# Does Zeno's Paradox hold the simplest key to explain the widely-held continuous spacetime belief in physics? [closed]

Richard Feynman in this 60's public lecture claimed it's easy to prove "physical space cannot be discrete automata", otherwise it will soon violate existing physical observations towards the end of this lecture. But he didn't mention any proof or explanation there.

I thought about this further and reasoned under classical analysis framework, only continuous space and time make velocity (not the other feature - position) possible. If space or time is really discrete ontologically, then like Zeno's logic in his famous Zeno's Paradox, an arrow can never move without violating metaphysical "Principle of Continuity"! The essence of Zeno paradox's resolution lies in time and space are continuous, thus you can have possible velocity notion via its position's change along with "measuring" its corresponding time interval, and Zeno tacitly avoided the required velocity notion in any motion to arrive at his famous paradox. If time is discrete automata, then you can only have position along with "counting" its time instants, via counting and summing "countably while even infinitely many" instants with freezing positions, the final position is thus still frozen as logically correctly claimed by Zeno. It's hard to imagine a way here to derive a velocity-like concept without some (infinitesimal) interval notion as later formally introduced as differential with its integral in calculus and coined by Leibniz as dx/dt.

Does Zeno's Paradox hold the simplest key to understand and thus explain the widely-held continuous spacetime belief in physics?

• Possible duplicate: physics.stackexchange.com/questions/28259/… – BioPhysicist Mar 7 at 14:07
• – Sandejo Mar 8 at 6:31
• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Mar 9 at 17:11
• "Is X the simplest way to explain Y?" is an opinion-based question that may or may not invite answers with alternative explanations and therefore may also be too broad/poorly defined to be answered usefully here (see: all the existing answers criticizing the content of your question or providing alternative arguments, but not actually answering the question you asked). – ACuriousMind Mar 9 at 17:13

## 5 Answers

One objection to discrete spacetime for me is that it requires some degree of anisotropy. For example, if the space is formed by a grid with spacements at the Planck scale, some directions should be more compact than others.

It is what happens in crystal lattices, and affect its macroscopic properties.

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Mar 9 at 17:14

There are several presentations of Zeno's paradoxes. One version is as follows (taken from Wikipedia):

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

The resulting sequence can be represented as:

$${\displaystyle \left\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}}$$ This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

There are a number of resolutions to the paradox. The simplest, perhaps, is just to ask "why is it impossible to complete an infinite number of tasks?" Without this premise, the conclusion of the argument doesn't hold.

If you now ask about spacetime being discretized, then it is not even clear what argument there is left to be made. We wouldn't need an infinite number of tasks to be completed, so the argument towards the impossibility of motion becomes even less clear.

Another version, the arrow version, is (again taken from Wikipedia):

In the arrow paradox, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

A possible resolution to this paradox is to simply ask, why must it be the case that "at every instant of time there is no motion occurring?" For example, from the point of view of classical physics the velocity of particle can be defined at any instant. The paradox is trading off on a number of ambiguities surrounding the definition of motion without ever defining what motion is. As written, the argument is full of non-sequiturs.

Again, it is not clear what discretizing space-time has to do with this.

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Mar 9 at 17:14

Zeno's idea of movement was different from ours. That is, as seen from his arrow paradox (I'm sure that he was very well aware of the non-reality of his thoughts). In his mental image of an arrow standing still there isn't the possibility for the arrow to go on. It has to remain forever at the position he images. The arrow hasn't got the internal quality "velocity" (maybe "impetus" came close, but he didn't mention that).
Of course, velocity has to be defined wrt to the external qualities, as there are space and time. Basically, Zeno mentally takes these two away in defining his paradox. But in reality, this can't be done (or can it? Quantum loop gravity makes an attempt) obviously. So the arrow will move.
Do we take spacetime away when discretizing it? Of course. Will an arrow get stuck when frozen for a Planck time? That is, frozen according to a clock that is not discretized in small Planck times. A continuous clock, external to the situation. The Planck time is not an interval of time with duration, but rather represents two points in time which are separated by one Planck unit (time). And in between those points, no time exists. The bulk of spacetime is taken away (the same argument can be applied to space) and we are left with separate points of spacetime.
So, if no spacetime exists between two successive spacetime points, is the situation equivalent to the arrow being stopped in Zeno's mind?
The difference is of course that in this case spacetime is taken away in reality (even the major of), and not just mentally. So one would expect that this time the arrow will stop in reality too, and not just mentally. As Feynmann may have thought too.
Of course one cannot envision this by simply taking away Planck-sized chunks of spacetime from spacetime surrounding us. Here you can read a very readable and interesting piece on quantum loop gravity that attempts to take away these chunks of spacetime and in this way tries to formulate a theory of quantum gravity. Evidence for the discrete Nature of spacetime hasn't been provided yet. Observation on photons coming from distant regions in the Universe (they are predicted to arrive here at different times for different frequencies) yields a negative outcome. And if space is non-discrete, so is time (I guess).

So, does taking away spacetime in the real world stop the arrow (not in the imagination only, like in Zeno's mind, but in real spacetime)? I think it's a big problem to envision how time can be taken away between two separated points of time while still being able to say that the duration between these points is a Planck unit of time. And how can a particle jump to other points when there is no space between them? I'm inclined to say that Zeno's paradox indeed stops the arrow in this case, from which we may conclude that spacetime is not discrete.
To answer your question, I think that Zeno's paradox indeed holds the key, but only if the paradox refers to the real world and not to the imaginary situation of an arrow from which the surrounding spacetime is taken away while in reality there is a continuum of spacetime surrounding it. Transgression of matter to neighboring spacetime points will be a problem (I'm not sure how quantum loop gravity resolves this). Exactly because of Zeno's paradox (which will not be a paradox anymore), which becomes real in this case. I think this is what Feynmann had in mind, more or less, when he claimed that it's easy to prove that "physical space cannot be discrete automata". What he meant by automata though I'm not sure of.

