Relevant to Zeno’s argument?! It is the heart of his argument.

“Zeno’s argument is that if an object is resting at every point in time then it is not moving.”

Those points in time are durationless time points. Zeno’s argument is not about what rest is; it aims to show the impossibility of motion.

You don’t seem thoroughly familiar with Zeno’s arrow argument. You may want to consult good sources on Zeno’s paradox, such as Wesley Salmon’s classic anthology ZENO’S PARADOXES. You can look at the entry “instants” in the index of the book and especially the sub-entry “zero duration” under “instants.” Let me quote Adolf Grünbaum’s words from that book:

“It is a commonplace in the analytic geometry of physical space and time that an EXTENDED straight-line segment … is treated as ‘consisting of’ UNEXTENDED points, each of which has zero length. Analogously, time intervals of positive duration are postulated to be aggregates of instants, each of which has zero duration.” (p.176)

“You have yet to show me an example of a thing with ‘no duration’. Not merely declare that there is such a thing as ‘no duration’. You have to show it.”

You could direct your challenge to Einstein. His relativity theories postulate “point events,” i.e. events with zero temporal duration, according to which ANY physical process is made up of point events. You may want to look at a good book on special theory of relativity.

]]>The concept of motion is meaningful because it refers to something that can be experienced. However, the concept of “no duration” is meaningless because it does not refer to something that can be experienced.

You have yet to show me an example of a thing with “no duration”. Not merely declare that there is such a thing as “no duration”. You have to show it.

]]>They interpret it the way I said, i.e. an instant lasts exactly zero seconds literally. So does Zeno, otherwise he couldn’t claim the arrow is at rest at every instant in setting up his paradox.

“What does ‘no duration’ mean in real life? Can you show me something that has no duration?”

It means the following. From 3 o’clock to 4 o’clock, say, there are a continuum of infinitely many zero-second instants. Similarly with spatial points. Any spatial interval contains a continuum of infinitely many zero-sized spatial points. (Unless those scientists are atomists about time and space.)

“If physicists take it literally, the way you do, then both you and physicists are wrong. If physicists don’t take it literally then it means that only you do, and therefore, that it is only you who’s wrong here. Which one is the case?”

Physicists, Zeno and I are making exactly the same claims. In my article I assume that an instant has zero duration, because that is the premise Zeno rests his arrow paradox on. Then, using that assumption I expand Zeno’s paradox into my “new twist.”

“What I had to do in order to resolve your paradox is to think of situations, no matter how weird, when stating that something has no duration is actually meaningful. An example of such a situation is when you want to measure the magnitude of an object in terms of a whole number of units of some kind e.g. the height of a man in terms of a whole number of kilometers. Michael Jordan is 0.00198 km tall. However, making such a measurement, within the context of our needs, would be an overkill. We only want to know how many whole kilometers are there in the height of Michael Jordan. And the answer is exactly ZERO whole kilometers. When something is so small that measuring it in terms of commonly used units would be an overkill in relation to what we need, we simply say that its size is zero.”

I think I have said enough—perhaps even more than enough—on that issue. I have nothing to add to them.

“Can you show me a physical object that has no size? How can we measure something that has no size? How do we come to a conclusion that it has no size?”

I explained it above in my remarks about temporal and spatial continua.

On a side note, which I didn’t include in my article (but am currently working on it), I think one way of resolving Zeno’s arrow paradox (and some other puzzles) would be to deny the existence of extensionless instants of time and points of space in physical world. They are best viewed as mathematical fictions and limiting notions used in scientist’s mathematical models of reality. But the denial of the existence of instants and points is not easy; you have to deal with some thorny issues such as infinite divisibility and the status of infinitesimals.

]]>What I had to do in order to resolve your paradox is to think of situations, no matter how weird, when stating that something has no duration is actually meaningful. An example of such a situation is when you want to measure the magnitude of an object in terms of a whole number of units of some kind e.g. the height of a man in terms of a whole number of kilometers. Michael Jordan is 0.00198 km tall. However, making such a measurement, within the context of our needs, would be an overkill. We only want to know how many whole kilometers are there in the height of Michael Jordan. And the answer is exactly ZERO whole kilometers.

When something is so small that measuring it in terms of commonly used units would be an overkill in relation to what we need, we simply say that its size is zero.

Can you show me a physical object that has no size? How can we measure something that has no size? How do we come to a conclusion that it has no size?

]]>If I want to measure someone’s height in kilometers—for some weird reason—and find that height to be 1.80 meters, I don’t express my measurement result by saying “This person is zero kilometers tall.” Instead, I say “This person is 0.0018 kilometers tall.” Again, I follow the crowd in doing that. Of course somebody might adopt a different convention and prefer saying “This person is zero kilometers tall,” by which she means the person’s height is not expressible in a whole counting number. There is nothing wrong with that, except she is not following the crowd.

]]>One way to interpret the question “how tall is Michael Jordan?” is as “how many whole meters are there in the height of Michael Jordan?” The answer to such a question would be exactly one. There is only one whole meter in the height of Michael Jordan.

Another way to interpret the question is as “how many whole meters and how many tenths of a whole meter are there in the height of Michael Jordan?” The answer would be exactly one whole meter and exactly nine tenths of a whole meter. This can be represented with a real number as 1.9 meters.

Yet another way to interpret the question is as “how many whole meters, how many tenths of a whole meter and how many hundredths of a whole meter are there in the height of Michael Jordan?” The answer would be exactly one whole meter, exactly nine tenths of a whole meter and exactly eight hundredths of a whole meter. Or, using real numbers, 1.98 meters.

