The attempt of build up the Fundamentals of a new mathematical direction, which is called Harmony Mathematics, is addressed in the present article. The article has a «strategic» importance for development of computer science and theoretical physics. © 2005 Published by Elsevier Ltd.
I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts; the rest are details. Albert Einstein
As is well known, mathematics is one of the outstanding creations of the human intellect; a result of centuries of intensive and continuous creative work of man's geniuses. What is the goal of mathematics? The answer is not simple. Probably, the goal of mathematics is to discover «mathematical laws of the Universe» and to construct models of the physical world. It is clear that the progress of the human society depends on the knowledge of these laws.
During historical progress, mankind realized that it is surrounded by a huge number of different «worlds»: the «world» of geometric space, the «world» of mechanical and astronomical phenomena, the «world» of stochastic processes, the «world» of information, the «world» of electromagnetism, the «world» of botanical and biological phenomena, and the «world» of art, etc. For simulation and mathematical description of each of these «worlds», mathematicians created the appropriate mathematical theory most suitable to the phenomena and processes of this or that «world». To describe a geometric space, Euclid wrote his book The Elements. To simulate the mechanical and astronomical phenomena, Newton created the theory of gravitation and differential and integral calculus. MaxwelTs
theory was created to describe the electromagnetic phenomena; the theory of probabilities was created for simulation of the stochastic «world». In the 19th century, Lobachevsky created non-Euclidean geometry which is a deeper model of the geometric space. We can go on mentioning infinitely many such examples.
Modern mathematics is a complex set of different mathematical concepts and theories. One of the major problems of mathematical research is to find connections between separate mathematical theories. This always leads to the deepening of our knowledge about Nature and shows a deep connection between Nature and Universal laws.
Modern mathematics experienced a complex stage in its development. The prolonged crisis of its bases was connected to paradoxes in Cantor's theory of infinite sets. The passion of mathematicians for abstractions and generalizations broke the contact with natural sciences that are a source of mathematical origin. This has compelled many outstanding mathematicians of the 20th century to talk about a serious crisis in modern mathematics and even about its isolation from the general course of scientific and technical progress. In this connection the publication of the book Mathematics, The Loss of Certainty , written by Moris Kline, Professor Emeritus of Mathematics of Courant Institute of Mathematical Sciences (New York University), is symptomatic.
In this situation, the representatives of other scientific disciplines, namely, physics, chemistry, biology, engineering and even arts, began to develop what may be called natural mathematics, which can be used effectively for mathematical simulations of physical, biological, chemical, engineering and other processes. The idea of soft mathematics gained more and more attractiveness. Humanitarization of mathematics is being discussed as a tendency in the development of modern science . In this connection the book Meta-language of the Living Nature  written by the famous Russian architect, Shevelev, can be considered as an attempt to create one more variant of natural mathematics.
Harmony Mathematics that was developed by the author for many years [4-22] belongs to a category of similar mathematical directions. In its sources this new mathematical theory goes back to the Pythagorean Doctrine about Numerical Harmony of the Universe. Since the antique period, many outstanding scientists and thinkers like Leonardo da Vinci, Luca Pacioli, Johannes Kepler, Leibnitz, Zeizing, Binet, Lucas, Einstein, Vernadsky, Losev, Florensky paid a great attention to this scientific doctrine.
The main goal of the present article is to state the fundamentals of the Harmony Mathematics [4-22], that is, to describe its basic concepts and theories and to discuss its applications in modern science.