Any operation requires a duration of time in order to occur. Therefore in a finite amount of time that sequence of operations would not be completed. A supertask is logically impossible. — Metaphysician Undercover

This is why I find the use of the aforementioned geometric series to address the paradox to be nothing more than trickery. It assumes from the start that it takes a finite amount of time to travel some finite distance (e.g. 10 seconds to reach the half way point), and then extrapolates from there. But obviously if your reasoning assumes that it takes 20 seconds to get from A to B (which you have done if you've also assumed a constant speed), then you're going to conclude that it takes a finite amount of time to get from A to B.

So instead of halving the unit of time for each successive half way point, why not double the unit of time for the previous half way point (e.g. by defining a new unit of time for each successive half-way point and considering that to be the unit that is used to measure the time spent)? The logic is the same, but the maths doesn't work out the way the "solution" wants it to.

E.g. when considering 0 - 0.5m, define the time as 1 unit. But then when considering 0.5m to 0.75m, define that time as 1 unit and so the time from 0 - 0.5m is 2 units, and so on. What's the sum then?

But even then, the "it would take an infinite amount of time" claim isn't really what the paradox is implying (contrary to my account of it at the start). It's more concerned with the logic of actually being able to make a move at all.

And I believe atomic electron transition is a known example of discrete motion in nature. — Michael

If you are thinking of discrete quantum states of electrons in an atom, that is not an obvious example of discrete motion (except in a generalized sense of "motion" as "change").

If you are thinking of discrete quantum states of electrons in an atom, that is not an obvious example of discrete motion (except in a generalized sense of "motion" as "change"). — SophistiCat

Why not? The electron's position is a value in its quantum state. So going from one quantum state to another involves [or can involve] going from one position to another.

Why not? The electron's position is a value in its quantum state. — Michael

No, it isn't.

This is why I find the use of the aforementioned geometric series to address the paradox to be nothing more than trickery. It assumes from the start that it takes a finite amount of time to travel some finite distance (e.g. 10 seconds to reach the half way point), and then extrapolates from there. But obviously if your reasoning assumes that it takes 20 seconds to get from A to B (which you have done if you've also assumed a constant speed), then you're going to conclude that it takes a finite amount of time to get from A to B. — Michael

We could begin with the assumption that there is no such thing as a finite amount of time. I think this is a reasonable assumption, and those who argue that time is continuous would agree. Any application of a point, to divide time, is an arbitrary application, and is not actually being applied to any real point in time, so there is no real indication as to where that point actually is.

So instead of halving the unit of time for each successive half way point, why not double the unit of time for the previous half way point (e.g. by defining a new unit of time for each successive half-way point and considering that to be the unit that is used to measure the time spent)? The logic is the same, but the maths doesn't work out the way the "solution" wants it to.

E.g. when considering 0 - 0.5m, define the time as 1 unit. But then when considering 0.5m to 0.75m, define that time as 1 unit and so the time from 0 - 0.5m is 2 units, and so on. What's the sum then? — Michael

Now I don't understand what you are doing here. A unit of time is set out, determined, by some physical activity. You cannot just randomly change your unit of time, so that the same activity which takes 1 unit of time will later take 2 units of time, and then 4, etc.. What kind of measurement of time is that? We will just end up with an infinite amount of time in a finite motion, which resolves noting.

Now I don't understand what you are doing here. A unit of time is set out, determined, by some physical activity. You cannot just randomly change your unit of time, so that the same activity which takes 1 unit of time will later take 2 units of time, and then 4, etc.. What kind of measurement of time is that? — Metaphysician Undercover

I mean something akin to going from saying "0.001 second" to saying "1 millisecond". We just define new units of time (e.g. "nanosecond", "picosecond", etc.) so that the time taken is always considered in natural numbers and not in fractions/decimals.

That's interesting because then you are trying to convert the uncountable infinity to a countable infinity. I don't think that this is possible, and it may demonstrate that there is a real difference between the uncountable and countable, and I was earlier arguing against any such difference.

What this would require is a starting point, a definable, discrete, unit of time. Where would you get that from? An arbitrary designation would not do, because it would be just like saying we're making 1/64 our smallest divisible unit, arbitrarily. We would need an actual smallest unit, demonstrably indivisible, as the starting point. The Fourier transform indicates that the smaller the period of time, the more uncertainty there is in determining it, so you would have to get beyond that problem.

It might be indivisible at a certain scale, but it's not indivisible at every scale — Michael

This just unnecessarily maintains the paradox. There is no half at any scale. There is just a fleeting representation of such which is necessitated by a desire to measure.

Even if we are to assume an operation which requires a zero duration of time, — Metaphysician Undercover

There is no such thing as zero duration. If there was, then the flow of duration (time) would have to stop and then what. Stop for how long? How does it restart? Duration (real time) is continuous and heterogeneous. It never stops and cannot be seen as stopping. Scientific time (clock time) is just a movement in space (not real time) that is symbolic and is used to approximately establish simultaneity. This is something different and shouldn't be given ontological significance. Doing so leads to all kinds of paradoxes such as those associated with Zeno's and Relativity's.

