition from 0 to 1 (or any other length) cannot exist. This is the dichotomy paradox that we actually saw earlier in the claim that a certain size that can be divided in some way can be divided infinitely (paradoxically) and therefore basically all things are one thing. This is indeed a fantastic claim.

The second paradox is the arrow and describes the situation of an arrow shooting from a bow and moving towards the target. This distance is actually like a transition from 0 to 1 that we showed earlier, and if the arrow was at certain stages in all kinds of points between 0 and 1 then this means that the length of this segment can be divided at least once, but we have already argued earlier that if a certain segment can be divided into two then it can be divided into infinite parts. If this is the case, then during its flight the arrow passed through an infinite number of points and a frozen image of each such point will show that the arrow actually did not move, and from this it follows that it did not move throughout its flight and therefore the conclusion is that there was no movement at all.

I will try to explain it another way - when we say that a body moves we actually mean that it was at point A at a certain point and then moved to point B, this is one clear fact since we saw the transition, but the form in which the body traveled this distance can be in an infinite number of ways, each of which is called By us - time. We can say that the arrow moved from one point to another in a certain time, for example one second and we can say that it traveled this distance in two seconds. Where does the difference come from? Another thing called speed or the rate of change in time. That is, the definition of the term speed depends on two points in time, the one when the body started its flight and the one where it ended. As we all know, a line runs between two points, but at the beginning of the article we argued that a line is actually a certain size and if we wanted to cross it we wouldn't be able to because we would never get from our point to another point. From this it follows that time displacement (or a difference in time) cannot exist and if time does not move then velocity does not exist, meaning it is always equal to zero, which causes the arrow to always be at zero speed - i.e. at constant rest.
The third paradox presented by Zeno is probably the most famous of all and is known as "Achilles and the Tortoise".

The problem sounds somewhat the same as that of the dichotomy and describes a situation in which Achilles, who is the fastest man in the world, competes against the tortoise but gives the latter a "split" of several meters in order for the competition to be fair. Again, here too Zeno explains that Achilles will never defeat the tortoise because he will never complete the initial distance between him and the tortoise similar to the previous problem.

How much did these paradoxes really affect the world of mathematics? Bertrand Russell (Russell), one of the greatest logicians of mathematics, claimed in one of his books that although in Zeno's time these paradoxes were considered logical nonsense, many attempts were nevertheless made over time to solve the problems he posed, mainly by important areas of mathematics known as the theory of sets and convergent infinite series, however In the end, the basic problems of these paradoxes repeat themselves over and over again because the person perceives the concept of continuity (say, of the line) in two forms that are incompatible with each other.

Whatever the effects, today these paradoxes can be looked at as a preoccupation with two questions: one wonders what happens to some thing at a given moment, now, without any connection between what was before or what will be after, and the second question wonders what will happen to the same thing at a later stage, Later.
These are questions to which the answer is not easy, but in retrospect it can be said that these are actually the basics of the thought that was behind the development of the differential and integral calculus, the two most important concepts in the world of mathematics in particular and in science in general.