## Main Question or Discussion Point

Is motion continuous then? I read in my philosophy textbooks that Xeno's paradoxes were resolved in recent centuries (such as by Newton and Cantor). Does string theory or some other theory come into play in the real universe?

Related General Discussion News on Phys.org
verty
Homework Helper
Well since the real numbers are dense, I imagine that would refute it, but I'm not the person to ask.

D H
Staff Emeritus
The resolution to the paradox is quite simple. The time intervals involved get progressively smaller, by halves. The paradox implicitly assumes that because the number of steps is infinite, then so is the total amount of time. However, $\frac 1 2+\frac 1 4+\frac 1 8 + \cdots = 1$. The total time is finite.

To me, Xeno's paradoxes are nothing but wordplay.

At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest at that instant. Now, in following instants, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible.

There is no reason to assume such things as "moments" exist other than in our minds. Ontological considerations are much much more complex than what Xeno's paradoxes can reach.

Regarding "Achilles & Tortoise", you just have to remember that a mathematical description of nature is not the nature itself. Being able to mathematically divide a space to ad infinitum just means you can mathematically describe as small a number as you please. We have to ask, what does this have to do with the nature of reality?

Note that we can define just about any kind of quantity at all, like "viscosity" or "friction" or "acceleration", and then claim that no value of these quantities can be reached by the virtue that before getting to the value you can divide the difference to ad infinitum. But first you have to know how the given value correlates to reality, and how does it change in reality, for reality does not change its properties by dividing numbers.

In a nutshell, it's just wordplay.

AnssiH:

Thanks for the post. It makes a lot of sense.

AnssiH: I see, then. Well...hmmm. It was my impression that Newton and Cantor tried to figure out what happens with infinite sub-division, mathematically, along the lines of integral calculus and alpha-null vs. C. At least, those are the concepts by which I know their work. Does anyone know about their work on continuous motion?

o| Hiram

chroot
Staff Emeritus
Gold Member
The sum of an infinite series can be finite. That's the resolution to Zeno's paradoxes: calculus of series.

- Warren

xantox
Gold Member

The first one is that he is not really 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 by Aristotle could be merely his interpretation.

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 being 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 which brought to quantum mechanics. If infinite divisibility is mathematically consistent, it is not necessarily physically meaningful (see also the Banach-Tarski paradox). 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 : 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 more strong, 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.

Last edited by a moderator:
Hurkyl
Staff Emeritus
Gold Member
We can understand the paradoxes by Zeno of Elea in two ways.
Before we can understand Zeno's paradoxes, we have to first state them. What do you think is the argument that leads to an actual, logical contradiction?

Virtually every presentation I have seen of them is a non sequitor. IMHO that's why controversy exists: not because there is trouble refuting Zeno, but because people can't even agree upon what Zeno was thinking.

(Personally, I think it's an issue with the non-arrow paradoxes is simply about order-types: the assumption that any collection of events must have a first and a last)

Last edited:
xantox
Gold Member
Before we can understand Zeno's paradoxes, we have to first state them. What do you think is the argument that leads to an actual, logical contradiction?
There is no precise agreement upon what Zeno was thinking because there is no actual text from Zeno and these arguments are rather subtle, so, the presentation by Aristotle and others (which may be already an interpretation) has to be necessarily interpreted, and indeed many different interpretations exist. It appears however that these interpretations may be classified in two families.

Aristotle's statement of the first two paradoxes is roughly :
• DICHOTOMY: Motion is impossible, because before arriving at the end, that which is moved must first arrive at the middle, 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.​

So, there is an argument about infinite divisibility (argument A) and absence of movement at any moment (argument B), and it is concluded that there is no movement. Two key interpretations may arise depending on what one assumes as being postulated and as being demonstrated.

1. Both A and B, as well as the existence of movement, are postulated as true. In this case, the statement that they are mutually inconsistent is the actual paradox, and it must arise from a logical fallacy. The solution of the paradox shall be its refutation. It may be for example assumed, like Aristotle did, that Zeno's fallacy is to think that time is not infinitely divisible. Or that Zeno's fallacy is to think that the sum of an infinite series is always infinite. If we assume such fallacy, dropping it allows to restore a consistent reasoning where A and B imply "existence of movement". However, it should be noted that these fallacies are merely assumed by the interpreter and that they are not stated unambiguously by Zeno, who never says that time is not infinitely divisible or that the sum of an infinite series is always infinite.

2. Only A and B are postulated as true. The statement that A or B imply that there is no movement is accepted and is not considered a paradox. Now the paradox is no longer Zeno's statement, which appears perfectly consistent, but it lies outside of it, in the experience of reality: "how can we experience movement, if the truth we just demonstrated is that there is really no movement?". Now there is nothing to "refute", as we are addressing a purely physical problem, requiring a complete ontological theory of time and dynamics, until which we're left with a phenomenological paradox.

