Based on the above study of the Milesian school, one can only be convinced of the active influence of the world view on the process of mathematical knowledge only with a radical change in the socio-economic conditions of society. However, questions remain open about whether the change in the philosophical basis of society’s life influences the development of mathematics, whether mathematical knowledge depends on a change in the ideological orientation of the worldview, or whether the reverse effect of mathematical knowledge on philosophical ideas takes place. You can try to answer the questions posed by referring to the activities of the Pythagorean school.
Pythagoreanism as a direction of spiritual life has existed throughout the history of ancient Greece, beginning in the 6th century BC. and went through a series of stages in its development. The question of their temporal duration is complex and has not yet been unambiguously resolved. The founder of the school was Pythagoras of Samos (c. 580-500 BC). Not a single line written by Pythagoras has been preserved; it is not known at all whether he resorted to writing his thoughts. What was done by Pythagoras himself, and what his students, it is very difficult to establish. The testimony of ancient Greek authors is contradictory; to some extent, the various assessments of his activities reflect the diversity of his teachings.
In Pythagoreanism, there are two components: practical (“Pythagorean way of life”) and theoretical (a certain set of exercises). In the religious teachings of the Pythagoreans, the ceremonial side was considered the most important, then it was intended to create a certain mental state, and only then did the beliefs take precedence, in the interpretation of which various options were allowed. Compared with other religious movements, the Pythagoreans had specific ideas about the nature and fate of the soul.
The soul is a divine being, it is enclosed in a body in punishment for transgressions. the ultimate goal of life is to free the soul from bodily imprisonment, not to allow it into another body that is supposedly done after death. The way to achieve this goal is to implement a specific moral code, the “Pythagorean way of life”. In the numerous system of prescriptions that regulated almost every step of life, a prominent place was given to music and scientific studies.
The theoretical side of Pythagoreanism is closely related to the practical. In the theoretical studies, the Pythagoreans saw the best means of liberating the soul from the circle of births, and their results were aimed at rationally justifying the proposed doctrine. Probably, in the activities of Pythagoras and his closest disciples, scientific positions were mixed with mysticism, religious and mythological ideas. All this “wisdom” was stated as the oracle sayings, which gave a hidden meaning to divine revelation.
The main objects of scientific knowledge of the Pythagoreans were mathematical objects, primarily the number of the natural number (recall the famous “The number is the essence of all things”). A prominent place was given to the study of the connections between even and odd numbers. In the field of geometric knowledge, attention is focused on the most abstract dependencies. The Pythagoreans built a significant part of the planimetry of rectangular shapes; The highest achievement in this direction was the proof of the Pythagorean theorem, special cases of which, 1200 years before, are given in the cuneiform texts of the Babylonians. The Greeks prove it in a general way. Some sources attribute to the Pythagoreans even such outstanding results as the construction of five regular polyhedra.
The numbers of the Pythagoreans are the fundamental universal objects, which were supposed to reduce not only mathematical construction, but the whole diversity of reality. Physical, ethical, social and religious concepts have been mathematically matched. The science of numbers and other mathematical objects has a fundamental place in the worldview system, that is, in fact, mathematics is declared philosophy.
As Aristotle wrote, “… among the numbers, they saw seemingly many similarities with what exists and happens, more than with fire, earth and water … They, apparently, take the number as the beginning both as a matter for things, and as an expression for their states and properties … For example, such-and-such property of numbers is justice, and such-and-so soul and mind, the other is good luck, and you can say in each of the other cases in the same way. “If we compare the mathematical studies of the early Pythagorean and Milesian schools, we can identify a number of significant x differences.
Thus, the mathematical objects were considered by the Pythagoreans as the first essence of the world, that is, the very understanding of the nature of mathematical objects changed radically. In addition, the Pythagoreans turned mathematics into a component of religion, into a means of purifying the soul, achieving immortality. Finally, the Pythagoreans limit the area of mathematical objects to the most abstract types of elements and consciously ignore the applications of mathematics for solving production problems.
But what caused such global differences in the understanding of the nature of mathematical objects in schools that existed almost at the same time and drew their wisdom, apparently from the same source – the culture of the East? However, Pythagoras most likely enjoyed the achievements of the Milesian school, since, like Thales, he had found the main signs of mental activity that were different from the pre-Greek era; However, the mathematical activity of these schools was of a significantly different nature.
Aristotle was one of the first who tried to explain the reasons for the emergence of the Pythagorean concept of mathematics. He saw them within mathematics itself: “The so-called Pythagoreans, taking up the mathematical sciences, moved them forward for the first time and, having brought up on them, began to consider them to be the beginnings of all things.”
This viewpoint is not without reason, if only because relations between different phenomena. On this basis, it is possible, improperly expanding a given moment of mathematical knowledge, to come to the statement about the expressibility of all things with the help of mathematical dependencies, and if we consider numerical relations to be universal, then “the number is the essence of all things.” In addition, by the time the Pythagoreans acted, mathematics had come a long way in historical development; the process of formation of its main provisions was lost in the darkness of the ages. Thus, there was a temptation to neglect them and declare mathematical objects to be something primary in relation to the existing world. That is exactly what the Pythagoreans did.
Thus, already in the starting point of its development, theoretical mathematics was influenced by the struggle between two types of world outlook — the materialistic and religious-idealistic. We were convinced that along with the influence of the world view on the development of mathematical knowledge, there is a reverse effect.
