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Continuity in the Development of Science

Scientific and technological progress has a long history and passed in its historical development several stages: The Babylonian and Ancient Egyptian culture, the culture of Ancient China and Ancient India, the Ancient Greek culture, the Middle Ages, the Renaissance, the Industrial Revolution of the 18th century, the Great Scientific Discoveries of the 19th century, the Scientific and Technological Revolution of the 20th century and finally the 21st century, which opens a new era in the history of mankind, the Era of Harmony.

Although each of the mentioned stages has its own specifics, at the same time, every stage necessarily includes the content of the preceding stages. This is called the continuity in the development of science.

It was during the ancient period, a number of the fundamental discoveries in mathematics were made. They exerted a determining influence on the development of the material and spiritual culture. We do not always realize their importance in the development of mathematics, science, and education. To the category of such discoveries, first of all, we must attribute the Babylonian numeral system with the base 60 and the Babylonian positional principle of number representation, which is the foundation of the, decimal, binary, ternary, and other positional numeral systems. We must add to this list the trigonometry and the Euclidean geometry, the incommensurable segments and the theory of irrationality, the golden section and Platonic solids, the elementary number theory and the mathematical theory of measurement, and so on.

The continuity can be realized in various forms. One of the essential forms of its expression are the fundamental scientific ideas, which permeate all stages of the scientific and technological progress and influence various areas of science, art, philosophy, and technology. The idea of Harmony, connected with the golden section, belongs to the category of such fundamental ideas.

According to B.G. Kuznetsov, the researcher of Albert Einstein’s creativity, the great physicist piously believed that science, physics in particular, always had its eternal fundamental goal “to find in the labyrinth of the observed facts the objective harmony”. The deep faith of the outstanding physicist in the existence of the universal laws of the Harmony is evidenced by another well-known Einstein’s statement: “The religiousness of the scientist consists in the enthusiastic admiration for the laws of the Harmony” (the quote is taken from the book Meta-language of Living Nature [1], written by the outstanding Russian architect Joseph Shevelev, known for his research in the field of Harmony and the golden section [1–3]).

Pythagoreanism and Pythagorean MATHEM’s

By studying the sources of the origin of mathematics, we inevitably come to Pythagoras and his doctrine, named the Pythagoreanism (see Wikipedia article Pythagoreanism, the Free Encyclopedia). As mentioned in Wikipedia, the Pythagoreanism originated in the 6th century BC and was based on teachings and beliefs of Pythagoras and his followers called the Pythagoreans. Pythagoras established the first Pythagorean community in Croton, Italy. The Early Pythagoreans espoused a rigorous life and strict rules on diet, clothing and behavior.

According to tradition, Pythagoreans were divided into two separate schools of thought: the mathematikoi (mathematicians) and the akousmatikoi (listeners). The listeners had developed the religious and ritual aspects of Pythagoreanism; the mathematicians studied the four Pythagorean MATHEMs: arithmetic, geometry, spherics, and harmonics. These MATHEMs, according to Pythagoras, were the main composite parts of mathematics. Unfortunately, the Pythagorean MATHEM of the harmonics was lost in mathematics during the process of its historical development.

Proclus Hypothesis

The Greek philosopher and mathematician Proclus Diadoch (412–485 AD) put forth the following unusual hypothesis concerning Euclid’s Elements. Among Proclus’s mathematical works, his Commentary on the Book I of Euclid’s Elements was the most well known. In the commentary, he puts forth the following unusual hypothesis.

It is well known that Euclid’s Elements consists of 13 books. In those, XIII^th book, that is, the concluding book of the Elements, was devoted to the description of the geometric theory of the five regular polyhedra, which had played a dominant role in Plato’s cosmology and is known in modern science under the name of the Platonic solids.

Proclus drew special attention to the fact that the concluding book of the Elements had been devoted to the Platonic solids. Usually, the most important material, of the scientific work is placed in its final part. Therefore, by placing Platonic solids in Book XIII, that is, in the concluding book of his Elements, Euclid clearly pointed out on main purpose of writing his Elements. As the prominent Belarusian philosopher Edward Soroko points out in [4], according to Proclus, Euclid “had created his Elements allegedly not for the purpose of describing geometry as such, but with purpose to give the complete systematized theory of constructing the five Platonic solids; in the same time Euclid described here some latest achievements of mathematics”.

It is for the solution of this problem (first of all, for the creation of geometric theory of dodecahedron), Euclid 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.

Another mention about the Mathematics of Harmony in the connection to the ancient Greek mathematics is found in the book by Vladimir Dimitrov, A New Kind of Social Science, published in 2005 [44]. It is important to emphasize that in Ref. [44], the concept of Mathematics of Harmony is directly associated with the golden section, the most important mathematical discovery of the ancient science in the field of Harmony. This discovery at that time was called “dividing a segment into the extreme and mean ratio” [32].