Since we know that Zeno intended to teach something about Parmenide's philosophy of a timeless and unchanging world, I tend to favor the second interpretation, which seem also the most interesting one to debate.

Last edited:
Hurkyl
Staff Emeritus
Gold Member
Why would we postulate A and B are true?

xantox
Gold Member
Because calculus works, and infinite divisibility is mathematically consistent.

Last edited:
Hurkyl
Staff Emeritus
Gold Member
Because calculus works, and infinite divisibility is mathematically consistent.
And just how does that show motion is impossible, and arrows are motionless?

xantox
Gold Member
And just how does that show motion is impossible, and arrows are motionless?

Kinematically as in #1 it does not show that, so the paradox may be refuted by assuming some or some other error in Zeno's mathematical analysis (eg he was possibly considering that time is not also infinitely indivisible, or that an infinite series cannot converge, etc.)

However, dynamically as in #2 the statement has a different meaning. Here movement is taken as a physical process, and a continuous movement, as in the dichotomy paradox, would mean that the body assumes a non countably infinite number of states, so that the problem is, more than the absence of a first and last member as you suggested above, rather that each state has no successor. We may believe that such infinite separation between any two events would mean their effective physical dissociation, though of course quantum mechanics even prevent us to allow for infinitely sharp localizations of moving particles. So, considering now the arrow paradox in a discrete setting, we see the world as a collection of "now" states, and we find that movement may not be any possible state, so that newtonian time cannot exist. Here we see that the answer cannot be mathematical, but physical, as what is being questioned is the physical basis of time and motion.

Last edited:
D H
Staff Emeritus
That philosophers are still struggling Zeno's paradox illustrates, to me, the utter arrogance and utter uselessness of philosphy. If a physicist, after scribbling on a white board for the better part of a day, finally arrived at the conclusion $v_{\text{light}}=0$, said physicist would say "#@\$&! Where did I make my stupid mistake?" If a mathematician, after building a new callus by fiddling with math on paper all day long, finally arrived at the conclusion $0=1$, said mathematician would utter a similar four-word remark and begin a hunt for the stupid mistake. Fast vehicles overtake slow ones, arrows fly through the air. Why don't philosophers similarly say, "where is my stupid mistake?"

Last edited:
Hurkyl
Staff Emeritus
Gold Member
Kinematically as in #1 it does not show that, so the paradox may be refuted by assuming some or some other error in Zeno's mathematical analysis (eg he was possibly considering that time is not also infinitely indivisible, or that an infinite series cannot converge, etc.)
What is this "mathematical analysis" of which you speak? The argument

DICHOTOMY: Motion is impossible, because before arriving at the end, that which is moved must first arrive at the middle, and so on ad infinitum.​

is a non sequitur: the conclusion does not follow from the argument. If you are evaluating some different argument, then please share it.

than the absence of a first and last member as you suggested above, rather that each state has no successor.
A successor is the same thing as a first element of the set of all subsequent things.

We may believe that such infinite separation between any two events
Infinite separation? I suppose you mean that there are infinitely many points between them -- most people would use the word "separation" to refer to the distance or duration between the events. I hope you're not confusing anyone (or yourself!) with your choice of word.

would mean their effective physical dissociation,
In what sense would they be dissociated? (And why?)

So, considering now the arrow paradox in a discrete setting
Why would we do that?

and we find that movement may not be any possible state
And if we do, how would we find this?

Hurkyl
Staff Emeritus
Gold Member
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 being static and eternal.
At the risk of going off topic...

It sounds like the basic idea of Parmenides is how special and general relativity are studied: one looks at space-time, rather than space at different times. But it would be wrong to say that change is illusory: when one looks at a worldline in space-time, one can clearly see that a particle has different positions at different times.

It is a mistake to think that, just because we are looking at space-time as a whole, that words like "change" and "motion" should refer to some new (nonexistent) external notion of time.

Of course, formally, there is little to no difference between "space as a function of time" and "space-time". The difference is mainly in how it's studied.

xantox
Gold Member
What is this "mathematical analysis" of which you speak?
I'm referring to Zeno's mathematical assessment of the problem in the case of interpretation #1.

The argument
DICHOTOMY: Motion is impossible, because before arriving at the end, that which is moved must first arrive at the middle, and so on ad infinitum.​
is a non sequitur: the conclusion does not follow from the argument. If you are evaluating some different argument, then please share it.
This is apparently what I said in :
In this case, the statement that they are mutually inconsistent is the actual paradox, and it must arise from a logical fallacy. The solution of the paradox shall be its refutation.
A successor is the same thing as a first element of the set of all subsequent things.
Yes, I was meaning that it is more than the absence of a last element in a (countably infinite) set.

