Introduction
What are “Paradigm” and “Scientific Revolution”?
As it is known, the term paradigm is derived from the Greek word paradeigma (example, sample) and refers to a combination of explicit and implicit (and often not realized) prerequisites that define the main essence of scientific research at some stage of scientific development.
This concept, in the modern sense of this term, was introduced by the American physicist and historian of science Thomas Kuhn (1922–1996) in the 1962 book The Structure of Scientific Revolutions [139]. According to Thomas Kuhn, a paradigm means a set of fundamental scientific ideas, which unite members of the scientific community and, conversely, the scientific community consists of people, who recognize the certain paradigm. As a rule, the paradigm is fixed in the textbooks and works of scientists and over the years determines the circle of problems and methods of their solution in a particular field of science. According to Kuhn [139], the examples of paradigms are Aristotle’s views on education and ethics, Newtonian mechanics, etc.
A paradigm shift is a term also first introduced by Thomas Kuhn [139] for the description of changes in the basic assumptions within the framework of the leading theory in science (paradigm). Usually, a change of the scientific paradigm relates to the most dramatic events in the history of science. When a scientific discipline changes one paradigm for another, this is called the scientific revolution or paradigm shift, according to Kuhn’s terminology [139]. The decision to abandon the old paradigm is always at the same time the decision to adopt the new paradigm; the proposal, which leads to such a decision, includes both a comparison of both paradigms with Nature and a comparison of the paradigms with each other.
What is the “Golden” Paradigm?
To answer this question, let us turn once again to the well-known statement by the genius of Russian philosophy, the aesthetics researcher of ancient Greece and the Renaissance, Alexei Losev (1893–1988), which is given in the Preface. In this statement, Losev in a very distinct form had formulated the essence of the “golden” paradigm of the ancient cosmology [5]:
“From Plato’s point of view, and in general from the point of view of all the ancient cosmology, the world is some proportional Whole, obeying the law of harmonic division, the golden section (that is, the whole relates to the larger part, as the larger part to the smaller one).”
In Losev’s well-known statement [5], the essence of the “golden” paradigm of the ancient cosmology is formulated as follows. The “golden” paradigm is based on the most important ideas of ancient science, which in modern science are sometimes interpreted as a curious result of unrestrained and wild fantasy [5]. First of all, these are the Pythagorean doctrine of the numerical harmony of the Universe and Plato’s Cosmology, based on the golden section and Platonic solids.
It is important to emphasize that Losev put the golden section in the center of the “golden” paradigm of ancient science. Thus, by referring to the geometric relations and geometric concepts, which expressed the Universal Harmony, in particular, the golden section and Platonic solids, Plato, along with Pythagoras, anticipated the emergence of mathematical natural sciences, which began developing rapidly in the 20th century. The idea of Pythagoras and Plato about the Universe Harmony proved to be immortal.
The Relationship Between Scientific Paradigms in Mathematics and Mathematical Natural Sciences
One of the original contemporary ideas, expressed in the article “Pseudoscience: a disease that there is no one to cure”, written in 2011 by the talented Russian philosopher Denis Kleschev [128], is the fact that the processes of paradigm shift in mathematics and natural sciences are closely interrelated.
Kleschev notes as follows in [128]:
“Studying a history only for the sake of studying the history itself can hardly attract the attention of other researchers to it. Therefore, Kuhn’s concept must be supplemented by consideration of both the internal and external structures of the change of scientific paradigms. To cope with this task is impossible if we are interested in the natural sciences in isolation from the study of the history of mathematics, as practiced by Thomas Kuhn. But if we include into the consideration the history of mathematics, rich with dramatic events and crises, as it immediately becomes apparent that to every paradigm leap in physics was preceded by cardinal changes in mathematics, preparing the ground for changing the natural science paradigm.”
The examples of the successful usage of Platonic solids, the golden section, and Fibonacci numbers in modern theoretical natural sciences, considered in Vols. I and II, allow expression of the idea that the process of harmonization of natural sciences has been realized actively in modern theoretical natural sciences. Fullerenes and quasi-crystals, awarded by the Nobel Prizes, are the most prominent examples of such harmonization, and this process requires a corresponding response from mathematics.
The development of modern Fibonacci numbers theory [7–9, 11] is a convincing example of harmonization of mathematics. This process obtained further reflection in Stakhov’s book The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science [6] as a new interdisciplinary direction of modern science and mathematics.
Mathematization of Harmony and Harmonization of Mathematics
By considering the history of the development of mathematics since the ancient Greeks to the present time, we can distinguish the two processes that are closely related to each other, despite the more than 2000-year time distance between them. This connection is carried out through the “golden” paradigm of ancient Greeks as a fundamental conception that permeates the entire history of science. The first of these is the process of Mathematization of Harmony. This process began developing in ancient Greece in the sixth or fifth century BC (Pythagoras and Plato’s mathematics) and ended in the third century BC by creating the greatest mathematical work of the ancient era, the Euclidean Elements. All efforts of the ancient Greeks were aimed at creating the mathematical doctrine of Nature, in the center of which the ancient Greeks placed the Idea of Harmony, which, according to Proclus hypothesis, had been expressed in the Euclidean Elements through Platonic solids (XIIIth Book of the Elements) and the golden section (Book II, Proposition of II.11).
The process of Mathematization of Harmony in the ancient period ended with the creation of Euclidean Elements; the main purpose of this process was the creation of the complete geometric theory of Platonic solids (Book XIII of the Elements), which expressed the Universal Harmony in Plato’s cosmology. To create this theory, Euclid already in Book II introduced the task of dividing a segment in extreme and mean ratio (the Euclidian name for the golden section), which was used by Euclid by creating the geometric theory of the dodecahedron, based on the golden ratio.
Harmonization of Mathematics is a process opposite to Mathematization of Harmony [68]. This process began developing most rapidly in the second half of the 20th century in the works of the Canadian geometer Harold Coxeter [7], the Soviet mathematician Nikolay Vorobyov [8], the American mathematician Verner Hoggatt [9], the English mathematician Stefan Waida [11], and other famous Fibonacci mathematicians.
The creators of the modern Fibonacci number theory [7–9, 11] have acted very wisely and cautiously, not attracting attention to the fact that Fibonacci numbers are one of the most important numerical sequences, which together with the golden section actually express Harmony of Nature. They “euthanized” the vigilance of modern orthodox mathematicians, which allowed them to establish the Fibonacci Association, the mathematical journal The Fibonacci Quarterly and, starting from 1984, regularly (once every 2 years) holding the International Conference on Fibonacci Numbers and their Applications. Thanks to the active work of the Fibonacci Association,