Subjects: Mathematics, Arithmetic, Geometry.
Keywords: mathematical inheritance, question of the relationship of mathematics, mathematical documents, knowledge of all secrets, mathematical knowledge of the Egyptians, number of workers, philosophical analysis, main figures.
The question of the relationship of mathematics and philosophy was first asked a long time ago. Aristotle, Bacon, Leonardo da Vinci – many great minds of mankind were engaged in this issue and achieved outstanding results. This is not surprising: after all, the basis of the interaction of philosophy with any of the sciences is the need to use the apparatus of philosophy to conduct research in this area; Mathematics, undoubtedly, is most of all among the exact sciences amenable to philosophical analysis (by virtue of its abstractness). Along with this, the progressing mathematization of science has an active influence on philosophical thinking.
The joint path of mathematics and philosophy began in ancient Greece around the 6th century BC. Not constrained by the framework of despotism, the Greek society of that time was like a nutrient solution, on which a great deal grew that reached us in a form that was greatly modified by time, but retaining the basic idea that the Greeks laid down: theater, poetry, dramaturgy, mathematics, philosophy. In this paper, I tried to follow the process of formation, development and mutual influence of mathematics and philosophy of ancient Greece, and also to bring different points of view on the driving forces and results of this process.
It is known that the Greek civilization at the initial stage of its development was repelled by the civilization of the ancient East. What was the mathematical inheritance received by the Greeks?
From the mathematical documents that have come down to us, we can conclude that in ancient Egypt there were strongly branches of mathematics connected with the solution of economic problems. Reind’s Papyrus (ca. 2000 BC) began with a promise to teach “a perfect and thorough study of all things, an understanding of their essences, a knowledge of all secrets.” In fact, the art of calculating with integers and fractions is described in which government officials were dedicated in order to be able to solve a wide range of practical tasks, such as the distribution of wages among a known number of workers, the calculation of the amount of grain for making such and such amount of bread, the calculation of surfaces and volumes, etc. Apparently, the Egyptians did not go further than the equations of the first degree and the simplest quadratic equations. The entire content of the Egyptian mathematics known to us convincingly testifies that the mathematical knowledge of the Egyptians was intended to meet the specific needs of material production and could not be seriously connected with philosophy.
Babylonian mathematics, like the Egyptian one, was brought to life by the needs of industrial activity since problems were tied to the needs of irrigation, construction, economic accounting, property relations, and the calculation of time. The surviving documents show that, based on the 60-number system, the Babylonians could perform four arithmetic operations, there were tables of square roots, cubes and cubic roots, sums of squares and cubes, degrees of a given number, the rules of summation of progressions were known. Remarkable results were obtained in the field of numerical algebra. Although the Babylonians did not know algebraic symbolism, the solution of problems was carried out according to plan, the tasks were reduced to a single “normal” form and then were solved by general rules, and the interpretation of the “equation” transformations was not associated with the specific nature of the source data. There were problems that came down to solving equations of the third degree and special types of equations of the fourth, fifth and sixth degrees.
If we compare the mathematical sciences of Egypt and Babylon by the way of thinking, then it will not be difficult to establish their commonality by such characteristics as authoritarianism, uncriticality, following the tradition, extremely slow evolution of knowledge. The same features are found in philosophy, mythology, and religion of the East. As E. Kohlman wrote about this, “in this place, where the will of the despot was considered the law, there was no place for thinking, searching for reasons and justifications for phenomena, much less for free discussion.”
Analysis of ancient Greek mathematics and philosophy should begin with the Milesian mathematical school, which laid the foundations of mathematics as an evidence-based science.
Milesian School Milesian School is one of the first ancient Greek mathematical schools, which had a significant impact on the development of philosophical ideas of the time. It existed in Ionia at the end of the 5th – 4th centuries. BC. ; its main figures were Thales (c. 624-547 BC), Anaximander (c. 610-546 BC) and Anaximenes (c. 585-525 BC). ). Considering the example of the Milesian school, the main differences between Greek science from pre-Greek and analyze them.
