I don't see why I should accept that the mathematical problem of infinite divisibility should prevent movement from starting. — Luke

But you accept that it prevents counting from starting? What is the difference between counting each 1/(2n) number and moving through each 1/(2n)m mark? Each is a physical event. We can even tie them together and say that a machine "counts" each time it passes through a 1/(2n)m mark. If such a movement is possible then such a counting is possible, but we know that such a counting is impossible and so it must be that such a movement is impossible.

But counting is not required for motion (is it?), so it doesn't matter if the counting is impossible. It doesn't make the movement impossible.

But counting is not required for motion (is it?), so it doesn't matter if the counting is impossible. It doesn't make the movement impossible. — Luke

The thing that makes counting impossible is the thing that makes movement impossible, so it doesn't matter that motion doesn't depend on counting. What it depends on is having completed a sequential series of events with no first event – which doesn't make any sense. A sequential series of events with no first event cannot be started.

I still don't understand what you mean by a "first event" in practical terms in relation to motion. The first distance that I count/pass through? But that theoretical first distance has only been defined into existence (or undefined into existence) by purely mathematical operations which are imposed on the assumed total distance travelled (or to be travelled). In reality, nothing prevents me from passing through that distance; especially not having to count it first.

I believe that the overall sentiment expressed by Aristotle's view is vindicated by the Heisenberg Uncertainty principle, which expresses his sentiment by saying that if the position of a moving arrow is determined, then it no longer has a definite state of motion. Hence according to QM, Aristotle is right to distinguish a continuous line from a line divided into intervals.

At any moment in time, try to simultaneously observe both the position and motion of a moving object. In order to maximise one's observational precision of the object's position, one has to pay more attention to features in the visual field that establish the object's position. But this comes at the cost of being less precise when judging the state of the object's motion.

Of course, this experiment when interpreted along classical lines does not demonstrate the uncertainty principle, for it merely demonstrates one's ignorance of the total state of the arrow. But this is an irrelevant argument in the case of Zeno's paradox, for the arguments Zeno presents are phenomenological arguments that appeal only to thought experiments or practical demonstrations in which the motion of the arrow is temporarily ignored while it's position is determined and vice versa. I'm simply saying we have every reason to be phenomenologically suspicious of being able to imagine, or be literally aware of, an arrow's position simultaneous with it's state of motion.

Imagine if we had taken a video-recording of the object's motion in order to establish a per-frame analysis of the object's positions over time. No per-frame analysis will tell us about the object's motion, since for that we need to look at inter-frame differences which is a feature not present in individual frames. This is again, analagous to the uncertainty principle in that motion and position are estimated, or rather constructed, with respect to incompatible features.

The distances aren't just theoretical. When walking a metre the 0.5m mark is an actual point in space that has to be physically passed, as is the 0.25m mark before that, the 0.125m mark before that, and so on.

I don't see any reason to think that the physical act of passing the 0.5m mark is any different in kind to the physical act of saying "0.5", so I don't see any reason to think that the physical act of passing each 1/(2n)m mark can be done without there being a first mark to physically pass but that the physical act of saying each 1/(2n) number can't be done because there is no first number to say.

If I needed to say or count every distance that I moved (as I moved), then I agree that motion would be impossible. But I don't need to.

If I needed to say or count every distance that I moved (as I moved), then I agree that motion would be impossible. But I don't need to. — Luke

Such a motion is impossible even if you don't need to count each distance. The logic behind passing each 1/(2n)m mark is the same logic behind counting each 1/(2n) number; it's a sequential series of events with no start and so cannot be started.

Only if you accept Zeno's assumption that counting is identifiable with motion. But we appear to be going around in circles, as impossible as it may sound. Besides, if it's the same logic, then how can you ever make the first division in order to prove the impossibility of motion?

Only if you accept Zeno's assumption that counting is identifiable with motion. — Luke

You don't need to accept that at all. It just follows from claiming that motion is continuous and so that to get from A to B you have to pass each 1/(2n) mark. This is a sequential series of events with no start, and so logically can't be started.

then how can you ever make the first division in order to prove the impossibility of motion? — Luke

I don't know what you mean by this.

Imagine if we had taken a video-recording of the object's motion in order to establish a per-frame analysis of the object's positions over time. No per-frame analysis will tell us about the object's motion, since for that we need to look at inter-frame differences which is a feature not present in individual frames. This is again, analagous to the uncertainty principle in that motion and position are estimated, or rather constructed, with respect to incompatible features. — sime

The uncertainty principle is derived from the Fourier transform which involves the problem of "the start" (or however you want to call it), in the sense of a time period, which is similar to what Michael is arguing. A time period is defined by frequency, but the shorter the time period, the less accurate is the determination of frequency. The problem is reciprocal, if the time period is too short we can't determine the frequency, if we can't determine the frequency the time period is indefinite. "The start" is the first time period, and the shorter that time period is, the more indefinite any determination made from it is. This is very similar to the problem of acceleration. If a thing is at rest at one moment, then accelerating at the next moment, there must be a time of infinite acceleration.

