Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
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.
About the Author
Alexey Stakhov, born in May 7, 1939, is a Ukrainian mathematician, inventor and engineer, who has made a contribution to the theory of Fibonacci numbers and the golden section and their applications in computer science and measurement theory. He is a Doctor of Computer Science (1972) and a Professor (1974), and the author of over 500 publications, 14 books and 65 international patents. He is also the author of many original publications in computer science and mathematics, including algorithmic measurement theory [16, 17], Fibonacci codes and codes of the golden proportions [19], hyperbolic Fibonacci and Lucas functions [64, 75] and finally the Mathematics of Harmony [6], which goes back in its origins to Euclid’s Elements. In these areas, Alexey Stakhov has written many papers and books, which have been published in famous scientific journals by prestigious international publishers.
The making of Alexey Stakhov as a scientist is inextricably linked with the Kharkov Institute for Radio Electronics, where he was a postgraduate student of the Technical Cybernetics Department from 1963 to 1966. Here, he defended his PhD thesis in the field of Technical Cybernetics (1966) under the leadership of the prominent Ukrainian scientist Professor Alexander Volkov. In 1972, Stakhov defended (at the age of 32 years) his Grand Doctoral dissertation Synthesis of Optimal Algorithms for Analog-to-Digital Conversion (Computer Science speciality). Although the dissertation had an engineering character, Stakhov in his books and articles touched upon two fundamental problems of mathematics: theory of measurement and numeral systems.
Prof. Stakhov worked as “Visiting Professor” of different Universities: Vienna Technical University (Austria, 1976), University of Jena (Germany, 1986), Dresden Technical University (Germany, 1988), Al Fateh University (Tripoli, Libya, 1995–1997), Eduardo Mondlane University (Maputo, Mozambique, 1998–2000).
Stakhov’s Prizes and Awards
•Award for the best scientific publication by Ministry of Education and Science of Ukraine (1980);
•Barkhausen’s Commemorative Medal issued by the Dresden Technical University as “Visiting Professor” of Heinrich Barkhausen’s Department (1988);
•Emeritus Professor of Taganrog University of Radio Engineering (2004);
•The honorary title of “Knight of Arts and Sciences” (Russian Academy of Natural Sciences, 2009);
•The honorary title “Doctor of the Sacred Geometry in Mathematics” (American Society of the Golden Section, 2010);
•Awarded “Leonardo Fibonacci Commemorative Medal” (Interdisciplinary Journal “De Lapide Philosophorum”, 2015).
Acknowledgments
Alexey Stakhov expresses great thanks to his teacher, the outstanding Ukrainian scientist, Professor Alexander