The Society for the Diffusion of Knowledge P.O. Box 964, Kaunakakai, HI 96748

   Prior to the advent of post classical physics, theoreticians who studied the natural sciences, viewed elementary substance as real stuff, such as orange marmalade.  Slowly and inexorably, as the institutions and men of science became more splendid, such real stuff began to give way to the more abstract and esoteric jargon of mathematics, especially inclusive of topology, with strings-in-space, manifolds and continuums, notwithstanding.
    It all more less began with the failure of the Michelson-Morely experiment in the attempt to prove the existence of luminiferous ether, essential to the furtherance of classical theories and paradoxes concerning light propagation, such as the the aberration of starlight noticed by astronomers;  ether representing the last bastion of ontological theorization.
    No doubt the verdant field of post classical theorization and advancement in calculus and mathematical studies induced the cultural acceptance of non substance causality, despite the principle thrust of such studies to explain our material world in a material way.  Because of mathematical predictability, the language of mathematics displaced common sense, consequences amplified by Albert Einstein's quantum leap into relativistic premise.  Is it wrong to do this?  Well, it is no more right than the alchemist's theorization and attempt to synthesize Au from adobe.
    In the following gedanken, I call the Split-Pendulum Paradox, mathematics is exposed as a one-way street away from our reality, despite being a powerful tool and science's principle ontological vehicle.

    Given two pendulums swinging from a common axis, one of which is split into two masses passing to either side of a single pendulum swinging between them so that they miss.  What happens to their centers of gravity when they swing past each other?

    To understand this problem, the two blue masses are connected above the axis creating a single pendulum whose center of gravity lies somewhere between the two weights.  When both the red and blue pendulums are set in opposite motion, as they pass each other at the bottom, these respective centers of gravity will pass through each other.  When they do, an interesting paradox arises.
    Though these centers of gravity are not real in the same sense that their respective weights are real in a material sense, their position, in both time and space, reflects the actual movement of their respective pendulums, and thus from an analytical premise evaluating inertial behavior, they are quite real, inasmuch as they faithfully follow real world behavior.  From a mathematical standpoint, they are useful because as point infinitesimals, they represent optimum accuracy, even exceeding the manufacturing precision of the pendulum apparatus itself.  Clearly, though they precisely follow the pendulum masses, the pendulum masses are not in turn physically affected by them.
    Disregarding quantum wave mechanics and friction, heavy masses swinging under the influence of gravity, are thought to follow continuous motion and continuous acceleration.  Their centers of gravity however, cannot move this way, but follow instead, a sequence of discontinuous and jerky steps.  This is because, as the two centers of gravity which are about to pass through each other at coincidence, they can never find themselves in a grazing position essential to a progressive merging to coincidence, since such a condition necessitates their having a finite radius, an obviously disallowed condition, and thus the paradox.
    That from a purely mathematical logical perspective, mass velocity must be intermittent, whereas from a scientific perspective, in due consideration of both the logic and observation of momentum, intermittent motion is impossible.
    Despite the intellectual need and mathematical necessity to precisely track the respective masses, the centers of gravity, which exist only to fulfill an intellectual expectation, must follow instead intermittent motion, as depicted following. thus challenging the traditional and classical view that real objects undergo smooth motion.

    Besides such limitations, further analysis of infinitesimalpoint behavior, immutably reveals that fundamental motion produces random outcomes (Please see Two Point Paradox and Interchange.), a behavior reinforced by actual high energy studies, such as X-ray bombardment of materials, producing random displays of Laue spots which defy mathematical predictability.