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 understanding of the world 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. The following axioms were prominent among them: 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 precisely 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 opposite 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.