K. Marx called Aristotle (384-322 BC) “the greatest philosopher of antiquity”. The main questions of philosophy, logic, psychology, natural science, technology, politics, ethics and aesthetics, posed in the science of ancient Greece, received from Aristotle full and comprehensive coverage. In mathematics, he apparently did not conduct specific research, but the most important aspects of mathematical knowledge were subjected to a deep philosophical analysis, which served as the methodological basis for the work of many generations of mathematicians.
By the time of Aristotle, theoretical mathematics has come a significant way and reached a high level of development. Continuing the tradition of philosophical analysis of mathematical knowledge, Aristotle raised the question of the need to streamline the very knowledge about how to master science, about purposeful development of the art of cognitive activity, which includes two main sections: “education” and “scientific knowledge of the case.” Among the famous works of Aristotle, there are no specially devoted to the presentation of the methodological problems of mathematics. But on individual statements, on the use of mathematical material as illustrations of general methodological provisions, one can get an idea of what his ideal was for building a system of mathematical knowledge.
The initial stage of cognitive activity, according to Aristotle, is learning, which “is based on (some) already existing knowledge … Both the mathematical sciences and each of the other arts are acquired (precisely) by this method.” To separate knowledge from ignorance, Aristotle suggests analyzing “all those opinions that some thinkers expressed in their own way in this area” and pondering the difficulties that have arisen. The analysis should be carried out in order to clarify four questions: “what (the thing) is, why (she) is, is (she) and what (she) is”.
The basic principle that determines the entire structure of “scientific knowledge of the case” is the principle of reducing everything to the beginnings and reproducing everything from the beginnings. According to Aristotle, the proof is the universal process of the production of knowledge from the beginnings. “As a proof, I call the syllogism,” he writes, “which gives knowledge.” The exposition of the theory of evidentiary knowledge is entirely devoted to Aristotle’s Organon. The main provisions of this theory can be grouped into sections, each of which reveals one of the three main aspects of mathematics as a proving science: “what is being proved, what is being proved and what is being proved is based on.” Thus, Aristotle differentially approached the object, subject and means of proof.
The existence of mathematical objects was recognized long before Aristotle, however, the Pythagoreans, for example, assumed that they were in sensible things, while Platonists, on the contrary, considered them to exist separately.
According to Aristotle:
1. In sensual things, mathematical objects do not exist, since “two bodies are not in the same place in the same place”;
2. “It is also impossible for such a reality to exist in isolation.”
Aristotle considered the subject of mathematics “quantitative certainty and continuity.” In his interpretation, “quantity refers to what can be divided into its constituent parts, each of which … is something one that is present. This or that quantity is set, if it can be counted, it is a quantity if it can be measured “. At the same time, it is called “what is in a possibility (potentially) divided into parts not continuous, its magnitude – what is divided into parts continuous”.
Before giving a definition of continuity, Aristotle considers the concept of the infinite, since “it belongs to the category of quantity” and manifests itself primarily in the continuous. “That the infinite exists, confidence in this arises from researchers of five bases: from time (for it is infinite); from the separation of quantities …; further, only in this way will occur and destroy, if there is infinite, where does the arising occur. from the fact that the final always borders on something, since it is necessary that one always borders on the other.
But most of all – … on the basis that thinking does not stop: both the number seems endless and mathematical quantities. ” Does the infinite exist as a separate entity, or is it an accident of magnitude or set? Aristotle accepts the second option, since “if the infinite is neither a quantity nor a set, but is itself an entity …, then it will be indivisible, since the dividend will be either a quantity or a set. If it is not divisible, it is not infinite in the sense of impassable to the end. ” The impossibility of the mathematical infinite as indivisible follows from the fact that a mathematical object is a distraction from the physical body, and “the actually indivisible infinite body does not exist.
” The number “as something separate and at the same time infinite” does not exist, in fact “… if it is possible to recalculate the numeral, then it will be possible to go through to the end and infinite.” Thus, infinity exists here in potency, but not actually.