If one compares the original mathematical knowledge of the Greeks with the achievements of the Egyptians and Babylonians, one can hardly doubt that such elementary positions as the equality of angles at the base of an isosceles triangle, the discovery of which is attributed to Thales of Miletus, were not known to ancient mathematics. Nevertheless, the Greek mathematician already in its initial point had a qualitative difference from its predecessors.
Its originality consists primarily in an attempt to systematically use the idea of proof. Thales seeks to prove that empirically was obtained and without proper justification was used in Egyptian and Babylonian mathematics. It is possible that during the period of the most intensive development of the spiritual life of Babylon and Egypt, during the period of the formation of the foundations of their knowledge, the presentation of certain mathematical propositions was accompanied by substantiation in one form or another. However, as Van der Warden writes, “in the times of Thales, Egyptian and Babylonian mathematics were already dead knowledge. Thales could be shown how to calculate, but the course of reasoning underlying these rules was already unknown.”
The Greeks introduce the process of justification as a necessary component of mathematical reality, the evidence is indeed a distinctive feature of their mathematics. The technique of proving early Greek mathematics, both in geometry and arithmetic, was originally a simple attempt to give clarity. Specific types of such evidence in arithmetic were evidence with stones, in geometry, by means of imposing. But the fact that there is evidence suggests that mathematical knowledge is not perceived dogmatically but in the process of reflection. This, in turn, reveals a critical mindset, confidence (perhaps not always conscious) that by thinking it is possible to establish the correctness or falsity of the situation in question, confidence in the power of the human mind.
For one or two centuries, the Greeks were able to master the mathematical heritage of their predecessors, accumulated over thousands of years, which indicates the intensity and dynamism of their mathematical knowledge. The qualitative difference between the research of Thales and his followers from pre-Greek mathematics is manifested not so much in the specific content of the studied dependence, as in the new method of mathematical thinking. The source material was taken by the Greeks from their predecessors, but the method of assimilating and using this material was new. Distinctive features of their mathematical knowledge are rationalism, criticism, dynamism.
The same features are characteristic of philosophical studies of the Milesian school. The philosophical concept and set of mathematical positions are formed by means of a thinking process that is homogeneous in its general characteristics and qualitatively different from the thinking of the previous era. How did this new way of perception of reality form? Where does the desire for scientific knowledge come from?
A number of researchers declare the above-mentioned characteristics of the thought process “innate characteristics of the Greek spirit.” However, this link does not explain anything, since it is not clear why the same “Greek spirit” loses its qualities after the Hellenistic era. You can try to look for the reasons for this understanding of the world in the socio-economic sphere.
Ionia, where the activities of the Milesian school took place, was a fairly economically developed area. Therefore, it was she who, above all others, embarked on the path of overthrowing the primitive communal system and the formation of slaveholding relations. In the VIII-VI centuries. BC. the land was increasingly concentrated in the hands of the great clan aristocracy. The development of handicraft production and trade further accelerated the process of social and property stratification. The relationship between the aristocracy and the demos becomes strained; over time, this tension develops into an open struggle for power. A kaleidoscope of events in the inner life, a no less changeable external environment forms the dynamism, liveliness of social thought.
Tensions in the political and economic spheres lead to clashes in the field of religion, since the demos, without any doubt that religious and secular institutions are eternal, as they are given by the gods, require that they be recorded and made publicly available, because the rulers distort divine will and interpret it in their own way. However, it is not difficult to understand that the systematic presentation of religious and mythological ideas (an attempt to make such a statement was given by Hesiod) could not but deal a serious blow to religion. When checking religious fabrications by logic, the first would undoubtedly seem to be a conglomeration of absurdities.
“Thus, the materialistic worldview of Thales and his followers is not somehow mysterious, not a creature of the“ Greek spirit. ”It is the product of well-defined socio-economic conditions and expresses the interests of historically specific social forces, primarily trade and craft sections of society “- writes O. I. Kedrovsky.
