Dynamic Modeling and the New Mathematics and Physics
by
E. E. Escultura
University of the Philippines Pampanga, Clark Field, Pampanga 2019, Philippines

Using the long-standing unsolved problems of mathematics and physics as catalyst, the paper assesses the reach and limits of both disciplines. Physics has made tremendous breakthrough in the investigation of the very large and the very small. With advanced technology, spectroscopy, measurement, computation and some physical principles like Hubble’s law, it has provided much information about our universe. For example, at the very large, we know the radius of our universe, 10¹⁰ light years, its rate of radial expansion, 10²⁰ km/sec, and even its acceleration of 10^–10 km/secsec which says that it is at the youthful phase of its cycle. At the micro scale, physics has computed the mass of the electron and detected the quark and neutrino. However, it has long-standing unsolved problems such as the gravitational n-body problem and structure of the electron, what the basic constituent of matter is and what gravity is all about. This inadequacy is due to limitation of methodology, mathematical modeling, that describes nature in terms of numbers, equations, functions and statistical trends. Its principal components are measurement and computation. It has two levels of inadequacy: a) description of physical system yields neither insight on its internal dynamics nor predictive capability on its future behavior and b) computation is limited by the inadequacy of the real number system which we now refer to as the decimals for precision. With respect to a) we need to know how nature works; mathematical modeling does not provide this. The remedy is dynamic modeling that explains nature in terms of its laws. This shifts the task of the physicist from computing to discovering the laws of nature. Measurement and computation fail here. For example, Newton’s law of gravitation does not explain gravity but describes the motion of a body under its influence. Physics does not have a sense of what charge and gravity are although it can measure their effect. With respect to b) there is inherent limitation in computation. For example, we cannot enter all the digits of a nonterminating decimal into the calculator or computer. Therefore, we can neither add nor multiply decimals; we can only approximate their sum and product. This means that there is a level of ambiguity or uncertainty in a nonterminating decimal or, to be precise, computation is well-defined only on terminating decimals; even here, it fails on large and small numbers. Another computational problem is the fact that most differential equations are unsolvable. Thus, if dy/dx = f(x) is solvable, dy/dx = f(x)(sinn1/x)(cosn1/x) isn’t at the origin, for integers m, n, and yet oscillation is a universal motion. Another universal configuration is nested fractal, a sequence of objects at decreasing scale where each term is included in the previous term and similar to the first term with respect to a specific property or behavior. The root system of a tree is nested fractal. One can see the difficulty of modeling and solving it by a system of differential equations although this is surmountable since the system is finite. However, mathematical modeling of seismic wave presents insurmountable difficulty not only since cosmic wave is an infinite nested fractal sequence of waves but also since it involves motion of what is called dark matter, i.e., non-observable. Its effect is seen only in the softening of metal in building foundations and cracking and pulverization of concrete. Moreover, inadequacy, error and contradiction come from ambiguity of infinite set, large and small numbers (depending on context), vacuous concept and proposition, ambiguous proposition (proposition involving ambiguous concepts) and self-reference. This problem of computation which was recognized in the course of resolving Fermat’s last theorem catalyzed the critique-rectification of its underlying fields of foundations, particularly, mathematical reasoning, number theory and the decimals leading to their axiomatization as the discrete, contradiction-free new real number system and its extension to the new nonstandard calculus. This is the appropriate mathematics for physics and computing, particularly, simulation, both of which have discrete domains. This rectification only fixes computation but does not meet the requirements of dynamic modeling. The remedy is qualitative mathematics, the complement of computation. It includes abstract mathematical spaces, foundations, especially, mathematical reasoning, axiomatization of mathematical spaces, especially, the new real number system and its extensions, and the search for the laws of nature. This new methodology paved the way for the discovery of the basic constituent of matter that, together with the laws of nature (43 natural laws have been discovered), anchors the flux theory of gravitation or the new physics. The theory not only provides the solution of the long standing problems of physics but also explains natural phenomena and nature itself. Dynamic modeling gave birth to theoretical physics where only mathematical physics existed before. Initially, dynamic modeling, especially is principal component, qualitative mathematics, is quite shocking to physicists who did computing all their lives because of the absence of numbers, equations, graphs, functions, etc. However, once it is grasped that it was introduced precisely and solely for the purpose of serving physics where computation fails that initial shock should vanish. Dynamic modeling does not displace computation; on the contrary, it complements and provides validity to it. #61567;

Date received: June 20, 2006