If there are paradoxes one must immediately look at issues with assumptions in order to resolve them. In this case the assumption is that time is divisible and homogenous and that is what Bergson challenged.

There is no half at any scale. — Rich

Sure there is. If the object is to travel 10 metres then it passes the half-way point after 5 metres. And this is true even if we're not measuring it.

So what are you saying? That it doesn't have to first travel 5 metres, or that it's incorrect to say that 5 metres is the half-way point unless we're measuring? Because I think both are ridiculous.

There is no such thing as zero duration. If there was, then the flow of duration (time) would have to stop and then what. Stop for how long? How does it restart? Duration (real time) is continuous and heterogeneous. It never stops and cannot be seen as stopping. Scientific time (clock time) is just a movement in space (not real time) that is symbolic and is used to approximately establish simultaneity. This is something different and shouldn't be given ontological significance. Doing so leads to all kinds of paradoxes such as those associated with Zeno's and Relativity's. — Rich

Actually, my suggestion was a change (operation) which requires zero amount of time. This implies that state A is simultaneous with state B, but are contradictory, such that state A changes to state B without any time passing. I agree that this is random nonsense, and incomprehensible, just like your description of "zero duration", but I was just trying to make sense of the proposed "supertask", which also appears to be nonsense.

Sure there is. If the object is to travel 10 meters then it passes the half-way point after 5 meters. And this is true even if we're not measuring it. — Michael

Space and time must be thought of in a different way as not being divisible. An object doesn't travel half-way. It moves from here to there in one indivisible motion. There is no half in a continuously flowing and changing space.

I agree. Implying that zero duration has ontological meaning creates irresolvable issues as does trying to split time and space into a series of homogeneous, divisible units. One must cease using mathematical descriptions to describe living experiences. It doesn't work.

There is a half way point between the start of a 100m line and the end, and this is true even if we don't plot it, which is why I don't understand aletheist's and apokrisis' objection at the start. — Michael

We do not have to treat every halfway point as a discrete step in the motion from the start of that 100-m line to its end. We can traverse the one full interval (100 m) without individually traversing infinitely many half intervals (50 m, 25 m, 12.5 m, etc.).

We do not have to treat every halfway point as a discrete step in the motion from the start of that 100-m line to its end. We can traverse the one full interval (100 m) without individually traversing infinitely many half intervals (50 m, 25 m, 12.5 m, etc.). — aletheist

I don't get this. You do pass the half-way point (after 50m). And you do pass the quarter way point (after 25m). And so on, ad infinitum.

Space and time must be thought of in a different way as not being divisible. An object doesn't travel half-way. It moves from here to there in one indivisible motion. There is no half in a continuously flowing and changing space. — Rich

As above, I don't get this. You do travel half the way before you reach the end. That's just a fact that's entailed by continuous motion. You don't just teleport from the start to the end. That would be discrete motion.

We do not have to treat every halfway point as a discrete step in the motion from the start of that 100-m line to its end. We can traverse the one full interval (100 m) without individually traversing infinitely many half intervals (50 m, 25 m, 12.5 m, etc.). — aletheist

Yes, this is the whole issue. We are layering symbolic notions of divisibility onto a continuous flow, leading us to paradoxical concepts. We must stop using mathematics for describing life experiences. I saw Achilles moved from here to there. That is what happened. He didn't move half-way of anything.

As above, I don't get this. You do travel half the way before you reach the end. That's just a fact that's entailed by continuous motion. — Michael

But I didn't move half-way. I moved from here to there. In retrospect to may try to figure out what half-way might have been and you may be approximately correct with your measurements, but my motion was one motion as you viewed it and add I experienced it - the two being totally different.

To understand experiences one must understand from the point of consciousness, not via some mathematical symbol or equation. Observe what you see.

But I didn't move half-way. I moved from here to there. In retrospect to may try to figure out what half-way might have been and you may be approximately correct with your measurements, but my motion was one motion as you viewed it and add I experienced it - the two being totally different.

To understand experiences one must understand from the point of consciousness, not via some mathematical symbol or equation. Observe what you see. — Rich

As you continuously moved from A to B in one smooth motion you passed the half way point, and did so even if nobody was measuring. What's so problematic about this?

I don't get this. You do pass the half-way point (after 50m). And you do pass the quarter way point (after 25m). And so on, ad infinitum. — Michael

Yes, you pass each of those arbitrarily identified "points"; but each instance of doing so is not a separate, discrete step in the continuous motion of traversing the entire 100-m line.