The uncertainty principle is derived from the Fourier transform which involves the problem of "the start" (or however you want to call it), in the sense of a time period, which is similar to what Michael is arguing. A time period is defined by frequency, but the shorter the time period, the less accurate is the determination of frequency. The problem is reciprocal, if the time period is too short we can't determine the frequency, if we can't determine the frequency the time period is indefinite. "The start" is the first time period, and the shorter that time period is, the more indefinite any determination made from it is. This is very similar to the problem of acceleration. If a thing is at rest at one moment, then accelerating at the next moment, there must be a time of infinite acceleration. — Metaphysician Undercover

It depends on what you mean by "determine". The mathematics of quantum uncertainty refers, at least according to its most literal interpretation, to the logical inconsistency of two or more propositions, in this case that a particle simultaneously possesses a precise position and a precise momentum.

According to this interpretation, Zeno's paradox is a valid argument, and might even be useful in intuitively explicating some of the principle of quantum mechanics, but nevertheless does not prohibit motion, because Zeno's paradox is understood as referring to the modification of a particle so as for it to have a precise position, at the expense of the precision of it's state of motion.

Personally i don't think appealing to physics or mathematics is ultimately relevant in solving the paradox but that Quantum mechanics complements the vagueness of our phenomenological intuitions in many respects.

When I imagine zeno's paradox, I tend to imagine an arrow travelling for a bit and then I stop it momentarily in my imagination and say to myself "This is now the arrow's position. Now how did it get here?". But of course I am not allowed to mentally stop the arrow from moving, for I would no longer thinking of a moving arrow.

Is it even possible to imagine a moving object that has a precise velocity and/or position? Personally I don't think so. I always find myself either fantasising that I have mentally stopped the arrow in order to measure it's position, or that I am entirely ignoring it's position when thinking about it's motion.

It would appear that in order to move from A to B, one would need to arrive at 1/2 the distance between them, and so on and so on — frank

Okay, consider this scenario:
1. So the distance between Mars and Earth is a finite distance of X feet.
2. The ship that we use to go there is a finite size of Y feet.
3. The time it takes to travel the finite distance of X feet is a finite amount of time.
4. The finite distance of X feet is infinitely divisible.
5. The time it takes to travel a finite distance of X feet is also infinitely divisible.
Therefore, the ship changing its location is impossible?

Isn't the mistake in interpreting 4. to mean that a finite size of X feet, since it is infinitely divisible, is also infinitely long if each part of the distance is decreasing and continuing to decrease infinitely small rate? Consider, that the time it takes to travel those smaller distances is also changing at the same rate.

Isn't this just confusing different kinds of infinity?

Isn't the mistake in interpreting 4. to mean that a finite size of X feet, since it is infinitely divisible, is also infinitely long if each part of the distance is increasingly infinitely small? — Walter Pound

I don't think so.

Imagine if in our scenario we used an infinitely small thing instead of a finitely small space ship.

The size of X distance and the time it takes to travel X distance is still finite.

Would this mean that the infinitely small thing takes an infinitely long time to travel a finite distance?
(since we have already assumed that the time taken to travel a finite distance is also finite, this seems false.)
Or would such an infinitely small thing have to travel at infinite speed to travel through a finite distance?

You can use a one dimensional point to mark the location of any object.

There's no need to change the speed of anything. Zeno's paradox just starts with common intuitions about movement through space.

Notice that counting and passing through are not the same?

I don't know what you mean by this. — Michael

This was a bit of a throwaway line for a separate argument (to my 'motion doesn't require counting' argument) that occurred to me late last night of trying to turn Zeno's logic against itself. I'm not sure whether it works, but I was thinking something along these lines:

If you identify counting with motion, or if you define counting as a necessarily physical act (e.g. machine countability), then how can the physical act of halving a distance be achieved if you accept Zeno's conclusion that motion is impossible? Halving distances is required for Zeno's argument, yet it is apparently an impossible act. Therefore, either mathematical tasks are not necessarily physical acts or else halving distances is impossible.

What it depends on is having completed a sequential series of events with no first event – which doesn't make any sense. — Michael

Then it doesn't depend on it.

Maths is just language. We can choose any description we want for movement. You choose descriptions that do not work, and conclude that what we see is impossible. It's wrong-headed.

But think about the theory of motion you've adopted. It allows you to move from one end of Australia to the other without ever being at any point in between. Now that's magical!

Why would you think that?

Or you could be at every point simultaneously. Or time could be discrete and you come into existence and then disappear, only to reappear in a different place.

Or it could all be holographic. Or a product of software.

Which do you favor?

You've lost me entirely.

Just alternatives to continuous motion.

But we already made the point that it is continuous.

Is it? How do you know that?

Halving distances is required for Zeno's argument, yet it is apparently an impossible act. Therefore, either mathematical tasks are not necessarily physical acts or else halving distances is impossible. — Luke

There's a difference between the act of halving a number and there being half a number. We can say that there is an infinite number of rationals between 0 and 1 but that it is impossible to count them all. We can say that the space between two points is infinitely divisible but that it is impossible to move through them all.

Then it doesn't depend on it.

Maths is just language. We can choose any description we want for movement. You choose descriptions that do not work, and conclude that what we see is impossible. It's wrong-headed. — Banno

That there's a half-way point between any A and any B is, allegedly, a physical fact about space rather than just a constructed mathematical premise. So I have no idea what you're trying to say here. That there isn't an infinite series of half-way points between A and B except as a matter of language?

OK, if you prefer, if you do not treat it as continuous, you get a whole bunch of philosophers playing silly buggers with it. :joke:

Well, we can't have that. :smile: