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.
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
Volume II is devoted to the discussion of two fundamental problems of science and mathematics, the problem of measurement and the problem of numeral systems, their relationship with the development of science and their historical role in the development, first of all, of contemporary mathematics and computer science by taking into consideration the contemporary achievements in mathematical theory of measurement and numeral systems.
As it is known, a set of rules, used by the ancient Egyptian land surveyors, was the first measurement theory.From this measurement theory, as the ancient Greeks testify, there originated the geometry, which takes its origin (and title) in the problem of earth measuring.
Already in the ancient Greece, the mathematical problems of geometry (that is, earth measuring) were the main focus of ancient mathematics. The science of measurement, related to geometry, was developing primarily as a mathematical theory. It is during this period that the discovery of the incommensurable segments and the formulation of Eudoxus’ exhaustion method, to which the number theory as well as the integral and differential calculus go back in its origin, were made.
By basing on these important mathematical discoveries, which had the relation to measurement, the Bulgarian mathematician academician Ljubomir Iliev, the leader of the Bulgarian mathematical community, asserted that “during the first epoch of its development, from antiquity to until the discovery of differential calculus, mathematics, by studying primarily problems of measurement, did created the Euclidean geometry and number theory” [137].
In 1991, the Publishing House “Science” (the main Russian edition of the physical and mathematical literature) has published the book, Mathematics in its Historical Development [102], written by the outstanding Russian mathematician academician Andrey Kolmogorov (1903–1987). By discussing the period of the origin of mathematics, academician Kolmogorov pays attention to the following features of this period:
“The counting of objects at the earliest stages of development of culture led to the creation of the simplest concepts of arithmetic of natural numbers. Only on the basis of the developed system of oral numeration, the written numeral systems arise, and gradually methods of performing four arithmetic operations over natural numbers are developed …
The demand for measurement (the amount of grain, the length of the road, etc.) leads to the appearance of the names and symbols of the most widespread fractional numbers and the development of methods for performing arithmetical operations over fractions. Thus, there was accumulating material, which is added gradually to the most ancient mathematical direction, arithmetic. Measurement of space and volumes, the needs of construction equipment, and a little later, astronomy, cause the development of the beginnings of geometry.”
By comparing the views of the academicians Iliev (Bulgaria) and Kolmogorov (Russia) on the period of the origin of mathematics, it should be noted that these views mostly coincided to and were reduced to the following. At the stage of the origin of mathematics, two practical problems influenced the development of mathematics: the counting problem and the measurement problem.
The study of the counting problem ultimately led to the formation of such an important concept of mathematics as the natural numbers and to the creation of the elementary number theory, which solved important mathematical task in studying the properties of the natural numbers as well as solved the problem of creation of the elementary arithmetic that satisfied the needs of practice in performing the simplest arithmetic operations. The study of the problem of measurement led to the creation of geometry, and within this direction, the existence of the irrational numbers, the second most significant fundamental ancient mathematical discovery, which caused the first crisis in the foundations of mathematical science is proved.
By discussing the origins of mathematical science, we should not forget about another outstanding mathematical discovery of ancient mathematics, Eudoxus’ exhaustion method, which, on the one hand, was created to overcome the first crisis in the foundation of mathematics, associated with the introduction of the irrational numbers into mathematics and, on the other hand, underlies the Euclidean definition of the natural numbers, which represents the same natural number as the sum of the “monads”
Volume II pursues two goals. The first goal is to set forth the foundation of the new mathematical measurement theory, the Algorithmic Measurement Theory, worked out by Alexey Stakhov in his doctoral dissertation, Synthesis of Optimal