Based on all of the above, it is still not possible to say with great certainty that it was the impact of the worldview that was the decisive factor for the emergence of evidence; It is possible that this happened due to other reasons: the needs of production, the demands of elements of natural science, the subjective motives of researchers. However, it can be verified that each of these reasons has not changed its fundamental character as compared with the pre-Greek era does not directly lead to the transformation of mathematics into evidential science. For example, to meet the needs of technology was quite enough practical science of the ancient East, in the validity of the provisions of which could be seen empirically. The very process of identifying these provisions has shown that they provide sufficient accuracy for practical needs.
It can be considered one of the motivations for the emergence of evidence of the need for understanding and summarizing the results of predecessors. However, this factor does not play a decisive role, since, for example, there are theories that we perceive as obvious, but have received a strict justification in ancient mathematics (for example, the theory of divisibility by 2).
The appearance of the need for proof in Greek mathematics receives a satisfactory explanation, given the interaction of the world view on the development of mathematics. In this regard, the Greeks are significantly different from their predecessors. In their philosophical and mathematical research, faith in the power of the human mind, a critical attitude to the achievements of predecessors, dynamism of thinking are manifested. Among the Greeks, the influence of the worldview turned from a deterrent factor of mathematical knowledge into a stimulating, effective force of the progress of mathematics.
In that the substantiation took exactly the form of evidence and did not stop at empirical verification, the appearance of a new, worldview function of science is decisive. Thales and his followers perceive the mathematical achievements of their predecessors primarily to meet technical needs, but science for them is more than an apparatus for solving production problems. Separate, the most abstract elements of mathematics are woven into the natural-philosophical system and here they serve as an antipode to mythological and religious beliefs. Empirical evidence for the elements of the philosophical system was insufficient due to their common character and scarcity of the facts confirming them. Mathematical knowledge, by that time, had reached such a level of development that it was possible to establish logical connections between individual provisions. This form of justification turned out to be objectively acceptable for mathematical positions.
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 different 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 regulations 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 a 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 is given a fundamental place in the worldview system, that is, in fact, mathematics is declared philosophy. As Aristotle wrote, “… among the numbers they saw, it would seem, many similarities with what exists and happens – more than fire, earth, and water … They, apparently, take the number beginning as a matter for things, and as an expression for their states and properties … For example, such a property of numbers is justice, and such and such is soul and mind, the other is good luck, and one can say – in each of the rest of the cases are exactly the same. “If we compare the mathematical studies of the early Pythagorean and Milesian schools, we can identify a number of GOVERNMENTAL 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 has changed. 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 at almost 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 he, like Thales, found the main signs of mental activity that are 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.” The viewpoint is not without foundation, if only because of the applicability of the mathematical propositions to express relations between different phenomena. On this basis, it is possible, improperly extending 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.
In Soviet philosophical science, the problem of the emergence of the Pythagorean concept of mathematics was considered, naturally, from the standpoint of Marxist-Leninist philosophy. So, O. I. Kedrovsky writes “… The concept developed by him (Pythagoras) objectively turned out to be an ideology of quite certain social strata. They were … representatives of the aristocracy, crammed by demos … They are characterized by a desire to escape from earthly life, appeal to religion and mysticism.
” This point of view, like the first, is not devoid of meaning; the truth is probably somewhere in the middle. However, in my opinion, the collapse of the Pythagorean doctrine should be associated primarily not with the degeneration of the aristocracy as a class, but with the attempt of the Pythagoreans to distort the very nature of the process of mathematical knowledge, depriving mathematics of such important sources of progress as production applications, open discussion of research results, collective creativity, to keep the progress of mathematics in the framework of the refined teaching for the initiated. By the way, the Pythagoreans themselves undermined their fundamental principle “the number is the essence of all things”, discovering that the ratio of the diagonal and the side of a square is not expressed by integers.
