1. Jan 14, 2007

### eehiram

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?

2. Jan 14, 2007

### verty

Well since the real numbers are dense, I imagine that would refute it, but I'm not the person to ask.

3. Jan 14, 2007

### 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.

4. Jan 15, 2007

### eehiram

5. Jan 15, 2007

### AnssiH

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.

6. Jan 15, 2007

### mace2

AnssiH:

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

7. Jan 15, 2007

### eehiram

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

8. Jan 15, 2007

### chroot

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

- Warren

9. Jan 16, 2007

### xantox

We can understand the paradoxes by Zeno of Elea in two ways.

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.

10. Jan 16, 2007

### Hurkyl

Staff Emeritus
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: Jan 16, 2007
11. Jan 17, 2007

### xantox

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: Jan 17, 2007
12. Jan 17, 2007

### Hurkyl

Staff Emeritus
Why would we postulate A and B are true?

13. Jan 18, 2007

### xantox

Because calculus works, and infinite divisibility is mathematically consistent.

Last edited: Jan 18, 2007
14. Jan 18, 2007

### Hurkyl

Staff Emeritus
And just how does that show motion is impossible, and arrows are motionless?

15. Jan 18, 2007

### xantox

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: Jan 18, 2007
16. Jan 18, 2007

### 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: Jan 18, 2007
17. Jan 18, 2007

### Hurkyl

Staff Emeritus
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.

A successor is the same thing as a first element of the set of all subsequent things.

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?)

Why would we do that?

And if we do, how would we find this?

18. Jan 18, 2007

### Hurkyl

Staff Emeritus
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.

19. Jan 19, 2007

### xantox

I'm referring to Zeno's mathematical assessment of the problem in the case of interpretation #1.

This is apparently what I said in :
Yes, I was meaning that it is more than the absence of a last element in a (countably infinite) set.

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?

We do that because the arrow paradox also applies to discrete spacetime, and we wanted to check whether it applies.

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?

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: Jan 19, 2007
20. Jan 21, 2007

### Hurkyl

Staff Emeritus
The point of my comment is that, as far as I can tell, no such thing has been exhibited.

Why do you put "seem" in there?

What does hypercomputation have to do with anything?

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?

No we didn't:

(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.