Infinite separation? I suppose you mean that there are infinitely many points between them -- most people would use the word "separation" to refer to the distance or duration between the events. I hope you're not confusing anyone (or yourself!) with your choice of word. In what sense would they be dissociated? (And why?)
I don't use here "separation" to refer to the distance or duration, as we're no longer analyzing the problem in kinematical terms. I refer to the dynamics of the problem : here you have a physical process which seem to require the arrow to go through infinite states, like in hypercomputation models. Is that physically meaningful and if not, would you agree that requiring such condition could mean that events become separated, in the sense of being disjoint?

So, considering now the arrow paradox in a discrete setting
Why would we do that?
We do that because the arrow paradox also applies to discrete spacetime, and we wanted to check whether it applies.

And if we do, how would we find this?
To use a simplistic metaphor, take a still picture of a moving object from a movie. Does it contain in itself any physical property allowing to distinguish it from the still picture of a non-moving object? Do the dynamical relations between frames of the actual movie exist physically as the structure of the pictures themselves, or as additional entities?

It sounds like the basic idea of Parmenides is how special and general relativity are studied: one looks at space-time, rather than space at different times. But it would be wrong to say that change is illusory: when one looks at a worldline in space-time, one can clearly see that a particle has different positions at different times.
It is a mistake to think that, just because we are looking at space-time as a whole, that words like "change" and "motion" should refer to some new (nonexistent) external notion of time.
Of course, formally, there is little to no difference between "space as a function of time" and "space-time". The difference is mainly in how it's studied.
I consider that, since Parmenides was addressing reality as a whole, the remark that there is no time at this level is completely pertinent and again, you should not consider that he was also negating change as we observe it. The paradox is similar to the "problem of time" in quantum gravity, eg how do you derive time from a timeless hamiltonian constraint. This is something openly debated in physics and I'm impressed by the modernity of those who were raising these arguments over 2500 years ago.

Last edited:
Hurkyl
Staff Emeritus
Gold Member
I'm referring to Zeno's mathematical assessment of the problem in the case of interpretation #1.
The point of my comment is that, as far as I can tell, no such thing has been exhibited.

here you have a physical process which seem to require the arrow to go through infinite states,
Why do you put "seem" in there?

like in hypercomputation models.
What does hypercomputation have to do with anything?

Is that physically meaningful and if not, would you agree that requiring such condition could mean that events become separated, in the sense of being disjoint?
Why wouldn't it be physically meaningful? Our current theories (GR, QFT) certainly assert that it is. And what do you mean by "disjoint" here? When did we start considering events to be like sets that may or may not have elements in common?

We do that because the arrow paradox also applies to discrete spacetime, and we wanted to check whether it applies.
No we didn't:
Opening Post said:
Is motion continuous then? I read in my philosophy textbooks that Xeno's paradoxes were resolved in recent centuries...

To use a simplistic metaphor, take a still picture of a moving object from a movie. Does it contain in itself any physical property allowing to distinguish it from the still picture of a non-moving object? Do the dynamical relations between frames of the actual movie exist physically as the structure of the pictures themselves, or as additional entities?
(1) They are part of the picture. (see phase space)
(2) Even if they weren't part of the picture, I don't see how that would prove anything relevant.

Hurkyl
Staff Emeritus
Gold Member
I consider that, since Parmenides was addressing reality as a whole, the remark that there is no time at this level is completely pertinent and again
But it's an equivocation fallacy. When others are discussing the notion of change as we observe it, it is entirely incorrect to invoke some new notion of change when one joins the discussion.

I'm impressed by the modernity of those who were raising these arguments over 2500 years ago.
I think you're reading into things too much. I think this position is just as silly as trying to credit Democritus instead of Dalton with the founding of modern atomic theory -- there is a vague resemblance, but that's it.

arildno
Homework Helper
Gold Member
Dearly Missed
I think you're reading into things too much. I think this position is just as silly as trying to credit Democritus instead of Dalton with the founding of modern atomic theory -- there is a vague resemblance, but that's it.
I agree.
Similarly, there were people in antiquity (Strabo was one of them, I think), that speculated that "little creatures" might cause diseases in people.
Since they did not do any experiments to ascertain whether they were right or not, nor for that matter thought it important to develop instruments that could be used in such experiments, these people don't deserve the recognition that is due to guys like Pasteur and Koch.