• I believe Feynman was thinking about the famous Weyle's Tile Argument to disprove space can be discrete (en.wikipedia.org/wiki/Weyl%27s_tile_argument) which was basically using isotropy requirement similar to @ClaudioSaspinski answer, but Tile Argument is not settled yet. However, I highly agree with your "a big problem to envision how time can be taken away between two separated points of time while still being able to say that the duration between these points is a Planck unit of time", I think this is a useful argument for my time contimuum hypothesis above. – Double Knot Mar 8 at 5:36
• ... Most critiques will say u can still define average velocity like in computer screen, but like two unrelated parallel universes, since they're totally unrelated, how even to make sense to define their space interval without any underlying manifold curvature connection linking them which seems must require some intermediate info? The implicit assumption in defining any duration actually is based on Lebesgue measure-theoretic notion which is hard to apply to a meager (nowhere dense) topological set. So based on Tile Argument and Zeno's Paradox I do believe spacetime are contimuum for now... – Double Knot Mar 8 at 5:44
• @Double Knot I don't think that QLG exhibits a spacetime anisotropy. This is only the case if you take spacetime away in a trivial manner – Deschele Schilder Mar 8 at 7:52
• @DoubleKnot Weyl's argument makes space discrete in a trivial way. Quantum loop gravity makes spacetime discrete in a non-trivial way. All spacetime points have a distance of one "Plank four-vector" wrt each other. This means the argument of anisotropy doesn't hold. – Deschele Schilder Mar 8 at 11:39
• I know nothing about QLG except I read some place it predicts Big Bounce instead of Big Bang, and it predicts Black Hole is just worm hole which links to another multiverse... seems hotter these days than any string theory? – Double Knot Mar 8 at 17:23

Zeno's paradox is based on incorrect logical reasoning, as he did not take into account that shorter length steps are associated with shorter time steps as well. And incorrect logical reasoning can not hold the key to anything.

• Of you want to resolve that in correctness then you can add to the reasoning that the shorter the length gets slower the turtle and Achilles moves – Ranjit Kumar Sarkar Mar 7 at 17:41
• @Thomas Your argument is what later infinitesimal integration (calculus) shows under continuous spacetime model, dt+dt+... uncountably infinitely such summation (called definite integral or more correctly Lebesgue measure) = finite interval. So the convergence of this uncountably transfinite summation somehow indicates although in our world level it doesn't make sense to accumulate infinite steps within a final time duration, however there's no trouble for those contained infinitesimal intervals (not instants) in a transcendental Platonic world to accumulate in such an infinite manner. – Double Knot Mar 8 at 5:19
• ... these math and the continuum properties (with modern different Baire Categories clearly defined for rationals vs irrationals) are not well understood in Zeno's time, so I wouldn't judge it's a simple logical reasoning error. Zeno ultimately wanted to support his teacher Parmenides's monistic view, believed some perceived plurality is just an illusion of some oneness, definitely there's some deep truth behind it... – Double Knot Mar 8 at 5:22
• @DoubleKnot If you see/experience something moving whilst your logic tells you that it could not move then you don't have to be a rocket scientist to conclude that something is wrong with your logic. I tend to think sometimes that these ancient philosophers made this only up as some form of entertainment. – Thomas Mar 8 at 22:06
• If someone just want to entertain then I'll label them as real sophists playing with superficial word games (actually I view most so called logicians' works as such games). But here Zeno stimulated so many famous philosophers and scientists to try to argue against his borderline metaphysical/scientific question (see this from its wikipedia page) shows its value. Its "reduction up to infinite" method definitely influenced later calculus formulation from Archimedes onward... – Double Knot Mar 8 at 22:15

No. Consider a physical pinball machine, and a very good computer simulation thereof. In the latter, the ball moves in discrete space-time (discretized by CPU frequency and pixels); yet from how the player experiences the ball kinematics, there is no difference between the two.

Clearly, Zeno-style reasoning can be applied to the physical machine's ball instead of arrow, and hence to the computer ball (since there's no perceptible difference between the two). If such a reasoning ruled out discrete space-time, it would therefore also show that a computer pinball simulation cannot exist. And yet they do exist.

• A computer simulation doesn't get rid of spacetime really. Only in the simulation is spacetime discretized. If spacetime is really discretized it still the question if the computer can run its program. According to Zeno, no. – Deschele Schilder Mar 7 at 22:16
• @DescheleSchilder, I don't see how the real spacetime is relevant here, since I'm speaking about applying Zeno's argument to the pixel ball and its simulation spacetime. I see nothing in it that doesn't apply verbatim (unsurprisingly, since for the observer the simulation is indistinguishable from the real world). – Kostya_I Mar 7 at 22:34
• Real spacetime is relevant for the workings of the computer. If real spacetime is discretized it might be that the computer can't function. – Deschele Schilder Mar 7 at 22:38
• In your example, there is nothing physically tangible moving through space from A to B. There are just many things going on that create the illusion of moving if you don't look too closely. – Thomas Mar 8 at 22:22