The issue is attempting to find a half-way point when there isn't any. The problem begins with attempting to use a symbolic, 1/2, in a continuous flowing and changing motion. Can you find anything called half-way in the universe? No. Only when you start trying to symbolically attempting to dissect in memory does such things begin to emerge. I don't actually experience 1/2. I may label it as such for some practical application sometimes after the motion is accomplished.

It order to relieve oneself of mathematical symbolism of life experiences, one must stop layering symbolic concepts and observe what one is experiencing. I run from here to there. I don't run half-way. This is a very important observation to make. No one runs half-way. I don't hear letters. I hear sounds. I don't hear notes. I hear sounds. Symbols are awful representation of experiences.

We must stop using mathematics for describing life experiences. — Rich

I still think that this is an overreaction. We can still use mathematics for describing certain life experiences, depending on our purpose in doing so.

Yes, you pass each of those arbitrarily identified "points"; but each instance of doing so is not a separate, discrete step in the continuous motion of traversing the entire 100-m line. — aletheist

It doesn't matter if you don't consider the movement to be in separate, discrete steps. The problem is that it has to pass through discrete points (which exist even if I'm not measuring; it would be absurd to say that there isn't a half-way point between A and B unless I'm there with a ruler).

The issue is attempting to find a half-way point when there isn't any. The problem begins with attempting to use a symbolic, 1/2, in a continuous flowing and changing motion. Can you find anything called half-way in the universe? No. Only when you start trying to symbolically attempting to dissect in memory does such things begin to emerge. I don't actually experience 1/2. I label n it as such for practical reasons sometimes after the motion is accomplished.

It order to relieve oneself of mathematical symbolism of life experiences, one must stop and observe what one is experiencing. I run from here to there. I don't run half-way. This is a very important observation to make. No one runs half-way. — Rich

There is a half-way point. If one object is 10 metres away from another then the half-way point is 5 metres between them, and this is true even if nobody is there to measure.

you can try your best to represent life experiences using symbols, but try as you might "1" does not in any way describe the experience of going from a bed to a bathroom and neither does the words (though early modernist writers tried their best the result being unreadable novels).

Symbols are not experiences.

It doesn't matter if you don't consider the movement to be in separate, discrete steps. — Michael

It has nothing to do with how I consider it. The movement does not actually consist of an infinite series of separate, discrete steps. It is simply a single, continuous motion from the start of the 100-m line to its end. This is what I mean when I say that the line itself does not actually consist of infinitely many separate, discrete points; it is simply a single, continuous line.

It has nothing to do with how I consider it. The movement does not actually consist of an infinite series of separate, discrete steps. It is simply a single, continuous motion from the start of the 100-m line to its end. This is what I mean when I say that the line itself does not actually consist of infinitely many separate, discrete points; it is simply a single, continuous line. — aletheist

It doesn't matter if the movement doesn't actually consist of an infinite series of separate, discrete steps. It still has to pass through an infinite series of separate, discrete points. There is a half-way point that must be passed before the end is reached, there is a quarter-way point that must be passed before the half-way point is reached, and so on ad infinitum.

you can try your best to represent life experiences using symbols, but try as you might 1 does not in any way describe the experience of going from a bed to a bathroom. — Rich

We agree that the phenomenal experience cannot be modeled adequately by mathematics, or even by other symbols like narratives; but various other aspects of it can be - again, depending on the purpose of the model.

It stills has to pass through an infinite series of separate, discrete points. — Michael

Only if all of those points actually exist, which is precisely what I deny. The line does not consist of separate, discrete points; it can only be modeled as having separate, discrete points.

Your claim, as I understand it, is that the line does consist of infinitely many separate, discrete points, and thus can only be modeled (or "considered') as continuous. This seems to be our basic disagreement.

Only if all of those points actually exist, which is precisely what I deny. The line does not consist of separate, discrete points; it can only be modeled as having separate, discrete points.

Your claim, as I understand it, is that the line does consist of infinitely many separate, discrete points, and thus can only be modeled (or "considered') as continuous. This seems to be our basic disagreement. — aletheist

Of course the points actually exist. There actually is a half way point between the start and the end of a 100m line. There actually is a quarter way point. And so on. Are you denying this?

I am not sure Michael is even clear about what kind of argument he is trying to make. Syntactically, it is a purely logical argument, and it stems from his misunderstanding of mathematical concepts. He is, in effect, committed to the view that the only collections that are allowed to exist are those that are isomorphic to the natural numbers, and anything else is conceivable.

Of course the points actually exist. There actually is a half way point between the start and the end of a 100m line. There actually is a quarter way point. And so on. Are you denying this? — Michael

Points and lines do not actually exist; they are mathematical abstractions that we use to model things that do actually exist, like objects moving from one place to another. A line is simply the path through space over time that an object would trace if it were to move with constant velocity. In that sense, the concept of motion is more fundamental than the concept of a line; and as such, the object's path through space over time is more accurately modeled by an unbroken continuum than by an infinite series of separate, discrete locations.