The Zeno Paradoxes, Metaphysics and Modern Quantum Mechanics


Zeno was an ancient Greek thinker whose mathematical paradoxes are of greater importance to modern thought than is realized. He was a disciple of Parmenides of Elea whose followers were of the school of philosophers known as the Eleatics. The Eleatics produced brilliantly insightful arguments showing how other thinkers’ premises led to contradictory conclusions and could, therefore, not be true.

The Zeno paradoxes have interested mathematicians for ages, especially with the fundamental questions about the infinite divisibility, or otherwise, of space and time that it raises. What is the smallest unit or instant of time? Is there a smallest unit of space? Is possible to locate a mathematical point in space and time?

Tolstoy, in his War and Peace, had argued from the standpoint of the common assumption that space and time is a continuum. Einstein’s Relativity Theory, and his efforts at a Unified Field Theory assumed a spacetime continuum, though, ironically, he was the one who built on Planck’s work by the conjecture that electromagnetic radiation is released in energy quanta localized at points in space and which (according to Einstein) come in discrete packets and can only be absorbed and emitted in wholes. It was only after Einstein’s work that the word “quantum” and the Planck Constant (h) came to be used to refer to the smallest amount in which physical quantity exists in nature and in multiples of which it increases or decreases.

Quantum Mechanics was founded in the late 1920s from a reconciliation of the interpretation of Heisenberg’s Uncertainty Principle and Schrodinger’s wave equation; and has, since that time, been the subject of extensive metaphysical and philosophical debate, for Quantum Mechanics raises basic philosophical questions about our universe which are of the same essential nature as those raised by the Zeno Paradoxes.

One of the best known of Zeno’s Paradoxes goes forth: Consider a race between Achilles and the tortoise. Achilles allows the tortoise a headstart because he is faster. The race starts with Achilles at point A and the tortoise at point B. By the time Achilles gets to point B, at which the tortoise started off, the tortoise has moved a little further to point C. When Achilles reaches point C, the tortoise has moved further still to point D closer to C than C was close to B. When Achilles reaches point D, the tortoise has moved to point E closer to point D than D was close to point C; and so ad infinitum such that Achilles will never catch up with the tortoise.

Zeno’s argument is more that just amusing, for if our assumptions of a spacetime continuum were correct then it is difficult to explain why Zeno’s argument should not be true! But the fact that we do not observe this paradox in nature raises questions about our assumptions that spacetime is a continuum. The significance of the Zeno Paradox is that we had had, for centuries, conceptual theoretical grounds, before Planck and Einstein, to believe in the idea of a quantized spacetime universe. The discovery of Quantum Mechanics should only have confirmed our smart hunch from the Zeno paradox that we live in a broken or fragmented spacetime universe. The question which Zeno unwittingly raised about whether spacetime is a real or apparent continuum appears to have been settled by Quantum Physics.

Heisenberg, in a very important paper in the late 1920s, showed that if the basic assumptions of Quantum Mechanics were right then it should be impossible to determine accurately the position and momentum of a particle at a given time. Some had misinterpreted his argument to mean that the experimenter cannot determine the position and momentum of the subatomic particle only as a result of the limitations of the experimenter and the type of experimental set up he must, by necessity, make do with. Physicists have, however, stressed that the indeterminacy principle is not a consequence of the limitations of observer but a fundamental property of nature due to the fact of the finiteness of quantum action in nature.

About the same time as Heisenberg’s work in the 1920s, Schrodinger developed what came to be known as wave mechanics (in contrast to “matrix mechanics”). In his wave mechanics, he addressed himself to the problem of developing an equation for “matter waves.” He introduced the famous Schrodinger wave equation which, according to Bohr’s Copenhagen interpretation, measures the probability that certain observable quantities would take certain values at a specified location. The so-called quantum “jump” in the mechanics is a probabilistic event to the effect that the motion of particles came to be seen as obeying laws of probability.

The general philosophical implications of Quantum Mechanics are profound. To begin with, it would appear, from the Zeno Paradox, that we live in a quantized spacetime universe because we have to. There simply is no way for magnitude to increase or decrease in physical actions except by a unit, h, greater than zero. Our impression of an unbroken flow of transformations in spacetime may be compared to the illusory appearance of unbroken flow in a cartoon animation. One has not fully absorbed the implication of the comparison of a broken stream or flow in spacetime to a cartoon animation until it begins to dawn that our naive materialistic notions of being, reality and existence might, after all, be mistaken.

The unfolding of the bizarre world of subatomic particles, in the field of quantum mechanics, stretches the imagination and challenges long held and cherished materialistic philosophies. If a fundamental particle constituent of nature assumes a proper state only when a measurement is taken, to what extent can we speak of the particle as “real” in our naive understanding of that word? What sense does it make to speak, as some do, of a distinction between the classical world of macroscopic objects, in which things are “real,” and the microscopic world of “quantum particles” in which it is admitted that things are not “so real?”

Could phantom particles add up to ontologically substantial “things?” Why do some “mind theorists” continue to assert dogmatically that dualistic philosophies have been consigned once and for all to the trash bin of history when physicists, like Stapp, borrowing insight from quantum mechanics are proposing interactive subject-object dualistic theories which are merely sophisticated versions of the old Berkeleyan-type idealism. Some leading physicists, such as Wolfram and Deutsch, have even suggested that we might actually be something like consciousness-brains immersed in the output of a virtual reality generator.

Everett’s “many world” solution of the “measurement problem” was the pioneer attempt, in what are now “multiverse” theories, which propose that our world is a virtual reality projection. In his original “many world” theory, Everett suggested that the universe might be constantly splitting into a stupendous number of branches, all the result of “measurement” interactions, and (in his view) because there exists no entity outside the system that can designate which branch is the “real world” we must regard all branches as “real.”

The multiplication of variations of the basic Young double-slit experiment (the delayed choice and quantum eraser, for instance), using subatomic particles, gives us a peep, from a fresh angle, into a world of causality we’d never dreamed of. In the crazy world of subatomic particles, one could actually take a decision in the future to determine an event in the past!

Indeed, there is more under the sun than we had ever dreamed of in our materialistic philosophies. How far physicists have come from the naive materialism of the nineteenth century world!

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