Abstract
The theme of this essay is that the time of dominance of Newton’s world view in science is drawing to a close. The harbinger of its demise was the work of Poincaré on the three-body problem and its culmination into what is now called chaos theory. The signature of chaos is the sensitive dependence on initial conditions resulting in the unpredictability of single particle trajectories. Classical determinism has become increasingly rare with the advent of chaos, being replaced by erratic stochastic processes. However, even the probability calculus could not withstand the non-Newtonian assault from the social and life sciences. The ordinary partial differential equations that traditionally determined the evolution of probability density functions (PDFs) in phase space are replaced with their fractional counterparts. Allometry relation is proven to result from a system’s complexity using exact solutions for the PDF of the Fractional Kinetic Theory (FKT). Complexity theory is shown to be incompatible with Newton’s unquestioning reliance on an absolute space and time upon which he built his discrete calculus.
Highlights
Three centuries ago, Newton transformed Natural Philosophy into today’s Science by focusing on change and mathematical quantification and he did so in a way that resonated with the scientific community of his day
Some of the world’s leading mathematicians have been working on what on what might be a proper replacement for, or extension of, Newton’s physics. They typically begin with the notion that a conservative nonlinear dynamical system with three or more degrees-of-freedom is chaotic [13], which means that its dependence on initial conditions is so sensitive that an infinitesimal change in the initial state will produce a trajectory that exponentially diverges from the trajectory predicted by the original state
The first generalization of the historical kinetic theory argument is made by taking into account the fractal nature of the set generated by the ensemble of chaotic trajectories initiated by an underlying non-integrable Hamiltonian
Summary
Newton transformed Natural Philosophy into today’s Science by focusing on change and mathematical quantification and he did so in a way that resonated with the scientific community of his day. The community agreed that the ordinary differential calculus of Newton and Leibniz, along with the analytic functions entailed by solving the equations resulting from Newton’s force law, are all that is required to provide a scientific description of the macroscopic physical world. In his Mathematical Principles of Natural Philosophy [1], Newton introduced mathematics into the study of Natural Philosophy. We express the purpose of this paper in the form of a hypothesis and present arguments in support of the Complexity Hypothesis (CH): Complex phenomena entailing description by chaos theory, fractional Kinetic Theory, or the fractional calculus in general, are incompatible with the ordinary calculus and are incompatible with Newtonian Physics
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