an urgent need to return to the “harmonious ideas” of Pythagoras, Plato and Euclid that permeated across the ancient Greek science and culture. The Harmony idea, formulated in the works of the Greek scholars and reflected in Euclid’s Elements turned out to be immortal!
We can safely say that the above-mentioned books by Stakhov and Aranson (2009, 2016, 2017) [6, 51–53], the book by The Prince of Wales with the coauthors (2010) [49] and book by Arakelian (2014) [50] are the beginning of a revolution in modern science. The essence of this revolution consists, in turning to the fundamental ancient Greek idea of the Universal Harmony, which can save our Earth and humanity from the approaching threat of the destruction of all mankind.
It was this circumstance that led the author to the idea of writing the three-volume book Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science, in which the most significant and fundamental scientific results and ideas, formulated by the author and other authors (The Prince of Wales, Hrant Arakelian, Samuil Aranson and others) in the process of the development of this scientific direction, will be presented in a popular form, accessible to students of universities and colleges and teachers of mathematics, computer science, theoretical physics and other scientific disciplines.
Structure and the Main Goal of the Three-Volume Book
The book consists of three volumes:
•Volume I. The Golden Section, Fibonacci Numbers, Pascal Triangle and Platonic Solids.
•Volume II. Algorithmic Measurement Theory, Fibonacci and Golden Arithmetic and Ternary Mirror-Symmetrical Arithmetic.
•Volume III. The “Golden” Paradigm of Modern Science: Prerequisite for the “Golden” Revolution in the Mathematics, the Computer Science, and Theoretical Natural Sciences.
Because the Mathematics of Harmony goes back to the “harmonic ideas” of Pythagoras, Plato and Euclid, the publication of such a three-volume book will promote the introduction of these “harmonic ideas” into modern education, which is important for more in-depth understanding of the ancient conception of the Universal Harmony (as the main conception of ancient Greek science) and its effective applications in modern mathematics, science and education.
The main goal of the book is to draw the attention of the broad scientific community and pedagogical circles to the Mathematics of Harmony, which is a new kind of elementary mathematics and goes back to Euclid’s Elements. The book is of interest for the modern mathematical education and can be considered as the “golden” paradigm of modern science on the whole.
The book is written in a popular form and is intended for a wide range of readers, including schoolchildren, school teachers, students of colleges and universities and their teachers, and also scientists of various specializations, who are interested in the history of mathematics, Platonic solids, golden section, Fibonacci numbers and their applications in modern science.
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