Thus, already in the starting point of its development, theoretical mathematics was influenced by the struggle of two types of materialist and religious-idealistic worldviews. We were convinced that along with the influence of the world view on the development of mathematical knowledge, there is a reverse effect.
The Eleatic school is quite interesting for research, since it is one of the oldest schools, in the works of which mathematics and philosophy interact quite closely and diversified. The main representatives of the Eleatic school are Parmenides (end of the 6th – 5th centuries BC) and Zeno (the first half of the 5th century BC).
The philosophy of Parmenides is as follows: all sorts of systems of world outlook are based on one of three premises:
1) There is only being, there is no non-being;
2) There is not only being but also non-being;
3) Being and non-being are identical. True Parmenides recognizes only the first parcel. According to him, being is one, indivisible, unchangeable, timeless, complete in itself, only it is the true being; multiplicity, variability, discontinuity, fluidity – all this is a lot of imaginary.
With the defense of the teachings of Parmenides from objections made by his student Zeno. The ancients attributed to him forty evidence to defend the doctrine of the unity of things (against the multiplicity of things) and five proofs of his immobility (against the movement). Of these, only nine reached us. The greatest fame at all times enjoyed the Zenon evidence against the movement; for example, “movement does not exist on the grounds that a moving body must first reach half than the end, and to reach half, half of this half must be passed, etc.”
Arguments of Zeno lead to paradoxical, from the point of view of “common sense”, conclusions, but they could not be simply discarded as untenable, because both in form and content they met the mathematical standards of that time. By decomposing the aporia of Zeno into its component parts and moving from conclusions to parcels, it is possible to reconstruct the initial positions, which he took as the basis of his concept. It is important to note that in the concept of the Eleatics, as in the Dozenon science, fundamental philosophical ideas were essentially based on mathematical principles. Prominent among them were the following axioms:
1. The sum of an infinitely large number of any, albeit infinitely small, but extended values should be infinitely large;
2. The sum of any, albeit an infinitely large number of unextended quantities, is always zero and can never become some predetermined extended value.
It is precise because of the close interrelation of general philosophical ideas with fundamental mathematical propositions that the blow struck by Zeno on philosophical views substantially affected the system of mathematical knowledge. A number of the most important mathematical constructions that were previously considered undoubtedly true in the light of Zeno’s constructions looked like contradictory. Zeno’s reasoning led to the need to rethink such important methodological issues as the nature of infinity, the relationship between continuous and discontinuous, etc. They drew the attention of mathematicians to the fragility of the foundation of their scientific activity and thus had a stimulating effect on the progress of this science.
Attention should also be paid to feedback – to the role of mathematics in the formation of Eleatic philosophy. Thus, it was established that the aporias of Zeno are associated with finding the sum of an infinite geometric progression. On this basis, the Soviet historian of mathematics, E. Kohlman, made the assumption that “it was on the mathematical basis of the summation of such progressions that the logical and philosophical aporia of Zeno grew.” However, this assumption seems to be devoid of sufficient grounds, since it too tightly links the teachings of Zeno with mathematics, while having historical data do not give grounds for asserting that Zeno was a mathematician in general.
Of great importance for the subsequent development of mathematics was an increase in the level of abstraction of mathematical knowledge, which was largely due to the activity of the Eleatic. A specific form of manifestation of this process was the emergence of indirect evidence (“by contradiction”), a characteristic feature of which is the proof not of the statement itself, but of the absurdity of the opposition to it. Thus, a step was taken towards the development of mathematics as a deductive science, and certain prerequisites were created for its axiomatic construction.
So, the philosophical arguments of the Eleatics, on the one hand, were a powerful impetus for a fundamentally new formulation of the most important methodological questions of mathematics, and on the other hand, were the source of a qualitatively new form of substantiation of mathematical knowledge.
Arguments Zeno revealed internal contradictions that have occurred in the prevailing mathematical theories. Thus, the existence of mathematics was questioned. In what ways were the contradictions revealed by Zeno resolved?
