Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
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, the idea of Harmony, which underlies the ancient Greek doctrine of Nature, was the main “paradigm” of the Greek science, starting from Pythagoras and ending with Euclid. This paradigm relates directly to the golden section and the Platonic solids, which are the most important Greek geometric discoveries for the expression of the Universal Harmony.
Johannes Kepler (1571–1630), the prominent astronomer and the author of “Kepler’s laws”, expressed his admiration with the golden ratio in the following words [6]:
“Geometry has the two great treasures: the first of them is the theorem of Pythagoras; the second one is the division of the line in the extreme and mean ratio. The first one we may compare to the measure of the gold; the second one we may name the precious stone.”
We should recall again that the ancient task of dividing line segment in extreme and mean ratio is Euclidean language for the golden section!
The enormous interest in this problem in modern science is confirmed by the rather impressive and far from the complete list of books and articles on this subject, published in the second half of the 20th century and the beginning of the 21st century [1–100].
Ancient Greeks Mathematical Doctrine of Nature
According to the outstanding American historian of mathematics, Morris Kline [101], the main contribution of the ancient Greeks is the one “which had the decisive influence on the entire subsequent culture, was that they took up the study of the laws of Nature”. The main conclusion, from Morris Kline’s book [101] is the fact that the ancient Greeks proposed the innovative concept of the Cosmos, in which everything was subordinated to the mathematical laws. Then the following question arises: during which time this concept was developed? The answer to this question is also addressed in Ref. [101].
According to Kline [101], the innovative concept of the Cosmos based on the mathematical laws, was developed by the ancient Greeks in the period from VI to III centuries BC. But according to the prominent Russian mathematician academician A.N. Kolmogorov [102], in the same period in ancient Greece, “the mathematics was created as the independent science with the clear understanding of the uniqueness of its method and with the need for the systematic presentation of its basic concepts and proposals in the fairly general form.” But then, the following important question, concerning the history of the original mathematics arises: was there any relationship between the process of creating the mathematical theory of Nature, which was considered as the goal and the main achievement of ancient Greek science [101], and the process of creating mathematics, which happened in ancient Greece in the same period [102]? It turns out that such connection, of course, existed. Furthermore, it can be argued that these processes actually coincided, that is, the processes of the creation of mathematics by the ancient Greeks [102], and their doctrine of Nature, based on the mathematical principles [101], were one and the same processes. And the most vivid embodiment of the process of the Mathematization of Harmony [68] happened in Euclid’s Elements, which was written in the third century BC.
Introduction of the Term Mathematics of Harmony
In the late 20th century, to denote the mathematical doctrine of Nature, created by the ancient Greeks, the term Mathematics of Harmony was introduced. It should be noted that this term was chosen very successfully because it reflected the main idea of the ancient Greek science, the Harmonization of Mathematics [68]. For the first time, this term was introduced in the small article “Harmony of spheres”, placed in The Oxford Dictionary of Philosophy [103]. In this article, the concept of Mathematics of Harmony was associated with the Harmony of spheres, which was, also called in Latin as “harmonica mundi” or “musica mundana” [10]. The Harmony of spheres is the ancient and medieval doctrine on the musical and mathematical structure of the Cosmos, which goes back to the Pythagorean and Platonic philosophical traditions.
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