Concept of the Harmony Mathematics The humanity became aware of a long time that it is the participant and the witness of huge number of different "worlds" surrounding it. And their own laws act in every from the "worlds". First of all there are the "mechanical world" and the "astronomical world" where "Newton's gravitation laws" act, the "electromagnetic world" where "Maxwell' equations" act, the world of "living nature", the world of "information", the world of "business", the "social world", the world of the "Art" and so on. And the human science created the corresponding mathematical theories adapted very well to modeling of processes flowing in that or another "world". And this was the answer of mathematics to the "social need". The calculus created for modeling the "movement processes" and "gravitation laws" of mechanical objects was the brightest example of this. The electromagnetic theory created by Maxwell for modeling of the electromagnetic processes is the other brightest example. We have described in our Museum a big number of examples where Fibonacci numbers and golden section play an important role. And the botanic phenomenon of phyllotaxis is the brightest example of this. And there arises a question: possibly there exist some "Fibonacci's world" subjected to Fibonacci numbers and golden section? Most likely the world of plants, animals and the man as the biological object is "Fibonacci's one". Recently the Ukrainian architect Oleg Bodnar showed that the geometry of living nature is the hyperbolic one. At that the growth processes are subjected to the hyperbolic Fibonacci and Lucas functions (we will tell about Bodnar' discovery more in detail). But possibly is the "world of business" the "Fibonacci world" too? And the highly unusual investigations of the American scientist Ralf Elliott ("Elliott's Waves") confirms this. And like to the fact that the investigation of the "movement problem" brought into creation of the calculus, the most important mathematical apparatus of modern mathematics, and the investigation of the electromagnetic phenomena brought into the "Maxwell's equations", the modern scientific discoveries based on Fibonacci numbers and golden section demand on development of the corresponding mathematical apparatus adequate to the studying physical phenomena. We showed in our Museum that the Fibonacci numbers theory was supplemented recently by some new mathematical results (the Analysis of the fundamental ideas underlying the classical mathematics fundamentals was used for the creation of the Harmony Mathematics fundamentals. We take for the basis the following mathematics definition given by the famous Russian mathematician academician Kolmogorov: "Mathematics is the science of the quantitative relations and the space forms of the real world".
It follows from Kolmogorov's definition that just concepts of the "Number" and the "Value" are fundamental concepts of mathematics. Developing such approach one may select three important mathematical concepts, which were put historically to the mathematics foundation: - The concept of the
*Natural Number*aroused from the*"count problem"*and brought into the*"Number Theory"*. - The concept of the
*Irrational Number*, which arose at studying the concept of the*"Value"*and is the result of the*"Mathematical Measurement Theory"*development. - The concept of the
*"fundamental mathematical constants", the p-number and Euler's number of e*, are the next important mathematical concepts. They express the most important quantitative relations of the real world and "generate" the most important class of the*"Elementary Functions"*used widely for modeling of the quantitative regularities of the real world.
Analysis of these fundamental ideas allows us to advance the following fundamental ideas, which can underlay the Fibonacci Mathematics or "Harmony Mathematics": - (1) The first idea is the new measurement theory called the
*Algorithmic Measurement Theory*. The foundations of the Algorithmic Measurement Theory are stated in Stakhov's book "Introduction into Algorithmic Measurement Theory" (1977) and Stakhov's brochure "Algorithmic Measurement Theory" (1979).
Later Prof. Sytakhov published the article "The Golden Section in the Measurement Theory" in the highly prestigious International Journal "Computers & Mathematics with Applications". This article was devoted to the algorithmic measurement theory as the new direction in the mathematical measurement theory. - The second idea consists of the following. It is suggested to add the golden ratio to the p and
*e*-numbers, the main mathematical constants, and together with the golden ratio to add the new class of the elementary functions, the*hyperbolic Fibonacci and Lucas function*, to the classical elementary functions. - The next idea is to give the
*New Number Geometric Definition*based on the*golden p-ratios*concept. This one generalizes the classical Euclidean number definition and "generates" the new positional number systems, the*number systems with irrational bases*.
Table 1 gives comparison between the foundations of the "Classical Mathematics" and "Harmony Mathematics".
We can see from this comparison that the foundation of the "Harmony Mathematics" is similar to the foundation of the "Classical Mathematics" but we have the The algorithmic measurement theory, ascending to the Fibonacci "weighing problem", is the heart of the Harmony Mathematics (Fig.1) The algorithmic measurement theory "generates" an infinite number of the new number series, in particular the The algorithmic measurement theory results in the new approach to the positional number systems ascending to the Babylonian sexagesimal number system. Due to this approach there arises an infinite extension of the number systems and this "oldest" part of mathematics turns into the new mathematical theory developing and adding the classical theoretical arithmetic. Binet's formulas generate the new fundamental system of elementary functions. These are the The hyperbolic Fibonacci and Lucas functions are the heart of the new Thus it follows from this consideration at least two important modern applications of the Fibonacci numbers and the golden ratio theory namely: - Simulation of biological processes (Bodnar's geometry).
- New computer theory (Fibonacci and "golden" computers).
As is well known the classical mathematical analysis based on the p- and Applications of the golden section in the Art are widely known. The Harmony Mathematics generates the new geometric proportions (the golden |