Basically, questions are generic. They do not specify all of the details, so they must be interpreted. In the above question, what is left unspecified is the precision with which we should measure the height of Michael Jordan. In fact, many other things are left unspecified, such as, for example, what kind of unit should be used.

When I say that Michael Jordan is 1.98 m tall does that mean he’s exactly 1.98 m tall? In other words, does that mean that his real height is 1.980000… m? Does it mean that his height has zero 1/10^3 of a whole meter, zero 1/10^4 of a whole meter, zero 1/10^5 of a whole meter and so on ad infinitum? Of course not. That’s not what it means. What it means is that his height is [i]less than[/i] 1.99 m but [i]greater than[/i] 1.979999… m. Basically, it means it is within certain range of values.

We cannot know the height of a man, or the magnitude of any physical object in general, with precision. We can only know the upper bound and the lower bound of the magnitude.

When we say that Michael Jordan is zero whole kilometers tall what that means is that his height is less than one whole kilometer. It does not mean he has no height.

The very concept of “no height” is meaningless. I am sure you’d have a lot of trouble demonstrating what it refers to in reality.

Your paradox reveals a problem, I agree with that, but the question is what exactly is the problem. You are suggesting that what is problematic is the concept of motion together with the concept of rest. I disagree. I think that what’s problematic is your premise that an instant has no duration in the literal sense of the word.

]]>It is possible to measure the magnitude of a thing in terms of any other thing. For example, you can measure the height of a man in terms of nanometers, milimeters, centimeters, meters, hectometers, kilometers, megameters, gigameters and so on. Also, you can measure the height of a man in terms of other man. You can say one man is one another man plus a half of him. It’s your choice. In fact, it is possible to measure the magnitude of a thing in terms of a combination of several different things. In plain terms, you can mix different units. For example, you can measure the height of a man in terms of the combination of meters and centimeters. You can say Michael Jordan is 1 meter and 98 centimeters tall. You don’t have to say Michael Jordan is 198 centimeters tall. It’s your choice how you’re going to do it.

You might say that units such as centimeters, meters, kilometers, etc are abstract. And you would be right. They are abstract in the sense that they are not particulars but universals. But you would be wrong if you said that because they are abstract they do not refer to anything concrete. For example, you might say that there are no centimeters in reality. You cannot, for example, touch a centimeter. While that’s true in the sense that universals are not physical objects it is not true that universals do not refer to particulars such as physical objects. A universal is simply a reference to ANY particular within some range of particulars. This means that the concept of centimeter, provided that it is a meaningful concept, which it is, has a reference point in physical world. For example, I can show you centimeters that you can physically interact with.

I understand that there are people who claim that there is such a thing as the smallest possible unit. They are ontological reductionists and physical atomists. I do not belong to this camp. I think that all reasoning is fundamentally inductive. This means that our beliefs are only more or less probable. As such, we can only say that it is very unlikely that we will find a smaller unit than the smallest one that is known to us. However, the universe can surprise us at any point in time. No matter how certain we are that the smallest unit known to us is the one that will remain the smallest one for the rest of eternity, the universe can still surprise us with an even smaller unit. As Hume said, the future is under no obligation to mimic the past.

With that in mind, an instant is not the smallest possible unit of time. Rather, it is simply a unit of time that is sufficiently small for our needs. It might be the smallest unit of time known to us. It might be very unlikely that a smaller unit will ever be discovered. However, none of that means that the unit is universally the smallest possible unit.

In order to measure the magnitude of a thing you have to choose how you’re going to measure it. For example, you might say you want to measure the height of a man in terms of a whole number of kilometers. If a man is shorter than a kilometer, which you will agree most men are, you will have to conclude that he’s “zero kilometers tall”. What this means is that there are no kilometers in his height. It does not mean that he has no height. You can’t use any other kind of number or any other kind of unit, not because it is impossible to measure his height in these other terms, but because you do not want to do so. You don’t want a real number of kilometers. You don’t want a whole number of meters. You want a whole number of kilometers.

]]>“When I want to measure the length of some physical object in terms of meters and I get zero meters as a result, what does that mean? Does that mean that that physical object has no length? Or does that mean that that physical object has no meters in its length?”

I’d say it is the former. I.e. when one measures the length of an object and says that she found its length to be zero meters, she ordinarily means that the object has exactly zero length, in other words the object has (somehow) no length. If she instead meant “The object is less than one meter long,” (as you claim in your previous comment) that would be a misleading way of talking, because she could have said “The object is 0.25 meters (or 0.81 meters long, 0.07 meters long, etc.)” rather than that it is zero meters long.

I also have to disagree with your claim that points can have any size. This is not true for geometric points (which have absolutely zero size) and for instants in the contexts of Zeno’s arrow paradox.

You can find many examples of authors who take an instant to have the length of zero seconds if you google up for the combined search terms: “instant ‘lasts zero seconds'”:

https://www.google.com.tr/search?rlz=1C1CAFA_enTR702TR702&q=instant+%22lasts+zero+seconds%22&oq=instant+%22lasts+zero+seconds%22&gs_l=psy-ab.12

If you start with wrong premises you will get wrong conclusions. The premise that is wrong in your (apparent) paradox is that “zero unit” means “zero magnitude”. That’s not true.

When I want to measure the length of some physical object in terms of meters and I get zero meters as a result, what does that mean? Does that mean that that physical object has no length? Or does that mean that that physical object has no meters in its length?

Your paradox is challenging the idea that instants are durationless in the literal sense of the word i.e. that they last exactly zero amount of time. It is not challenging the concept of motion. And then one has to ask whether you are challenging physicists or merely your own understanding of what physicists think. I think it is the latter.

When people say that “points have no size” what they are saying is that size is not the essential quality of points. In other words, what it means is that points can have ANY size. The same goes for instants.

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