The Mathematics of Harmony. Proclus’ Hypothesis and New View on Euclid’s Elements and History of Mathematics Starting since Euclid

We are discussing one of the most unlikely hypotheses in the history of mathematics—Proclus’ hypothesis, which overturns a traditional view on Euclid’s Elements and the history of mathematics, starting since Euclid. According to Proclus, the main goal of Euclid, when writing the Elements, was to build a complete geometric theory of Platonic solids (Book XIII), associated in the ancient philosophy (Pythagoras, Plato) with the Universe harmony. To construct this theory, Euclid introduced in Book II the problem of dividing a segment into extreme and mean ratio (the “golden section”). It follows from Proclus’ hypothesis that Euclid’s Elements are the first attempt to create the “Mathematical Theory of the Universe Harmony”, based on Platonic solids and the “golden section”.

Proclus＇ Ontological Arguments concerning the Objective Existence of the Forms

In his commentary on the Platonic Parmenides, the Neoplatonist philosopher Proclus （ca. 411-485） offers a systematic defense of the objective existence of the Forms by means of the use of distinctive arguments, four of which are ontological. In contrast with Aristotle, Proclus clearly accepts that the Forms exist as principles of the world and not as posterior concepts describing the common elements of the sensible entities. Thus, he argues first that the ontological presuppositions of the sensible beings are placed in the area of the self-subsistent beings, which correspond to the categories of the Second Hypothesis of the Parmenides. The Forms belong to this area as well. In this respect, Proclus refers to the way of production of the sensible beings, aiming at a strict definition of this process. Secondly, he proves that the existence of the Forms, which possess ontological completeness, is prior to the existence of the sensible things. This is possible since the Forms are generated by the Demiurge of the whole world, by means of an internal reflection upon himself. Thirdly, in order to prove that the sensible world cannot be attributed to mere chance, he argues that the external activity, i.e., the creation of the world, depends on the internal activity, i.e., the development of the Forms in the divine intellect. Fourthly, he implements the concept of hierarchy in the realm of the Forms, through an extensive reference to the ontological priority of the imparticipable over the participable. Thus, he shows how the existence of the primary and intelligible Forms is prior to that of the substantiated Forms. Interestingly enough, the above-mentioned arguments exercised considerable influence on the commentaries of the members of the school of Ammonius, son of Hermias.

On the Evolution of Approaches to the Space-Time Symmetry

The general space-time composition was analyzed starting in ancient times. During the previous three millenniums, to explain elementary physical phenomena, the world was twice admitted to be immersed into an “aether”. Presently the “aether” is assumed to be asymptotically close to the vacuum. This approximation is included in the foundation of the special theory of relativity.

The Scientific Concept of Later Neoplatonism

In the later Neoplatonism,natural science is fundamentally a theology,its origin and rationality of the basis,direction and final destination are gods.This concept represents the philosophical reflection of science by the Greeks,which inevitably has the limitations of the times,but it may provide a kind of enlightenment for today’s“modernity problem”and help us to understand science more deeply.