8. Stakhov A., Aranson S., The fine-structure constant as the physical-mathematical millennium problem. Physical Science International Journal 9(1), (2016).
9. Stakhov A., Aranson S., Hilbert’s fourth problem as a possible candidate on the millennium problem in geometry. British Journal of Mathematics & Computer Science 12(4), (2016).
Chapter 1
The Golden Section: History and Applications
1.1. The Idea of the Universal Harmony in Ancient Greek Science
1.1.1. What is Harmony?
V.P. Shestakov, the author of the book Harmony as an Aesthetic Category [10], notes the following:
“In the history of aesthetic teachings, various types of understanding of harmony were put forward. The very concept of “harmony” was used extremely broadly and multivalently. It denoted both the natural structure of Nature and the Cosmos, and the beauty of the physical and moral world of man and the principles of the structure of the work of art, and the laws of aesthetic perception.”
Shestakov singles out three basic understandings of Harmony that evolved in the process of development of science and aesthetics:
(1) Mathematical understanding of Harmony or mathematical Harmony. In this sense, harmony is understood as the equality or proportionality of the parts with each other and the part with the whole. In the Great Soviet Encyclopedia, we find the following definition of Harmony, which expresses the mathematical understanding of harmony:
“Harmony is the proportionality of parts and the whole, the fusion of various components of an object into a single organic whole. Harmony is the outer revealing of inner order and the measure of existence.”
(2) Aesthetic harmony. Unlike the mathematical understanding, the aesthetic understanding is no longer just quantitative, but qualitative, expressing the inner nature of things. The aesthetic Harmony is associated with aesthetic experiences, with aesthetic evaluation. This type of harmony is most clearly manifested in the perception of the beauty of Nature.
(3) Artistic harmony. This type of Harmony is associated with art. The artistic Harmony is the actualization of the principle of Harmony in the material of art itself.
The most important aspect that follows from the above reasonings is the following: Harmony is a universal concept that has relation not only to mathematics and science but also to fine arts.
1.1.2. Numerical harmony of Pythagoreans
Pythagoras and Heraclitus were philosophers and thinkers, whose names are usually associated with the beginning of the philosophical doctrine of Harmony. According to many authors, the key idea of Harmony as a proportional unity of opposites belongs to Pythagoras. Pythagoreans first put forward the idea of harmonious construction of the whole world, including not only Nature and Man but also the entire cosmos. According to Pythagoreans, “harmony is an inner connection of things, without which the cosmos could not exist” [10]. Finally, according to Pythagoras, harmony has a numerical expression, that is, it is integrally connected with the concept of number.
Pythagoras (570–500 BC) is perhaps one of the most famous scientists in the history of science. He is revered by every person who studies geometry and is familiar with the “Pythagoras theorem”, one of the most famous theorems of geometry. In ancient literature, Pythagoras has been described by his contemporaries as a well-known philosopher and scholar, a religious and ethical reformer, an influential politician, a demigod in the eyes of his disciples and a charlatan. His popularity was such that during his lifetime, coins with his image were issued in 430–420 BC. For the fifth century BC, this was an unprecedented case! Pythagoras was the first Greek philosopher awarded to a special assay (Fig. 1.1).
Fig. 1.1. Pythagoras.
The important role of Pythagoras in the development of Greek science consists in the fact that he fulfilled a historical mission in transferring the knowledge of the Egyptian and Babylonian priests into the culture of Ancient Greece. It was thanks to Pythagoras, who undoubtedly was one of the most educated thinkers of his time, that Greek science received a huge amount of knowledge in the fields of philosophy, mathematics and natural sciences, which, by getting into the favorable environment of ancient Greek culture, contributed to its rapid development.
Pythagoreans created the doctrine of the creative essence of the number. Aristotle in “Metaphysics” notes this particular feature of the Pythagorean doctrine:
“The so-called Pythagoreans, having engaged in mathematical sciences, first had put forth them forward and after their study began to consider them the beginnings of all things . . . Since, therefore, everything else was explicitly compared to numbers throughout their essence, and numbers took first place in the whole of nature, they had recognized harmony and number as the basis of all things and all Universe.”
1.1.3. The contribution of Heraclitus to the development of the doctrine of Harmony
Starting from antiquity to the present day, Heraclitus remains one of the most popular philosophers in the history of philosophy. In 1961, on the recommendation of the World Peace Council, the 2500th anniversary of the birth of Heraclitus was celebrated. Such an anniversary is usually celebrated to commemmorate the history of some world-famous ancient cities or countries, but to do so for a person is rare and unusual.
Heraclitus believed that everything is constantly changing. The idea of the eternal motion was presented by Heraclitus in the bright image of the ever-flowing river (Fig. 1.2). The postulate on the universal variability of the world, one of the cornerstones of all dialectics, is compressed by Heraclitus in the famous formula: “It is impossible to enter twice into the same river.”
Fig. 1.2. Heraclitus.
As Shestakov points out [10], “in the aesthetics of Heraclitus, ontological understanding of harmony is at the forefront. Harmony is inherent, above all, the objective world of things, the cosmos itself what is inherent to the nature of art. It is characteristic that when Heraclitus wants to reveal the nature of harmony most clearly, he turns to fine arts. Best of all, Heraclitus illustrated the harmony of the Cosmos by the image of the lyre, in which the differently strained strings create a perfect harmony.”
But, in the aesthetics of Heraclitus, there is also a moment of evaluation. This is especially pronounced in the doctrine of two kinds of harmony: “hidden” and “obvious”. Heraclitus prefers the “hidden” Harmony. Widely known is the following saying of Heraclitus: “The hidden harmony is stronger than the obvious.”
Cosmos, as the highest and most perfect beauty, is an example of the Hidden Harmony. Only at first glance the cosmos seems to be a chaos.