The works of Plato (427-347 BC) are a unique phenomenon in terms of highlighting a philosophical concept. This is a highly artistic, fascinating description of the very process of becoming a concept, with doubts and uncertainty, sometimes with unsuccessful attempts to resolve the question raised, with a return to the starting point, numerous repetitions, etc. It is rather difficult to single out any aspect in Plato’s work and systematically presenting it, since you have to reconstruct Plato’s thoughts from individual statements that are so dynamic that, in the process of evolution, thoughts sometimes turn into their opposite.
Plato repeatedly expressed his attitude towards mathematics and she was always highly appreciated by him: without mathematical knowledge, “a person with any natural properties would not be blissful”, in his ideal state he intended “to approve by law and convince those who intend to occupy high positions in so that they practice the science of numbering. ” The systematic widespread use of mathematical material takes place in Plato, starting with the Menon dialogue, where Plato leads to the main conclusion by means of a geometric proof. It was the conclusion of this dialogue that knowledge is a recollection that became the fundamental principle of Plato’s epistemology.
Significantly more than in gnoseology, the influence of mathematics is found in Plato’s ontology. The problem of the structure of material reality in Plato received the following interpretation: the world of things perceived through the senses is not the world of the truly existing; things continually arise and die. The world of ideas possesses true being, which are incorporeal, insensible, and act in relation to things as their causes and the images by which these things are created.
Further, in addition to sensory objects and ideas, he establishes mathematical truths that differ from sensory objects in that they are eternal and immobile, and from ideas in that some mathematical truths are similar to each other, but there is only one idea every time. In Plato, as the matter, the beginnings are big and small, and as the essence is one, for ideas (they are numbers) are obtained from big and small by introducing them to unity. The world of perception, according to Plato, is created by God. The process of building the cosmos is described in the Timey dialogue. After reading this description, it is necessary to recognize that the Creator was well acquainted with mathematics and at many stages of creation essentially used mathematical concepts, and sometimes carried out exact calculations.
Through mathematical relationships, Plato tried to characterize some of the phenomena of social life, as exemplified by the interpretation of the social relation “equality” in the dialogue “Gorgiy” and in the “Laws”. It can be concluded that Plato essentially relied on mathematics in developing the main sections of his philosophy: in the concept of “knowledge of recalling”, the doctrine of the essence of material existence, of the structure of the cosmos, in the interpretation of social phenomena, etc.
Mathematics played a significant role in the constructive design of his philosophical system. So what was his concept of mathematics?
According to Plato, mathematical sciences (arithmetic, geometry, astronomy, and harmony) are bestowed upon man by gods, who “produced numbers, gave the idea of time, and aroused the need to explore the universe.” The original purpose of mathematics is to “purify and revitalize that organ of the human soul, frustrated and blinded by other things,” which “is more important than a thousand eyes, because truth alone is contemplated by them.” “Only no one uses it (mathematics) correctly, as a science that invariably leads to reality.” The “incorrectness” of mathematics Plato saw above all in its applicability to the solution of specific practical problems. It cannot be said that he denied the practical applicability of mathematics at all.
So, part of the geometry is needed for the “location of the camps,” “for all constructions, both during the battles themselves and during the hikes.” But, according to Plato, “for such things … a small part of the geometric and arithmetic calculations is sufficient, some of which are large, extending further, should … contribute to the easiest assimilation of the idea of good.” Plato spoke negatively about the attempts to use mechanical methods for solving mathematical problems that took place in the science of that time. His dissatisfaction was also caused by his contemporaries understanding of the nature of mathematical objects.
Considering the ideas of their science as a reflection of the real connections of reality, mathematicians, along with abstract logical reasoning, widely used sensory images and geometric constructions. Plato tries his best to convince that the objects of mathematics exist separately from the real world, therefore, when studying them, it is illegal to resort to sensual evaluation.
Thus, in the historically established system of mathematical knowledge, Plato singles out only a speculative, deductively constructed component and assigns it the right to be called mathematics. The history of mathematics is mystified, the theoretical sections are sharply contrasted to the computing apparatus, the application area is narrowed to the limit. In such a distorted form, some real aspects of mathematical knowledge were one of the reasons for constructing the system of objective idealism of Plato. Indeed, mathematics in itself does not lead to idealism at all, and in order to build idealistic systems, it has to be significantly deformed.
The question of the influence exerted by Plato on the development of mathematics is rather difficult. For a long time dominated by the belief that the contribution of Plato to mathematics was significant. However, a deeper analysis led to a change in this assessment.
So, O. Neugebauer writes: “His own direct contribution to mathematical knowledge, obviously, was equal to zero … The exceptionally elementary nature of the examples of mathematical reasoning given by Plato and Aristotle does not confirm the hypothesis that Evdox or Teeth learned something Plato …
His advice to astronomers to replace observations with speculation could destroy one of the most significant contributions of the Greeks to the exact sciences. ” This argument is quite convincing; one can also agree that the idealistic philosophy of Plato as a whole played a negative role in the development of mathematics. However, we should not forget about the complex nature of this impact.
Plato belongs to the development of some important methodological problems of mathematical knowledge: the axiomatic construction of mathematics, the study of the relationship between mathematical methods and dialectics, the analysis of the basic forms of mathematical knowledge. Thus, the proof process necessarily links a set of proven positions to a system based on some unprovable provisions.
The fact that the beginnings of the mathematical sciences are “the essence of assumptions” may raise doubts about the truth of all subsequent constructions. Plato considered such a doubt unfounded. According to his explanation, although the mathematical sciences themselves, “using assumptions, leave them in a stillness and cannot give them grounds,” the assumptions find their foundations through dialectics. Plato expressed a number of other provisions that proved fruitful for the development of mathematics.
So, in the dialogue “Feast” the concept of the limit is put forward; the idea appears here as the limit of becoming things.
The criticism to which Plato’s methodology and ideological system were subjected by mathematicians, for all its importance, did not affect the very foundations of the idealistic concept. To replace the methodology of mathematics developed by Plato by a more productive system, his doctrine of ideas, the main sections of his philosophy and, consequently, his view of mathematics, should be subjected to critical analysis. This mission fell to the lot of Plato’s disciple – Aristotle.