**Grand Unified Natural Dynamics**

**(Formerly, Grand Unified Theory)
Edgar E. Escultura**

Research Professor, Lakshmikantham Institute for Advanced Studies and

Departments of Mathematics and Physics, GVP College of Engineering, Jawaharlal Nehru Technical University, Kakinada Campus

Madurawada, Visakhapatnam, India

Member, International Federation of Nonlinear Analysts

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This website offers state-of-the-art information on the new mathematics and physics catalyzed by the 365-year-old problem, Fermat’s last theorem (FLT), and the 200-year-old Laplace or gravitational n-body problem. FLT was shown to be undecidable in 1992 the proof presented at the Second International Conference on Dynamic Systems and Application, 1995, Atlanta, and published in its proceedings. Then the construction of counterexamples to FLT in 1998 proved that it is false, the counterexamples published in Nonlinear Studies, **5**(2), 1998. The gravitational n-body problem posed by Simon Marquiz de Laplace at the turn of the 18th Century was solved in 1995 the solution presented at the Second World Congress of Nonlinear Analysts, 1996, Athens, and published in Nonlinear Analysis, A-Series: Theory, Methods and Applications, **30**(8), 1997. The resolution of both problems has generated over 50 papers published in or accepted by a dozen peer-reviewed international journals and conference proceedings plus a new book co-authored with V. Lakshmikantham and S. Leela, The Hybrid Grand Unified Theory, published by Atlantis (a division of Elsevier Science, Ltd.), 2009. Its main chapter is the formulation of the Grand Unified Theory or GUT. GUT unifies not only gravity and the weak and strong forces of physics that was envisioned by Albert Einstein but also the forces and interactions of nature and all natural dynamics, i.e., motions and configurations of matter. Therefore, it unifies the natural sciences as well, the reason for the name or title of this website.

A book by this host in press at Bentham Science Publishers, Scientific Natural Philosophy, offers the most advanced formulation of the philosophy of natural science as materialist philosophy (Chapter 5). The book includes the mathematics of GUT (Chapter 2) and GUT's full formulation and development (Chapter 3). However, the book also includes the full solution of the gravitational n-body problem as application of GUT’s methodology – qualitative or non-quantitative modeling – that explains nature in terms of its laws. Its main tool is qualitative or non-quantitative mathematics. This new methodology is applied to the theoretical development of other fields: (a) The physics of the mind (physical psychology), (b) Theory of turbulence (atmospheric and geological sciences), (c) The unified theory of evolution (biology), (d) Global geology and oceanography and (e) Genetic alteration, modification and sterilization, with applications to the treatment of genetic diseases (medicine). Moreover, the book has an analysis of the disastrous final flight of the Columbia Space Shuttle of February 1, 2004.

The book provides both qualitative and quantitative models of macro and quantum gravity and thermodynamics, the three pillars of the new physics or GUT. All motions and configurations or forms of matter are physical systems. In this sense basic cosmic or electromagnetic waves generated by the normal vibration of atomic nuclei and seismic waves generated by the micro component of turbulence at its interface, e.g., at the inner core of a cosmological vortex, tectonic plate interface and interface of flowing compressed volcanic lava lamina are physical systems. They are propagated across dark matter by its suitably synchronized vibration. The macro envelopes of seismic waves generated at tectonic plate and geological faults interfaces are visible on the ground during earthquake as wave motion. Physical systems and their interactions and all the forces of nature such as charge and gravity are natural dynamics. Also included in this category are conversions from dark to visible matter and *vice versa* and from one form of physical system to another as well as their generation and transfer or conduction.

**The Resolution of FLT**

What made the difference in the quality of research output that pinned down this elusive 365-year-old problem is the orientation that the existence of long-standing unsolved problem reveals the inadequacy of its underlying fields. For FLT, the underlying fields are foundations, number theory and the real number system. Therefore, its resolution required their critique-rectification on which stands the appropriate mathematical space that resolves the problem. That space is the new real number system, the reconstructed real number system without the false trichotomy and completeness axioms. The new real number system is built on the additive and multiplicative identities 0 and 1 as integers, i.e., integral parts of decimals, and the addition and multiplication tables that well-define the integers and terminating decimals and the additive and multiplicative operations. Then the closure of the terminating decimals in the g-norm is developed as the new real number system **R*** that includes the nonterminating decimals, the nonstandard numbers d* (dark number) and u* (unbounded number). The g-norm of a decimal is the decimal itself. The new real number system has countably infinite counterexamples to FLT and proof of Goldbach’s conjecture. The resolution says that FLT is not only undecidable but also false (the questions of decidability and truth are separate). Undecidable propositions are characterized as ambiguous propositions. The critique-rectification of foundations played the crucial role in FLT’s resolution. Among the key points drawn from it are the following:

(1) Recognition that the subject matter of mathematics cannot be the subjective concepts of thought but their representation as symbols (we still refer to them as concepts) well-defined by a consistent set of axioms. This makes mathematics a language that describes physical phenomena but cannot resolve scientific questions and problems. This was the main contribution of David Hilbert almost a century ago.

(2) Avoidance of universal rules of inference, i.e., the rules of inference must be specific to the given mathematical space and specified by the axioms. In particular, universal rules of inference like formal logic are not valid. This requirement applies to physical theory where the axioms are the laws of nature.

(3) Only concepts well-defined by the axioms are admissible as basic concepts. A concept is well-defined if its existence, properties and relationship with other concepts are specified by the axioms. In particular, introduction of undefined terms brings in ambiguity and collapses a mathematical space. Therefore, while some undefined symbols may be introduced initially, the choice of the axioms of a mathematical space is not complete until all concepts are well-defined.

(4) Distinct mathematical spaces are independent. Therefore, any proof involving mapping or concepts coming from distinct spaces is flawed since some concept in one is not well defined in the other. In particular, any conclusion about nature drawn from its mathematical model is flawed and amounts to reasoning by analogy.

(5) With respect to the real number system, counterexamples to the trichotomy axiom were constructed by L. E. J. Brouwer and this host separately. The Banach-Tarski paradox is a counterexample to the completeness axiom (a variant of the axiom of choice). The trichotomy and completeness axioms are among the field axioms of the real number system.

Satisfying these requirements pinned down the major breakthrough in foundations: characterization of undecidable propositions from which sprung qualitative modeling that explains nature in terms of its laws. This is the new scientific methodology that made the development of both GUT and GUND possible and the resolution of the long-standing problems and fundamental questions of physics, in particular, the discovery of the superstring, the basic constituent of matter.

**The Solution of the Gravitational n-Body Problem**

A computational model (the current methodology of physics), e.g., system of differential equations, describes natural phenomena but does not shed light on their dynamics; nor does it explain them. Therefore, it neither provides insights into their internal dynamics and state nor their future course nor where a physical system such as n bodies in the Cosmos is headed for. This is the reason computational modeling alone could not solve the n-body problem. It required qualitative modeling to solve it; i.e., the search for the laws of nature on which to anchor physical theory that provides the solution. For the n-body problem that theory was the initial formulation of GUT called the flux theory of gravitation anchored on the initial 11 natural laws.

The characterization of undecidable propositions reveals a major flaw of quantitative modeling; it was resolved by qualitative modeling and proceeds as follows:

*To solve a scientific problem, a suitable physical theory is devised that provides the solution.*

The physical theory is anchored on some laws of nature that explain the underlying physical systems or natural phenomena. In the case of the n-body problem 11 natural laws were needed for the initial formulation of GUT that provided the solution in 1997.

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