From Refs. [44, 45], it is evident that prominent thinkers, scientists and mathematicians took part in the development of the Mathematics of Harmony for several millennia: Pythagoras, Plato, Euclid, Fibonacci, Pacioli, Kepler, Cassini, Binet, Lucas, Klein, and in the 20th century the well-known mathematicians Coxeter [7], Vorobyov [8], Hoggatt [9], Vaida [11], Knuth [123], and so on. And we cannot ignore this historical fact.

Fibonacci Numbers

The Fibonacci numbers, introduced into Western European mathematics in the 13th century by the Italian mathematician Leonardo of Pisa (known by the nickname Fibonacci), are closely related to the golden ratio. Fibonacci numbers from the numerical sequence, which starts with two units, and then each subsequent Fibonacci number is the sum of the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . . The ratio of the two neighboring Fibonacci numbers in the limit tends to be the golden ratio.

The mathematical theory of Fibonacci numbers has been further developed in the works of the French mathematicians of the 19th century Binet (Binet formula) and Lucas (Lucas numbers). As mentioned above, in the second half of the 20th century, this theory was developed in the works of the Canadian geometer, Donald Coxeter [7], the Soviet mathematician, Nikolay Vorobyov [8], the American mathematician, Verner Hoggatt [9] and the English mathematician, Stefan Vajda [11], the outstanding American mathematician, Donald Knuth [123], and so on.

The development of this direction ultimately led to the emergence of the Mathematics of Harmony [6], a new interdisciplinary direction of modern science that relates to modern mathematics, computer science, economics, as well as to all theoretical natural sciences. The works of the well-known mathematicians, Coxeter [7], Vorobyov [8], Hoggatt [9], Vaida [11], Knuth [123], and others, as well as the study of Fibonacci mathematicians, members of the American Fibonacci Association, became the beginning of the process of Harmonization of Mathematics [68], which continues actively in the 21st century. And this process is confirmed by a huge number of books and articles in the field of the golden section and Fibonacci numbers published in the second half of the 20th century and the beginning of the 21st century [1–100].

Sources of the Present Three-Volume Book

The differentiation of modern science and its division into separate spheres do not allow us often to see the general picture of science and the main trends in its development. However, in science, there exist research objects that combine disparate scientific facts into a single whole. Platonic solids and the golden section are attributed to the category of such objects. The ancient Greeks elevated them to the level of “the main harmonic figures of the Universe”. For centuries or even millennia, starting from Pythagoras, Plato and Euclid, these geometric objects were the object of admiration and worship of the outstanding minds of mankind, during Renaissance, Leonardo da Vinci, Luca Pacoli, Johannes Kepler, in the 19th century, Zeising, Lucas, Binet and Klein. In the 20th century, the interest in these mathematical objects increased significantly, thanks to the research of the Canadian geometer, Harold Coxeter [7], the Soviet mathematician Nikolay Vorobyov [8] and the American mathematician Verner Hoggatt [9], whose works in the field of the Fibonacci numbers began the process of the “Harmonization of Mathematics”. The development of this direction led to the creation of the Mathematics of Harmony [6] as a new interdisciplinary trend of modern science.

The newest discoveries in the various fields of modern science, based on the Platonic solids, the golden section and the Fibonacci numbers, and new scientific discoveries and mathematical results, related to the Mathematics of Harmony (quasicrystals [115], fullerenes [116], the new geometric theory of phyllotaxis (Bodnar’s geometry) [28], the general theory of the hyperbolic functions [75, 82], the algorithmic measurement theory [16], the Fibonacci and golden ratio codes [6], the “golden” number theory [94], the “golden” interpretation of the special theory of relativity and the evolution of the Universe [87], and so on) create an overall picture of the movement of modern science towards the “golden” scientific revolution, which is one of the characteristic trends in the development of modern science. The sensational information about the experimental discovery of the golden section in the quantum world as a result of many years of research, carried out at the Helmholtz–Zentrum Berlin für Materialien und Energie (HZB) (Germany), the Oxford and Bristol Universities and the Rutherford Appleton Laboratory (UK), is yet another confirmation of the movement of the theoretical physics to the golden section and the Mathematics of Harmony [6].

For the first time, this direction was described in the book by Stakhov A.P., assisted by Scott Olsen, The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science, World Scientific, 2009 [6].

In 2006, the Russian Publishing House, “Piter” (St. Petersburg) published the book, Da Vinci Code and Fibonacci numbers [46] (Alexey Stakhov, Anna Sluchenkova and Igor Shcherbakov were the authors of the book). This book was one of the first Russian books in this field. Some aspects of this direction are reflected in the following authors’ books, published by Lambert Academic Publishing (Germany) and World Scientific (Singapore):

•Alexey Stakhov, Samuil Aranson, The Mathematics of Harmony and Hilbert’s Fourth Problem. The Way to Harmonic Hyperbolic and Spherical Worlds of Nature. Germany: Lambert Academic Publishing, 2014 [51].

