first glance, it seems that the answer to this question is very simple: Euclid’s main goal was to set forth the main achievements of Greek mathematics during 300 years, preceding Euclid, by using the “axiomatic method”. Indeed, Euclid’s Elements is the main work in Greek science, devoted to the axiomatic description of geometry and mathematics. This view on Elements is the most common one in modern mathematics.
However, besides the “axiomatic” point of view, there is another point of view on the motives that guided Euclid in writing the Elements. This point of view was described by the Greek philosopher and mathematician Proclus Diadoch (412–485 AD), one of the best commentators of Euclid’s Elements.
First of all, a few words about Proclus (Fig. 1.8). Proclus was born in Byzantium in the family of a wealthy lawyer from Lycia. Intending to follow in the footsteps of his father, he left for Alexandria in his teens, where he studied at first rhetoric, then became interested in philosophy and became a disciple of the Neoplatonist Olympiodorus the Younger. It was here that Proclus began to study the logical treatises of Aristotle. At the age of 20, Proclus returned to Athens, where Plutarch of Athens headed the Platonic Academy. At the age of 28, Proclus wrote one of his most important works, Commentary on Plato’s “Timaeus”. About 450 AD, Proclus becomes the Head of the Platonic Academy.
Among Proclus’ mathematical works, the most famous is the Commentary on the first Book of Euclid’s Elements. In this commentary, he puts forward the following unusual hypothesis, which is called the Proclus hypothesis. Its essence is as follows. As we know, Book XIII, that is, the final book of the Elements, is devoted to the presentation of the theory of the five regular polyhedra, which played a dominant role in Plato’s cosmology and in modern science are known under the name of the Platonic solids. Proclus draws particular attention to this circumstance. As Edward Soroko points out in [4], according to Proclus, Euclid created the Elements allegedly not for the purpose of expounding geometry as such, but to give a complete systematized theory of constructing the five Platonic solids; in addition, he described here some of the latest achievements of Greek mathematics.”
Fig. 1.8. Proclus.
1.3.2. The significance of the Proclus hypothesis for the development of mathematics
The main conclusion from the Proclus hypothesis consists in the fact that Euclid’s Elements, the greatest Greek mathematical work, was written by Euclid under the direct influence of the Greek “idea of Harmony”, which was associated with the Platonic solids.
Thus, the Proclus hypothesis makes it possible to suggest that the well-known doctrines in ancient science “Pythagorean doctrine of the numerical harmony of the Universe” and Plato’s cosmology, based on regular polyhedra, were embodied in Euclid’s Elements, the greatest mathematical work of the ancient Greek mathematics. From this point of view, we can consider Euclid’s Elements as the first attempt in creating the Mathematical Theory of the Universal Harmony, which was associated in ancient science with the Platonic solids. This was the main idea of the ancient Greek science! This is the main secret of Euclid’s Elements, which leads to the revision of the history of the origin of mathematics, starting since Euclid.
Unfortunately, the original Proclus hypothesis, concerning the true Euclid’s goals in writing the Elements, was ignored by modern historians of mathematics, which led to a distorted view of the structure of mathematics and the whole mathematical education. This is one of the main “strategic mistakes” in the development of mathematics.
The Proclus hypothesis had a great influence on the development of science and mathematics. In the 17th century, Johannes Kepler, by developing Euclid’s ideas, built the Cosmic Cup, the original model of the Solar system, based on the Platonic solids.
In the 19th century, the eminent mathematician Felix Klein (1849–1925) proposed that the icosahedron, one of the most beautiful Platonic solids, is the main geometric figure of mathematics, which makes it possible to unite all the most important branches of mathematics: geometry, Galois theory, group theory, invariant theory, and differential equations [113]. Klein’s idea did not get further support in the development of mathematics, which can also be considered as another “strategic mistake”.
1.3.3. The Proclus hypothesis and “key” problems of ancient mathematics
As it is well known, the Russian mathematician academician Kolmogorov in his book [102] singled out two “key” problems, which stimulated the development of mathematics at the stage of its origin, the problems of counting and measurement. However, another “key” problem, which emerged from the Proclus hypothesis is the harmony problem which was connected with Platonic solids and the golden section, one of the most important mathematical discoveries of ancient mathematics (Proposition II.11 of Euclid’s Elements). It was this problem that Euclid laid in the foundation of Elements. According to Proclus, the main purpose of Elements was to create the geometric theory of the Platonic solids, which in Plato’s cosmology was expressed as Universal Harmony. This idea leads to a new view on the history of mathematics, as presented in Fig. 1.9.
The approach, shown in Fig. 1.9, is based on the following reasoning. Already, at the stage of origin of mathematics, a number of important mathematical discoveries were obtained; they fundamentally influenced the development of mathematics and all science on the whole. The most important of them are as follows:
(1) The positional principle of numbers representation. This was discovered by Babylonian mathematicians in the second millennium BC and embodied by them in the Babylonian positional numeral system with the base of 60. This important mathematical discovery underlies all subsequent positional numeral systems, in particular, the decimal system, the basis of mathematical education, and the binary system, the basis of modern computers. This discovery, ultimately, led to the formation of the concept of natural numbers
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