The simplest way out of this situation is the rejection of abstractions in favor of what can be directly verified with the help of sensations. This position was taken by the sophist Protagoras. He believed that “we can not imagine anything direct or round in the sense that these terms represent geometry; in fact, the circle touches a straight line at more than one point.” Thus, mathematics should be removed as surreal: ideas about an infinite number of things, since no one can count to infinity, infinite divisibility, since it is impracticable in practice, etc. In this way, mathematics can be made invulnerable to the reasoning of Zeno, but theoretical mathematics is practically eliminated. It was much more difficult to build a system of fundamental propositions of mathematics in which the contradictions revealed by Zeno would not exist. This problem was solved by Democritus, having developed the concept of mathematical atomism.
Democritus bal, according to Marx, “the first encyclopedic mind among the Greeks.” Diogenes Laertius (III c. AD) names 7O of his writings in which questions of philosophy, logic, mathematics, cosmology, physics, biology, social life, psychology, ethics, pedagogy, philology, art, technology, and others were covered. Aristotle wrote about him: “In general, except for superficial research, no one has established anything, except for Democritus. As for him, it seems that he provided for everything, and in the method of calculation it compares favorably with others.”
The introductory part of the scientific system of Democritus was the “canon”, in which the principles of atomistic philosophy were formulated and substantiated. Then followed physics, like the science of various manifestations of being, and ethics. The canon was part of physics as a starting point, while ethics was built as a product of physics. In the philosophy of Democritus, first of all, a distinction is established between the “truly existing” and the fact that it exists only in the “general opinion”. Only atoms and emptiness were considered truly real. As a true being, emptiness (non-being) is the same reality as atoms (being).
The “great emptiness” is limitless and encompasses everything that exists, it has neither top, nor bottom, nor edge, nor center, it makes matter discontinuous and its movement possible. Genesis consists of countless tiny qualitatively homogeneous first-ranks, differing from each other in external forms, size, position, and order, they are further indivisible due to absolute hardness and the absence of emptiness in them and “in size indivisible.” Atoms themselves inherently incessant motion, the diversity of which is determined by the infinite variety of forms of atoms. The movement of atoms forever and ultimately is the cause of all changes in the world.
The task of scientific knowledge, according to Democritus, is to reduce the observed phenomena to the area of “true being” and give them an explanation based on the general principles of atomistic. This can be achieved through joint activity of sensations and intelligence. Marx formulated the gnoseological position of Democritus as follows: “Democritus was not only not removed from the world, but, on the contrary, was an empirical naturalist.” The content of the initial philosophical principles and epistemological attitudes determined the main features of the scientific method of Democritus: a) In knowledge, it comes from the individual; b) Any object and phenomenon is decomposable into the simplest elements (analysis) and can be explained on the basis of them (synthesis); c) distinguish existence “in truth” and “according to opinion”; d) Phenomena of reality are separate fragments of an ordered cosmos that arose and functions as a result of actions of purely mechanical causality.
Mathematics must rightly be considered by Democritus as the first section of physics proper and follow immediately after canon. In fact, atoms are qualitatively homogeneous and their primary properties are quantitative. However, it would be wrong to interpret the teaching of Democritus as a kind of Pythagoreanism, since Democritus, although he retains the idea of dominance in the world of mathematical regularity, but criticizes a priori mathematical constructions of the Pythagoreans, believing that the number should not act as a legislator of nature, but be extracted from it. Mathematical regularity is revealed by Democritus from the phenomena of reality, and in this sense, he anticipates the ideas of mathematical natural science.
The basic principles of material existence appear in Democritus to a large extent as mathematical objects, and in accordance with this, mathematics is given a prominent place in the worldview system as a science about the primary properties of things. However, the inclusion of mathematics in the basis of the ideological system required its restructuring, bringing mathematics in accordance with the initial philosophical propositions, with logic, gnoseology, the methodology of scientific research. The concept of mathematics created in this way called the concept of mathematical atomism, turned out to be significantly different from the previous ones.
