The concept of eternity has been used throughout history, from physics to mathematics, from philosophical literature to mysticism. The contexts in which the idea of eternity is identified within these areas may, of course, be different. In this writing, we will talk about the eternity used in philosophy and mathematics. In doing so, we will talk about some mathematical concepts, without having to have any mathematical background to read.
The concept of eternity, as far as it is known, was first used by Anaximandro in Ancient Greece to express what is the essence of existence. According to Anaximandros, the essence of being is something without endless, everything in it. This is called apeiron. The meaning of the word means infinite, infinite. According to Anaximandros, apeiron has been seen as a positive thing with beauty, but apeiron, in the near term, according to Pythagorian, is seen as a bad, ugly, incomprehensible, negative thing. Of course, according to the Pythagoreans, apeiron is seen in this way because in his teachings, Pythagoras, apeiron can never be expressed in terms of natural numbers (0, 1, 2, 3, 4, ...). Pythagoreanism has argued that everything in the world can be explained by natural numbers or relations with natural numbers in relation to each other. In Pythagorean everything in the world and the universe can be explained in terms of the natural number. The object we call the natural number is, of course, an object with a final knowledge. For example, we can think of the number 100 as a combination of 100 bars or 100 rings. Every natural number is a concept that can be explained by ending information. Since Apeiron is the limitless essence of existence, something that is unlimited can not be described by a natural number with limited information. As we know from the Pythagorean theorem, when a right triangle in unit length is taken, the length of the hypotenuse is √2. √2 is not a fractional number, it can not be explained by the ratio of two natural numbers. It is an irrational number. Pythagorian, this number is alogos. So it can not be talked about, it can not be reasoned, it is inexplicable. Because the things that can be explained are, according to Pythagorian, only natural numbers should be related to each other or in proportion.
When we look at Plato, the concept of eternity is regarded as an absolute whole object, a totalit. According to Plato's idealary theory, just as every object is an absolute and perfect form in the world of ideals, the concept of eternity must exist in the world of ideals as an absolute totalitarian. However, for a long time after Aristotle, the notion of infinity has been treated as a potential object, not an absolute totalitarian. As it is known, one of the most important contributions of Aristotle to the history of philosophy is that he proposed a situation between existence and non-existence, that is, the idea of potentiality. In Plato's theory, we can think that the concept of eternity really exists as an absolute idea. Thinking of infinity as a totalitarian object has brought to accept the idea of absolute eternity. However, in Aristotle, eternity is not a complete and absolute state, but a potentially never ending process. Let's try to explain this with an example. If we regard the natural numbers as 0,1,2,3,4, ..., we can see this as a never-ending and continuous process. We can write a number that comes after each digit. This gives us a never-ending array of definitions. This series is a continuous ongoing potential, not complete. However, it is also possible to treat the natural numbers as a whole, perhaps in the form of a complete totalitem. Let's denote this as N = {0,1,2,3, ...}. Here N is a community of natural numbers. When we take this community as an object, just as it is in Plato's theory, the N object is seen as a complete community containing all the natural numbers. The series we wrote earlier is a never-ending, ongoing process. This process is not over, we can see that the index goes on forever, but we do not consider this process as a totalit object. If we take it, this finished object becomes the absolute infinity itself.
Another thinker who uses the idea of potential infinity was Eudoxus, an Ancient Greek mathematician, one of Platon's students. Eudoxus adopted the idea of potential eternity just like Aristotle. We developed a method of approach, which we can name the method of consuming, in order to calculate the area of the house. Accordingly, if we draw a square into the circle after drawing a circle, the area of the karen will be much smaller than the area of the natural circle. If we draw a hexagon instead of a square into the circle, we get closer to the circle shape. If we draw octagons and other polygons in a similar way instead of a hexagon, we see that the polygon area obtained by increasing the number of edges comes closer to the field of the field. Since the polygon with an infinite edge can not be drawn, the field of the field can never be precisely calculated by this method, but the more the number of edges, the closer we are to the field of the field. We have the possibility to increase the edge count forever. This method is an example of how potential infinity is used in mathematics.
One of the things about eternity is the paradoxes of Eleanor Zeno. Zeno is an ancient Greek thinker who lived on the southern coast of Italy. One of the paradoxes is concerned with whether or not he has reached the goal of a savage arrow. In order to reach the goal of a savvy arrow, as a rationale, it must have completed the half way to the goal that it will go first. Let's say this point. This time, the arrow A must complete the halfway between A and the target to reach the target. Let's call it B here. If we apply the same argument for point B and continue this way, we will never see the final goal of the arrow in the final step. The arrow must have an infinite number of steps to reach its final destination. Then the infinite step must be completed to go from one place to another. Actual action is impossible because the infinite step can not be completed. However, physically apparent movement is of course possible. This dilemma is called the Zeno paradox. This problem remained a paradox in the seventeenth century as Newton and Leibniz developed calculus (derivative and integral mathematics). With the development of the calculus, it has been shown that the sum of all these roads, which are kept in the constant radius of the Zeno paradox, is really equal to the final goal.
When we look at the medieval devrine we see that Islamic geo-philosophers are mostly influential. Kindi, Gazzali, Farabi, Ibn-i Sina, Omer Hayyam and many other philosophers have thought about logic science and eternity. One of these thought experiments turned into a probing, later known as the Galileo paradox. The Galileo paradox is concerned with determining whether two clusters containing infinite objects are equal in number to each other. The set of natural numbers is represented as N = {0,1,2,3, ...}. A set of double numbers is denoted by C = {0,2,4,6, ...}. The size of these two coops is the same. Because any number n from N can be defined as a pair that leads to 2n in O. How is it possible that the size of the two coops can be the same while the number of "less" is seen in Ç? This is called the Galileo paradox. It is natural that this problem has been seen as a paradox since the mathematical theory of infinity has not yet been developed at that time. In the 19th century, however, Georg Cantor developed his theory of clusters and developed his infinite mathematical theory.
Cantor is the first person to make the concept of eternity a mathematical theory. Consider the comparison of the sizes of the clusters. It is easy to compare the sizes of the clusters containing the finite number of elements. We can see that two clusters contain the same number of elements by counting the elements of the clusters. We can not compare infinite sets with counting in the same way. However, if there is a match between the elements of the two clusters, the sizes of these two clusters are the same. These are the N and C clusters we have given for infinite sets. The size of the N and C clusters is the same because there is an individual mapping between the elements.
One of Cantor's most spectacular theorems is that there is a larger cluster from each set. For the end clusters this result may not be very interesting. However, when this theorem is applied to infinite sets of clusters, it can be seen that every infinite is precisely a larger infinite, which is why there is no single infinite in mathematics. With Cantor, we can say that infinite clusters are used as objects in mathematics, and that the concept of absolute infinity is actually used in mathematics from this period. It is possible to accept the existence of infinite clusters, to deal with them, to join, to intersect, to multiply, or to perform similar operations, alone and only to accept the concept of absolute eternity.
The eternal cluster, although defined in mathematics, will continue to be a concept that protects the mystery of course, the eternal metaphysical nature.
