We are all prisoners of history. The incredible progress of science has produced an intimidating body of knowledge that is more than any one person can hope to grasp. For scientists, there are a number of basic concepts that are so deeply interwoven with every part of science as to eliminate any curiosity as to their ultimate validity. Most of the time, this is a very good thing. Witness the progress made. The primal such concept is the amazing applicability of mathematics to physics. In particular, there is the encompassing scope of the mathematics of continuous variables (this includes the Calculus invented by Newton and Leibniz).
Given a grand theory, such as Newtonian Mechanics, we find that its range of applicability extends both upwards to the motions of planets and stars, and downwards to the motions of atoms in a gas. Unfortunately no one mathematical framework seems to cover the fullest range of scales; we have quantum mechanics at the bottom and general relativity at the top. Further, as we go down in scale, we ought to expect greater simplicity, but instead, so far, we often find the opposite.
But once in a while, it’s healthy to question why we accept certain concepts as absolute fact. A good example from the past is the Copernican Hypothesis versus the Ptolemaic Hypothesis, that the Sun, rather than the Earth, is the center of the Universe. Many had a hard time giving up the Earth-Centric viewpoint. What few realize today is that the real reason for believing in the Copernican Hypothesis had nothing to do with the ultimate truth. Prior to the Foucault Pendulum, Newton’s Laws and other objective facts, it was simply a matter of esthetics, economy [1] and of passing the test of Occam’s Razor. From a mathematical point of view, the motions of the planets can be represented or calculated under the Copernican system or under the Ptolemaic system, without exception. Of course, the cost of sticking to the Ptolemaic system would grow and grow as the scope of astronomy expanded. Given two competing systems that are both in line with experimental results, we choose to believe as the truth, the one that results in the greatest overall economy of representation, thought and computation. Those issues aside, an accepted representational system may be no more correct than another competitive system that yields the same predictions regardless of the difficulties involved. What is true is that the factor that represents the relative economic efficiency of one system over another can grow to very large yet always-finite values.
We all would like to know the most basic, ultimate laws of nature, with the hope that all higher-level facts of physics, chemistry and even biology will be derivable in principle. If FN turned out to be true, then the ideas of DP and systems like DM might make sense. The problem is that it is very difficult to leap over the intellectual commitment everyone has to the continuum. Intuitively we easily accept, as absolutely correct, the idea that a vector can represent a velocity and that we can represent the resultant of two velocities by the simple addition of two vectors. We have learned how to suspend such beliefs in order to gain an intuitive understanding of the consequences of Relativity. In order to understand DP one might have to suspend belief in physical continuity, translational and rotation invariance, and the idea that things like motion just happen. Our experience with the awesome scope of the calculus makes our task hard, but we can be encouraged by the remarkable fact that the calculus is spectacularly successful at modeling many processes that we know to be basically discrete. Good examples are in electrical and nuclear engineering where users of the calculus assume that change in charge or rate of fission are continuous, differentiable quantities and nevertheless they get the right answers! Conversely, discrete computer programs are good at modeling the evolution of systems with continuity. So, try your best to imagine that the ages old process of admitting discrete models into physics (atoms, electrons, photons, spin, group theory in physics, DNA…) has continued until concepts of space and time come knocking, asking for admission.