•Alexey Stakhov, Samuil Aranson, Assisted by Scott Olsen, The “Golden” Non-Euclidean Geometry, World Scientific, 2016 [52].

•Alexey Stakhov, Numeral Systems with Irrational Bases for Mission-Critical Applications, World Scientific, 2017 [53].

These books are fundamental in the sense that they are the first books in modern science, devoted to the description of the theoretical foundations and applications of the following new trends in modern science: the history of the golden section [78], the Mathematics of Harmony [6], the “Golden” Non-Euclidean geometry [52], ascending to Euclid’s Elements, and also the Numeral Systems with Irrational bases, ascending to the Babylonian positional numeral system, the decimal and binary system and Bergman’s system [54].

These books discuss the problems, which in modern mathematics are considered long resolved and therefore are not included in the circle of the studies of mathematicians, namely the new mathematical theory of measurement called the Algorithmic Measurement Theory [16, 17], the Mathematics of Harmony [6] as a new kind of elementary mathematics that has a direct relationship to the foundations of the mathematics and mathematical education, the new class of the elementary functions called the hyperbolic Fibonacci and Lucas functions [64, 75, 82] and finally, the new ways of real numbers representation, and the new binary and ternary arithmetic’s [55, 72], which have the fundamental interest for computer science and digital metrology.

In 2010, the Odesa I.I. Mechnikov National University (Ukraine) hosted the International Congress on the Mathematics of Harmony. The main goal of the Congress was to consolidate the priority of Slavic science in the development of this important trend and acquaint the scientific community with the main trends of the development of the Mathematics of Harmony as the new interdisciplinary direction of modern science.

In the recent years, the new unique books on the problems of Harmony and the history of the golden section have been published:

•The Prince of Wales. Harmony. A New Way of Looking at our World (coauthors Tony Juniper and Ian Skelly). An Imprint of HarperCollins Publisher, 2010 [49].

•Hrant Arakelian, Mathematics and History of the Golden Section. Moscow, Publishing House “Logos”, 2014 [50].

For the last 30 years, Charles, The Prince of Wales, had been known around the world as one of the most forceful advocates for the environment. During that period, he focused on many different aspects of our lives, when we continually confront with the real life from new angles of view and search original approaches. Finally, in Harmony (2010) [49], The Prince of Wales and his coauthors laid out their thoughts on the planet, by offering an in-depth look into its future. Here, we see a dramatic call to the action and an inspirational guide on the relationship of mankind with Nature throughout history. The Prince of Wales’s Harmony (2010) [49] is an illuminating look on how we must reconnect with our past in order to take control of our future.

The 2014 book [50] by the Armenian philosopher and physicist Hrant Arakelian is devoted to the golden section and to the complexity of problems connected with it. The book consists of two parts. The first part is devoted to the mathematics of the golden section and the second part to the history of the golden section. Undoubtedly, Arakelian’s 2014 book is one of the best modern books devoted to mathematics and the history of the golden section.

The International Congress on Mathematics of Harmony (Odessa, 2010) and the above-mentioned books by The Prince of Wales and Armenian philosopher Hrant Arakelian are brilliant confirmation of the fact that in modern science, the interest in the mathematics of the golden section and its history increases and further development of the Mathematics of Harmony can lead to revolutionary transformations in modern mathematics and science on the whole.

Why did the author decide to write the three-volume book The Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science? It should be noted that the author and other famous authors in this field published many original books and articles in this scientific direction. However, all the new results and ideas, described in the above-mentioned publications of Alexey Stakhov, Samuil Aranson, Charles, The Prince of Wales, Hrant Arakelian and other authors are scattered in their numerous articles and books, which makes it difficult to understand their fundamental role in the development of the modern mathematics, computer science and theoretical natural sciences on the whole.

This role is most clearly reflected in the following citations taken from Harmony by the Prince of Wales (2010) [49]:

“This is a call to revolution. The Earth is under threat. It cannot cope with all that we demand of it. It is losing its balance and we humans are causing this to happen.”

The following quote, placed on the back cover of Prince of Wales’s Harmony [49], develops this thought:

“We stand at an historical moment; we face a future where there is a real prospect that if we fail the Earth, we fail humanity. To avoid such an outcome, which will comprehensively destroy our children’s future or even our own, we must make choices now that carry monumental implications.”

Thus, The Prince of Wales has considered his 2010 book, Harmony. A New Way of Looking at our World, as a call to the revolution in modern science, culture and education. The same point of view is expressed in the above-mentioned books by Stakhov and Aranson [6, 46, 51–53]. Comparing the books of Prince of Wales [49] and Hrant Arakelian [50] to the 2009, 2016 and 2017 books of Alexey Stakhov and Samuil Aranson [6, 51–53], one can only be surprised how deeply all these books, written in different countries and continents, coincide in their ideas and goals.

Such an amazing coincidence can only be explained by the fact that in modern science, there is 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.