Although each of the mentioned stages has its own specifics, at the same time, every stage necessarily includes the content of the preceding stages. This is called the continuity in the development of science.
It was during the ancient period, a number of the fundamental discoveries in mathematics were made. They exerted a determining influence on the development of the material and spiritual culture. We do not always realize their importance in the development of mathematics, science, and education. To the category of such discoveries, first of all, we must attribute the Babylonian numeral system with the base 60 and the Babylonian positional principle of number representation, which is the foundation of the, decimal, binary, ternary, and other positional numeral systems. We must add to this list the trigonometry and the Euclidean geometry, the incommensurable segments and the theory of irrationality, the golden section and Platonic solids, the elementary number theory and the mathematical theory of measurement, and so on.
The continuity can be realized in various forms. One of the essential forms of its expression are the fundamental scientific ideas, which permeate all stages of the scientific and technological progress and influence various areas of science, art, philosophy, and technology. The idea of Harmony, connected with the golden section, belongs to the category of such fundamental ideas.
According to B.G. Kuznetsov, the researcher of Albert Einstein’s creativity, the great physicist piously believed that science, physics in particular, always had its eternal fundamental goal “to find in the labyrinth of the observed facts the objective harmony”. The deep faith of the outstanding physicist in the existence of the universal laws of the Harmony is evidenced by another well-known Einstein’s statement: “The religiousness of the scientist consists in the enthusiastic admiration for the laws of the Harmony” (the quote is taken from the book Meta-language of Living Nature [1], written by the outstanding Russian architect Joseph Shevelev, known for his research in the field of Harmony and the golden section [1–3]).
Pythagoreanism and Pythagorean MATHEM’s
By studying the sources of the origin of mathematics, we inevitably come to Pythagoras and his doctrine, named the Pythagoreanism (see Wikipedia article Pythagoreanism, the Free Encyclopedia). As mentioned in Wikipedia, the Pythagoreanism originated in the 6th century BC and was based on teachings and beliefs of Pythagoras and his followers called the Pythagoreans. Pythagoras established the first Pythagorean community in Croton, Italy. The Early Pythagoreans espoused a rigorous life and strict rules on diet, clothing and behavior.
According to tradition, Pythagoreans were divided into two separate schools of thought: the mathematikoi (mathematicians) and the akousmatikoi (listeners). The listeners had developed the religious and ritual aspects of Pythagoreanism; the mathematicians studied the four Pythagorean MATHEMs: arithmetic, geometry, spherics, and harmonics. These MATHEMs, according to Pythagoras, were the main composite parts of mathematics. Unfortunately, the Pythagorean MATHEM of the harmonics was lost in mathematics during the process of its historical development.
Proclus Hypothesis
The Greek philosopher and mathematician Proclus Diadoch (412–485 AD) put forth the following unusual hypothesis concerning Euclid’s Elements. Among Proclus’s mathematical works, his Commentary on the Book I of Euclid’s Elements was the most well known. In the commentary, he puts forth the following unusual hypothesis.
It is well known that Euclid’s Elements consists of 13 books. In those, XIIIth book, that is, the concluding book of the Elements, was devoted to the description of the geometric theory of the five regular polyhedra, which had played a dominant role in Plato’s cosmology and is known in modern science under the name of the Platonic solids.
Proclus drew special attention to the fact that the concluding book of the Elements had been devoted to the Platonic solids. Usually, the most important material, of the scientific work is placed in its final part. Therefore, by placing Platonic solids in Book XIII, that is, in the concluding book of his Elements, Euclid clearly pointed out on main purpose of writing his Elements. As the prominent Belarusian philosopher Edward Soroko points out in [4], according to Proclus, Euclid “had created his Elements allegedly not for the purpose of describing geometry as such, but with purpose to give the complete systematized theory of constructing the five Platonic solids; in the same time Euclid described here some latest achievements of mathematics”.
It is for the solution of this problem (first of all, for the creation of geometric theory of dodecahedron), Euclid already in Book II introduces Proposition II.11, where he describes the task of dividing the segment in the extreme and mean ratio (the golden section), which then occurs in other books of the Elements, in particular in the concluding book (XIII Book).
But the Platonic solids in Plato’s cosmology expressed the Universal Harmony which was the main goal of the ancient Greeks science. With such consideration of the Proclus hypothesis, we come to the surprising conclusion, which is unexpected for many historians of mathematics. According to the Proclus hypothesis, it turns out that from Euclid’s Elements, two branches of mathematical sciences had originated: the Classical Mathematics, which included the Elements of the axiomatic approach (Euclidean axioms), the elementary number theory, and the theory of irrationalities, and the Mathematics of Harmony, which was based on the geometric “task of dividing the segment in the extreme and mean ratio” (the golden section) and also on the theory of the Platonic solids, described by Euclid in the concluding Book XIII of his Elements.
The Statements by Alexey Losev and Johannes Kepler
What was the main idea behind ancient Greek science? Most researchers give the following answer to this question: The idea of Harmony connected to the golden section. As it is known, in ancient Greek philosophy, Harmony was in opposition to the Chaos and meant the organization of the Universe, the Cosmos. The outstanding Russian philosopher Alexey Losev (1893–1988), the researcher in the aesthetics of the antiquity and the Renaissance, assesses the main achievements of the ancient Greeks in this field as follows [5]:
“From Plato’s point of view, and in general in the terms of the entire ancient cosmology, the Universe was determined as the certain proportional whole, which obeys to the law of the harmonic division, the golden section . . . The ancient Greek system of the cosmic proportion in the literature is often interpreted as the curious result of the unrestrained and wild imagination. In such explanation we see the scientific helplessness of those, who claim this. However, we can understand this historical and aesthetic phenomenon only in the connection with the holistic understanding of history, that is, by using the dialectical view on the culture and by searching for the answer in the peculiarities of the ancient social life.”
Here, Losev formulates the “golden” paradigm of ancient cosmology. This paradigm was based upon the fundamental ideas of ancient science that are sometimes treated in modern science as the “curious result of the unrestrained and wild imagination”. First of all, we are talking about the Pythagorean Doctrine of the Numerical Universal Harmony and Plato’s Cosmology based on the Platonic solids. By referring to the geometrical structure of the Cosmos and its mathematical relations, which express the Cosmic Harmony, the Pythagoreans had anticipated the modern mathematical basis of the natural sciences, which began to develop rapidly in the 20th century. Pythagoras’s and Plato’s ideas about the Cosmic Harmony proved to be immortal.
Thus,
