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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, it was possible to combine the efforts of a huge number of researchers, who found the Fibonacci numbers and the golden ratio in their scientific areas. Starting from the last decade of the 20th century, the so-called Slavic Golden Group began playing an active role in the development of this direction. The Slavic Golden Group was established in Kiev (the capital of Ukraine) in 1992 during the First International Workshop Golden Proportion and Problems of Harmony Systems. This scientific group included leading scientists and lovers of the golden ratio and Fibonacci numbers from Ukraine, Russia, Belarus, Poland, Armenia and other countries.

In 2003, according to the initiative of the Slavic Golden Group, the International Conference on Problems of Harmony, Symmetry and the Golden Section in Nature, Science and Art was held at Vinnitsa Agrarian University by the initiative of Professor Alexey Stakhov. According to the decision of the conference, the Slavic Golden Group was transformed into the International Club of the Golden Section.

In 2005, the Golden Section Institute was organized at the Academy of Trinitarism (Russia). In 2010, according to the initiative of the International Club of the Golden Section, the First International Congress on Mathematics of Harmony was held on the basis of the Odessa Mechnikov National University (Ukraine). All these provide evidence of the fact that the International Club of the Golden Section plays in the Russian-speaking scientific community the same role as the American Fibonacci Association in the English-speaking scientific community. The publication of Stakhov’s book The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science (World Scientific, 2009) [6] was an important event in harmonization of modern science and mathematics.

What is Harmonization of Mathematics?

This, first of all, refers to the wide use of fundamental concept of Mathematics of Harmony, such as the Platonic Solids, the golden proportion, the Fibonacci numbers and their generalizations (the Fibonacci p-numbers, the metallic proportions or the “golden” p-proportions, etc.), as well as new mathematical concepts (the Fibonacci matrices, the “golden” matrices, the hyperbolic Fibonacci and Lucas functions [64, 75], etc.) to solve certain mathematical problems and create new mathematical theories and models.

The brilliant examples are the solution of Hilbert’s 10th Problem (Yuri Matiyasevich, 1970), based on the use of new mathematical properties of the Fibonacci numbers, and the solution of Hilbert’s Fourth problem (Alexey Stakhov and Samuil Aranson), based on the use of Spinadel’s metallic proportions. The theory of numeral systems with irrational bases (Bergman’s system and the codes of the golden ratio) and the concept of the “golden” number theory, arising from them, are examples of the original and far from trivial mathematical results, obtained in the framework of Mathematics of Harmony [6].

The main merit of the modern mathematicians in the field of golden ratio and Fibonacci numbers consisted in the fact that their researches caused the spark, from which the flame had ignited. The process of Harmonization of Mathematics is confirmed by a rather impressive and far from complete list of modern books in this field, published in the second half of the 20th century and early 21st century [1–53].

Among them, the following three books, published in the 21st century, deserve special attention:

(1)Stakhov Alexey. Assisted by Scott Olsen. The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science (World Scientific, 2009) [6]

(2)The Prince of Wales with co-authors. Harmony. A New Way of Looking at our World (New York: Harpert Collins Publishers, 2010) [51]

(3)Arakelyan Hrant. Mathematics and History of the Golden Section (Moscow: Logos, 2014) [50] (Russian).

What Place Does Mathematics of Harmony Occupy in the System of Modern Mathematical Theories?

To answer this question, it is appropriate to consider a quote from the review of the prominent Ukrainian mathematician, academician Yuri Mitropolskiy on the scientific research of the Ukrainian scientist Professor Alexey Stakhov. In this review, Yuri Mitropolskiy reports the following:

“I have followed the scientific career of Professor Stakhov for a long time — seemingly since the publication of his first 1977 book, “Introduction into Algorithmic Measurement Theory”, which was presented by Professor Stakhov in 1979 at the scientific seminar of the Mathematics Institute of the Ukrainian Academy of Sciences. I became especially interested in Stakhov’s scientific research after listening his brilliant speech at the 1989 session of the Presidium of the Ukrainian Academy of Sciences. In his speech, Professor Stakhov reported on scientific and engineering results in the field of ‘Fibonacci computers’ that were conducted under his scientific supervision at the Vinnitsa Technical University. . .

One may wonder what place does take this work in the general theory of mathematics. As it seems to me, that in the last few centuries, as Nikolay Lobachevsky said, “Mathematicians have turned all their attention to the Advanced parts of analytics, and by neglecting the origins of mathematics and did not wishing to work in that field, which they passed and left behind. As a result, it was created a gap between ‘Elementary Mathematics’, the basis of modern mathematical education, and ‘Advanced Mathematics.’ In my opinion, the Mathematics of Harmony, developed by Professor Stakhov, fills out that gap. The Mathematics of Harmony is a big theoretical contribution to the development of the ‘Elementary Mathematics’, and the Mathematics of Harmony should be considered as great contribution to mathematical education.”

Note that Alexey Stakhov used Mitropolsky’s review as the Preface to Stakhov’s 2009 book The Mathematics of Harmony [6].

Thus, academician Mitropolsky in his review focuses on the historical aspect. His point of view is that Stakhov’s Mathematics of Harmony is, first of all, a new kind of elementary mathematics, based on the unusual interpretation of the Euclidean Elements, as historically the first version of Mathematics of Harmony, connected with the Platonic solids and the golden section.

But besides this, there are other aspects of Mathematics of Harmony: applied and aesthetical. First of all, we should note the applied nature of Mathematics of Harmony, which is the true Mathematics of Nature. Mathematics of Harmony is found in many natural phenomena, such as the movement of Venus across the sky (“Pentacle of Venus”), the pentagonal symmetry in Nature, the botanical phenomenon of phyllotaxis, the fullerenes, the quasicrystals, etc.

