## Modernity of Zeno’s Paradoxes

**xantox**, 16 January 2007 in

**Philosophy**

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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 paradoxes^{2} 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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12 April 2007, 4:44 am

Zeno’s thought experiments were right. But the assumption that space was continuous was wrong. But if they had done that we wouldn’t have calculus… You could find the number of positions an object was in by dividing the distance by the Planck Length.

16 April 2007, 2:53 am

Hello. It is in fact unknown if space is continuous or not, and this is independent from the existence of a minimum length. In addition, spatial states may be uncertain, allowing discrete geometries to describe continuous fields.

16 April 2007, 9:16 am

Space being continuous has everything to do with the existence of a minimum lenght. That was the main concern that brought up these ideas from Zeno. Don’t quote me on this but I think it was only through the work of Plato that we heard about this paradox, and was used to discredit him. But there was a big debate about if space was made of atoms or a smallest unit and if it was made of small units then you would not be able to move a smaller amount of lenght. I beleive this smallest unit is the Planck Lenght, and this is what allows movement without going through in infinite amount of points. You would just jump every 10^-33cm. This is because if it takes an infinite amount of energy to measure any smaller amount of distance, then that small of a distance could never be observed, because you could never have an infinite amount of energy in one point. If this distance is not observed then from our frame of reference it does not exist and you could not move distances smaller than this. And if something did move that far it would not be observable and would appear to not have moved at all…

17 April 2007, 9:20 pm

This is not a necessary condition in quantum theory, eg in Wheeler-DeWitt cosmology the fundamental Planck length coexists with continuous space and time. This point should be taken with great care: space is no longer assumed classical, and as such can’t be realistically modeled with a sort of “chessboard” where particles would jump from Planck Length 1 to Planck Length 2 and so on. Space should be considered quantum

beforequestioning its continuity or discreteness. First, there is an uncertainty in the position of particles, making their location in the “chessboard” fuzzy and unsharp. Second, quantumness of spacetime should imply that the “chessboard” is itself fuzzy and unsharp. Third, spacetime could not be a fundamental entity, and discrete properties of its underlining structure (not characterized by lengths or times, but still defining a geometry) should not automatically imply that such spacetime is lacking continuous properties. Fourth, some models in topological field theory show both discrete and continuous dual characteristics, so that some deep connections may well exist physically between these apparently opposed notions.18 April 2007, 4:36 am

Then how would an object move if it had to travel along an infinite amount of points? Based on the Zeno Paradox alone would show that it is not and this is how we are able to move. If space is truely continuos then there should be some kind of answer to this Paradox. You would basically have to show that an infinite amount of points that got increasingly smaller, would add up to one distinct value.

19 April 2007, 11:48 pm

This would be indeed physically problematic in a classical continuous model, but this whole picture has changed with quantum mechanics, where the position of a moving particle is undetermined and smeared into a probabilistic cloud. Moreover, if spatial coordinates are equally probabilistically unsharp, a continuous space may be defined which only allows for a finite number of degrees of freedom.

20 April 2007, 3:42 pm

I beleive this only only true for particles that travel close to the speed of light. Unless, they are makeing something up new from quantum loop gravity. But I don’t think anything would ever be confirmed about slow moveing particles reacting in this manner. Mainly because I think it is a property that is given from an object traveling close to the speed of light.

20 April 2007, 10:18 pm

This is true for any velocity, as it is a consequence of the uncertainty principle which states that:

So that when the uncertainty in momentum Δp tends to zero, the uncertainty in the position Δx tends to infinity (a particle with a perfectly defined momentum is a plane wave which is everywhere).

21 April 2007, 6:17 am

But then again, can’t every object traveling at a constant speed say that it is at rest? And then all of their momentums be “perfectly” defined.

21 April 2007, 5:43 pm

In fact, in quantum mechanics a particle is never at rest. Being at rest would mean to always have a perfectly defined position, but this would then imply according to the uncertainty principle to have a nonzero momentum, so that the particle would not be at rest any more. Quantum mechanics is weird and it is not possible to reduce it to the intuitive concepts of classical mechanics. Its reason is not agreement with intuition, but agreement with experiment.