Based on the above understanding of the infinite, Aristotle defines continuity and discontinuity. Thus, “the continuous is itself something adjacent. The adjoining is that which, following the other, touches it.” A number as a typical discontinuous (discrete) formation is formed by the combination of discrete, then indivisible elements – units.
The geometric equivalent of a unit is a point; at the same time, the connection of points cannot form a line, since “the points from which a continuous would have been composed must either be continuous or touch each other.” But they will not be continuous: “after all, the edges of the points do not form anything single, since the indivisible has no edge or other part.” The points cannot touch each other, since they touch “all objects either as a whole, or with their parts, or as a whole of parts. But since the indivisible has no parts, it is necessary to touch them all, but those that relate entirely do not form a continuous”.
The impossibility of compiling the continuous of the indivisible and the need to divide it into ever-divisible parts established for the quantity, Aristotle extends to motion, space and time, justifying (for example, in “Physics”) the validity of this step. On the other hand, he concludes that the recognition of indivisible quantities is contrary to the basic properties of motion. The separation of continuous and discontinuous as different kinds of being was the basis for the delimitation in the logical-epistemological field, for the sharp separation of arithmetic from geometry.
“The beginnings … in each genus, I call something for which it cannot be proved that it is. Therefore, what the primary means and what follows from it is accepted. Existence began to be accepted, another should be proved. For example, such a unit or what is a straight line or what is a triangle (should be accepted); that the unit and the value exists should also be accepted, the other should prove “. On the question of the emergence in people of the capacity for knowledge of beginnings, Aristotle does not agree with Plato’s point of view about the innateness of such abilities, but also does not allow the possibility of acquiring them; here he proposes the following solution: “it is necessary to have some possibility, but not one that would exceed these capabilities in terms of accuracy.” But this possibility is obviously inherent in all living beings; in fact, they have an innate ability to understand, which is called sensory perception. The formation of the beginnings goes “from the preceding and more famous for us,” that is, from that which is closer to sensory perception to the “previous and more famous unconditionally” (this is the general). Aristotle gives a detailed classification of the beginnings, based on various signs.
First, he identifies “the beginnings from which (something) is proved, and those about which (is proved)”. The first “are common (all beginnings),” the second “are peculiar (only to a given science), for example, number, size”. In the system, the beginnings of the common take the leading place, but they are not enough, since “among the common beginnings there can not be those from whom one could prove everything.” This explains that among the beginnings there should be “some peculiar to each science separately, others – common to all.” Secondly, the beginnings are divided into two groups, depending on what they reveal: the existence of an object or the presence of certain properties. Third, the complex of beginnings of proving science is divided into axioms, assumptions, postulates, and initial definitions.
The choice of beginnings in Aristotle is the defining moment of building a proving science; It is precisely the beginnings that characterize science as given, distinguish it from a number of other sciences. “What is proved” can be interpreted very broadly. On the one hand, it is an elementary proving syllogism and its conclusions. The building of the proving science in the form of a separate theory is built from these elementary processes. Of these, science is also created as a system of theories. However, not every set of evidence forms a theory. To do this, it must meet certain requirements, covering both the content of the proved sentences and the connection between them. Within the bounds of a scientific theory, a number of auxiliary definitions are necessary, which are not primary but serve to uncover the subject of the theory.
Although questions of the methodology of mathematical knowledge were not set forth by Aristotle in any particular work, but in terms of content, they together form a complete system. The basis of the philosophy of mathematics of Aristotle is the understanding of mathematical knowledge as a reflection of the objective world. This installation played an important role in the struggle of Aristotle with Plato idealism; after all, “if in the phenomena of the sensory world is not at all mathematical, then how is it possible that its properties are attached to them?” he wrote. Of course, the materialism of Aristotle was inconsistent, in general, his views more closely corresponded to the needs of mathematical knowledge, to those of Plato. In turn, mathematics was for Aristotle one of the sources for the formation of a number of sections of his philosophical system.