On the other hand, Mathematics of Harmony [6] by the name and by the contents fully satisfies Hutcheson and Dirac principles of mathematics beauty [154]. According to Dirac, the main mathematical ideas should be expressed in terms of excellent mathematics. This means that Mathematics of Harmony, which was aroused in the ancient Greek mathematics, is a beautiful mathematics, which must be embodied in the structures of Nature and contemporary science. This conclusion is confirmed by the modern scientific achievements, described in Vols. I and II and will be discussed in detail in this volume.

The harmonious combination of the applied aspect of Mathematics of Harmony, as the true Mathematics of Nature, with its aesthetic perfection (Hutcheson and Dirac principles [154]), gives us reason to suggest that it is Mathematics of Harmony that can become the “golden” paradigm of modern science, which will help overcome the crisis in modern mathematics [101]. Mathematics of Harmony, described in this three-volume book, is a very young mathematical theory, although in its origins it goes back to the Euclidean Elements.

The term Mathematics of Harmony was used first by Alexey Stakhov in the speech The Golden Section and Modern Harmony Mathematics [66], made in 1996 at the Seventh International Conference on Fibonacci Numbers and Its Applications (Austria, Graz, 1996).

The speech was perceived with great interest by the Fibonacci mathematicians, as evidenced by the fact that this speech was selected for publication in the collection of papers of the International Conference on Applications of Fibonacci Numbers, published by Kluwer Academic Publishers in 1998 [66]. Starting from this publication, the development of Mathematics of Harmony became the focus of Alexey Stakhov’s scientific interests, which led him to the publication of the book [6] and to the writing of this three-volume book.

The Goal of Vol. III

The main goal of Vol. III is to answer the following two questions:

(1)What place does Mathematics of Harmony occupy in the system of contemporary mathematical sciences and how does it influence the development of modern science and mathematics?

(2)Is Mathematics of Harmony the “golden” paradigm of modern science?

Volume III begins with discussion on the influence of Mathematics of Harmony on the course of the development of modern mathematics and computer science; a number of unusual ideas put forward in the first two volumes of this book. In particular, Proclus hypothesis, which was discussed in Volume I, is considered as a prerequisite to the “golden” revolution in the history of mathematics.

The influence of Mathematics of Harmony [6] on the development of two of the most ancient mathematical theories, the measurement theory and the elementary theory of numbers, is discussed. Next, the numeral systems with irrational bases (the Fibonacci codes and the codes of the golden proportion) are considered as a prerequisite for the “golden” revolution in computer science, as well as the elements of the “golden” theory of numbers, based on the golden ratio. In conclusion, the article by the famous Russian philosopher Sergey Abachiev “Mathematics of Harmony through the Eyes of the Historian and Expert of Methodology of Science” [156] is discussed.

Chapter 2 introduces a new class of hyperbolic functions, based on the classical golden proportion and its generalization, the golden p-proportions.

Chapter 3 is devoted to the discussion of the connection between Mathematics of Harmony and the Theory of elementary functions, which plays a fundamental role in mathematics and its applications in theoretical natural sciences. Here, a new class of “elementary functions” is introduced: the “golden” hyperbolic functions or the hyperbolic Fibonacci and Lucas functions [57, 58].

Chapter 4 discusses the applications of the “golden” hyperbolic functions in the new geometric theory of phyllotaxis, created by the Ukrainian researcher Oleg Bodnar [28], and also the function Golden Shofar and Shofar-like model of the Universe [77].

Chapter 5 is devoted to outlining the theory of Fibonacci numbers, which is the result of the collective creativity of several researchers from different countries and continents: Vera de Spinadel, Argentina [29]; Midhat Gazale, France [30]; Alexander Tatarenko, Russia [62]; Jay Kappraff, USA [33, 34]; Grant Arakelyan, Armenia [49, 63]; Victor Shenyagin, Russia [64]; Nikolay Kosinov, Ukraine [65]; Alexey Stakhov, Canada [66]; Spears, Bicknell-Johnson [67]), and others.

Chapter 6 is the central chapter from the point of view of the answer to the questions posed at the beginning of this Introduction. Chapter 6 addresses a wide range of issues relating to mathematics and its history. The crisis in modern mathematics, described in the book of the outstanding American historian of mathematics Morris Klein Mathematics. The Loss of Certainty [51], is analyzed. Further, in Chapter 6, special attention is paid to the analysis of “strategic mistakes” in the development of mathematics, described in Stakhov’s articles [71, 72]. The criteria of aesthetics and beauty of mathematics are considered, in particular, the Dirac principle of mathematical beauty. From the standpoint of these criteria, the most important mathematical results, obtained in the framework of Mathematics of Harmony [6, 46, 47], are analyzed.

In Chapter 6, special attention is paid to the analysis of the strategic errors in the development of mathematics conducted in Stakhov’s article [71]. The criteria of aesthetics and beauty of mathematics, in particular, Dirac’s principle of mathematical beauty are considered. From the standpoint of these criteria, the most important mathematical results, obtained in the framework of Stakhov’s 2009 book Mathematics of Harmony [6], are analyzed.

Mathematics of Harmony is discussed as the “golden” paradigm of modern science and also the interrelation of changes of the scientific paradigms in mathematics and theoretical natural sciences, and an attempt is made to answer the question about the place of Mathematics of Harmony in the system of the modern mathematical sciences.