In Democritus, all mathematical objects (bodies, planes, lines, points) appear in certain material images. Ideal planes, lines, points in his teaching are absent. The main procedure of mathematical atomism is the decomposition of geometric bodies into the thinnest leaves (planes), planes into the thinnest threads (lines), lines into the smallest grains (atoms). Each atom has a small but non-zero value and is further indivisible. Now the line length is defined as the sum of the indivisible particles contained in it. Similarly, the question of the relationship of lines on the plane and the planes in the body. The number of atoms in the finite volume of space is not infinite, although it is so large that it is inaccessible to the senses.
So, the main difference of the teachings of Democritus from those considered earlier is his denial of infinite divisibility. Thus, he solves the problem of the legitimacy of the theoretical constructs of mathematics, without reducing them to sensually perceived images, as Protagoras did. So, to the arguments of Protagoras about touching a circle and a straight line, Democritus could answer that feeling, which are the starting criterion of Protagoras, show him that the more accurate the drawing, the smaller the contact area; in reality, this area is so small that it is not amenable to sensual analysis, but relates to the field of true cognition.
Guided by the provisions of mathematical atomism, Democritus conducts a series of specific mathematical research and achieves outstanding results (for example, the theory of mathematical perspective and projection). In addition, he played, according to Archimedes, an important role in proving Euddox’s theorems on the volume of a cone and pyramid. It is impossible to say with certainty whether he used the methods of analysis of infinitely small in solving this problem. A.O. Makovelsky writes: “Democritus embarked on the path that Archimedes and Cavalieri went on. However, coming close to the concept of infinitely small, Democritus did not take the last decisive step. He does not allow an unlimited increase in the number of components that form in its sum volume. It takes only an extremely large, incalculable number because of its vastness, the number of these terms. ”
An outstanding achievement of Democritus in mathematics was also his idea of constructing theoretical mathematics as a system. In the germinal form, it represents the idea of the axiomatic construction of mathematics, which was then developed in methodological terms by Plato and obtained a logically expanded position from Aristotle.
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 present 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 assumed “to approve by law and convince those who intend to occupy high positions in the city, 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 is incorporeal, insensitive 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 that some mathematical truths are similar to each other, but there is only one idea each time.
In Plato, as the matter, the beginnings are big and small, and as the essence, they are 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, the 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 the truth is contemplated by them alone.” “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, in addition to 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 wrong to resort to sensual evaluation.
Thus, in the historically developed 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, “taking advantage of the assumptions, leave them in stillness and cannot give them grounds,” the assumptions find their foundations by means of 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 an 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 a lot of Plato’s disciple – Aristotle.
SYSTEM OF PHILOSOPHY OF MATHEMATICS OF ARISTOTLE
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 of how to master science, of 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 is 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 (thing) is there, why (she) is, is there (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 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, but the Pythagoreans, for example, assumed that they were insensible things, while Platonists, on the contrary, considered them to exist separately. According to Aristotle:
1. Insensible things, mathematical objects do not exist, since “two bodies are not able to be 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 the 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 a time (for it is infinite); from the separation of quantities …; further, only in this way will occur and destroy, if it is infinite, where does the arising occur. from the fact that the finite 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 the 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 another 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 entirely, 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 isolation of the continuous and intermittent as different kinds of life was the basis for the delimitation in the logical-epistemological field, for the sharp separation of arithmetic from geometry.
“The beginnings … in every genus I call something for which it cannot be proved that it exists. Consequently, what the primary means and what follows from it is accepted. Existence began to be accepted, another should be proved. For example, what unit or what is direct or what is a triangle (should be accepted); that unit and value exist should also be accepted, the other – to prove “. On the question of the emergence in people of the ability of knowledge of beginnings, Aristotle disagrees 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 “is 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 general beginnings there cannot 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, 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.