We can understand the paradoxes by Zeno of Elea (ca. 470 BC)1 in two ways.
The first interpretation is that Zeno is not denying movement, but rather questioning its continuity, which is what actually leads to the paradoxes. In this sense, we can consider that Zeno is experiencing a kind of technical difficulty, and that the problem can be easily solved with calculus or as a converging sum of an infinite series. This interpretation is however short-sighted in its way of arbitrarily postulating the existence of movement, and just concentrating on the technical argument of the consistency of continuity, which is truly a mathematical problem and not a physical nor philosophical one. It shall be noted that one cannot prove that Zeno intended to contradict that the sum of an infinite series can be finite, as the mention of ‘finite time’ appearing in the report of the paradoxes2 could be merely an interpretation by Aristotle.
The second interpretation is that Zeno basically denies movement, in the extraordinarily modern meaning of Parmenides, who considered change as illusory and the world as static and eternal. Zeno is not denying the appearance of movement, but rather its reality. The paradoxes thus appear at a deeper level, from the comparison between the phenomenon of movement and its disappearance implied by a thorough analysis of its model - either it being continuous (dichotomy paradox) or discontinuous (arrow paradox) -. The question becomes a purely physical question, which must be answered within a physical theory : why the experience of movement if movement appears logically impossible?
In the classical continuous model, the arrow must assume an infinite number of states in order to move from a point to another point. If such an infinite separation between two events, modelled by the absence of the successor of a real number, is equivalent or not to their physical dissociation, is a physical question, on the same level of reasoning as the ultraviolet catastrophe ideas which brought to quantum mechanics.3 If infinite divisibility is mathematically consistent, it is not necessarily physically meaningful (see also the Banach-Tarski paradox).4 This picture further changes with quantum mechanics as, per Heisenberg principle, a particle in a determined motion does not have a determined position. Interestingly, Zeno also gives his name to a quantum effect described by the Misra-Sudarshan theorem :5 if it is observed continuously whether a ‘quantum arrow’ has left the space it occupies, then it indeed never leaves this space.
In a discrete model (arrow paradox), Zeno’s argument is even stronger, and we can find a similar formulation of the argument in loop quantum gravity on the basis that time, being a pure gauge variable, is fundamentally nonexistent.6
- • DICHOTOMY: Motion is impossible, because before arriving to the end, that which is moved must first arrive at the middle, and so on ad infinitum.
• ACHILLES: The slower tortoise cannot be overtaken by the quicker Achilles, as he must first reach the point where the tortoise started, from which it has already left, and so on ad infinitum.
• THE ARROW: An arrow shot from a bow occupies an equal space when at rest, and when in motion it always occupies such a space at any moment, the flying arrow is therefore motionless. [↩]
- Aristotle, “Physics”, VI:9 [↩]
- A. Einstein, “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (”On a Heuristic Viewpoint Concerning the Production and Transformation of Light“), Annalen Der Physik, 1905. [↩]
- S. Banach, A. Tarski, “Sur la décomposition des ensembles de points en parties respectivement congruentes”, Fundamenta Mathematicae, 6, 244-277 (1924) [↩]
- B. Misra, E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory“, Journal of Mathematical Physics, 18, 4, 756-763 (1977) [↩]
- J. Barbour, “The end of time“, Oxford University Press (2001) [↩]
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