## INFINITE DESCENT

BY Edward R. Close, PhD, PE, DSPE

INTRODUCTION
The purpose of this discussion is to explain why the method of infinite descent, first articulated in modern times by the French Mathematician Pierre de Fermat, is so important to the paradigmatic shift from the simple materialism of the current scientific paradigm to TDVP, the Close-Neppe model of reality incorporating the organizing action of consciousness into the laws of physics. Application of the method of infinite descent to the mathematics of quantum physics has been largely overlooked and/or ignored by mainstream scientists because of the failure to recognize that ‘the calculus’ of Newton and Leibniz, while wonderfully efficient at solving problems on the macro-scale, does not apply at the quantum scale. Integral and differential calculus, called ‘the calculus’ for more than 300 years, has blinded mathematicians and physicists to the fact that it is just one of several possible calculi, and is not well suited to the quantum realm where the substance of reality is quantized.
Newton’s calculus depends upon variables that can be infinitesimally small, that is, they can approach zero. In the quantized world of particle physics, real variables cannot approach zero. Smaller and smaller values have a limit, that is to say they can only approach a finite limit: the smallest quantum. Another way of saying this is that there is a ‘bottom’ to their infinite descent.  The power of the method of infinite descent lies in application to finite, quantized reality.

THE CALCULUS OF DISTINCTIONS
Mathematics and science should consist of the simplest possible expressions reflecting the elegant structure of the extent of Space, Time and Consciousness, and the substance of matter, energy and mind. The calculus of distinctions (CoD) does this. Adapted from George Spencer Brown’s Laws of Form, the CoD sets the smallest possible quantum equal to one and describes everything in terms of that basic unit, which we call the TRUE quantum unit. This means that all mathematical descriptions of quantized reality are expressed by integer (whole-number) equations. The CoD differs from Brown’s calculus of indications in several ways, but the two most important differences are: 1. Distinctions are quantized and 2. Existence is central.

INFINITE DESCENT
Fermat used infinite descent very effectively to prove things wrong. It works like this: Suppose you suspect that no two whole numbers cubed (X3 and Y3) added together can equal a third whole number cubed. First assume the opposite of that which you wish to prove: Assume there are two numbers X and Y, that when cubed add up to a third number, Z cubed. If you can show that the existence of three such whole numbers leads to a second set of such whole numbers, that are smaller than the first three, then that set will lead to another smaller set, and so forth, until one of the numbers in a subsequent set is 1. But any number cubed plus 13 =1 is always smaller than the next whole number cubed, so we have a contradiction. Since the original assumption (the assumption that two integers cubed can add up to a third integer cubed) leads to a clear contradiction, that assumption is false, and you have proved that it is false by ‘infinite descent’.

This is an exceptionally powerful method for use in quantum physics where, if the right basic unit is used, all mathematical descriptions are expressible in equations of integer (whole-number) variables. Such equations are called Diophantine equations, after Diophantus, a Greek mathematician. When we say we are dealing with Diophantine equations, it just means that the equations of interest are expressed in terms of whole numbers. Applying the CoD with the minimum quantum unit as the unitary distinction avoids the quantum-level errors incorporated in the calculus of Newton and Leibniz, customarily used in the mathematical physics of the current paradigm. Using the CoD and a simple geometric procedure I call dimensional extrapolation, we have solved some of the puzzles of quantum mechanics that have baffled physicists for many years.

In a sense, “infinite descent” is a misnomer. The regular calculus of mainstream physics actually involves infinite descent, while the mathematical method called infinite descent depends upon the descent stopping at the quantum level. Fermat called it infinite descent because it involved the logical descent from any whole number, however large, and the integers (the natural numbers, 1,2,3,…) define what mathematicians call ‘ countable infinity’. So infinite descent is from any number, however large, to the smallest integer, or until a contradiction is obtained.

Stay tuned for more basics in the new consciousness-based scientific paradigm.