They were simply philosophers with fanciful ideas that happened to be right in a very superficial manner.

xantox
Gold Member
The point of my comment is that, as far as I can tell, no such thing has been exhibited.
You already said that, so please let me summarize again to exit from this loop. I said that Zeno's paradoxes may be interpreted as mathematical statements, or as physical statements. That we don't know for sure which is the correct interpretation, as there are at least some reasons to believe that it is the second one, and no certainty that it is the first. That if one decides to interpret them as mathematical statements, then, again:
the statement that they are mutually inconsistent is the actual paradox, and it must arise from a logical fallacy. The solution of the paradox shall be its refutation.
That is : if Zeno was indeed attempting to make a mathematical point, then he was fallaciously considering that A or B implies absence of movement, which, as I noticed above, corresponds to what you said about it being a non sequitur, so that I don't see what else there is to debate here.

Why do you put "seem" in there?
Because modeling motion as a continuous process leads us to an hypothesis, which is uncertain.

What does hypercomputation have to do with anything?
We're discussing about a possible physical process going through infinite states in finite time, and this could be considered computationally equivalent to a Turing machine performing an infinite computation in finite time.

Why wouldn't it be physically meaningful? Our current theories (GR, QFT) certainly assert that it is.
GR is a classical theory and should not be taken as an argument, as our best mechanics is not classical, which lead to suspect that spacetime may not be classical, too. A consistent theory of quantum gravity, not GR nor QFT, could possibly tell whether this is physically meaningful or not.

And what do you mean by "disjoint" here?
I mean physically separated or disconnected in a similar sense as one would say it causally when A is outside the lightcone of B, etc.

No we didn't
(1) They are part of the picture.

(2) Even if they weren't part of the picture, I don't see how that would prove anything relevant.
The point is that if the dynamical relations are additional entities, then trying to derive the dynamical relations from the pictures alone would necessarily lead to phenomenological paradoxes.

But it's an equivocation fallacy. When others are discussing the notion of change as we observe it, it is entirely incorrect to invoke some new notion of change when one joins the discussion.
Ignoratio elenchi. What we're discussing, is these ancient writings, and what they mean. Possibly you consider that there is no other possible meaning in them than the failure to write down a converging sum of a geometrical series. If that is not the case, then discussing any philosophical statement may require to take some risks in order to reach the relevant meaning beyond the simplest interpretations. And here you have one interpretation which considers that the apparent contradiction pointed out by Zeno is the one arising between change as we observe it and the view of a fundamentally unchanging world.

I think you're reading into things too much. I think this position is just as silly as trying to credit Democritus instead of Dalton with the founding of modern atomic theory -- there is a vague resemblance, but that's it.
I said nowhere that Parmenides or Zeno founded or even anticipated some particular modern scientific concept. First, I'm impressed by the courage of their physical abstraction (you may also consider Anaximander, who boldly concluded in ~600BC that earth is finite and floating into empty space). Second, and this is what matters for our discussion, the physical question "how motion and time emerge from an universe fundamentally motionless and timeless" (and, no matter who asks it) is addressing a valid and unsolved (and perhaps unsolvable) problem. Third, they followed certains abstract patterns of thought which happen to match others found at the heart of some modern physical theories, and this seemingly incidental fact, I consider remarkable, as it reveals the profound self-similarity of nature in its ways of allowing itself to be understood.

Last edited:
Hurkyl
Staff Emeritus
Gold Member
What we're discussing, is these ancient writings, and what they mean.
Maybe this is the crux of our disagreement. While studying ancient philosophers may be an interesting endeavor, that doesn't appear to me to be the topic started in the opening post! I've been actively trying to keep the discussion on the topic of whether there is a logical inconsistency in our physical theories, rather than let it turn into a historical survey of ancient Greek philosophy.

While think replying to the rest of your post is best deferred until we settle the above dispute, I do think it's worth responding to this:
GR is a classical theory and should not be taken as an argument, as our best mechanics is not classical, which lead to suspect that spacetime may not be classical, too. A consistent theory of quantum gravity, not GR nor QFT, could possibly tell whether this is physically meaningful or not.
We don't have a "best" mechanics; GR is far superior in some domains, and QFT is far superior in other domains. Incidentally, neither GR nor QFT suggest that spacetime is nonclassical; those suspicions only arise when we optimistically try to apply both theories and see what the combination suggests.

It sounds like the basic idea of Parmenides is how special and general relativity are studied: one looks at space-time, rather than space at different times. But it would be wrong to say that change is illusory: when one looks at a worldline in space-time, one can clearly see that a particle has different positions at different times.

It is a mistake to think that, just because we are looking at space-time as a whole, that words like "change" and "motion" should refer to some new (nonexistent) external notion of time.
It is interesting to observe that in relativity the length contraction between two objects at any point in time depends on the first derivative of the separation with respect to time as expressed by the Lorentz transformations. In other words, someone who moves with respect to A observes the distance to A to be shorter than someone who does not.