The arrow can move because time is not made up of zero-sized instances/moments; instead time is essentially an interval and so, the arrow can move. — TheMadFool

I like this. To me the temporal aspects of the paradox are nicely addressed and resolved in the OP.

All the same, my problem with Zeno’s arrow paradox is not so much temporal as spatial, which the OP’s resolution doesn’t address. Maybe you, or some other, can find a resolution to it; this in parallel to the temporal issues addressed by the OP.

To sum, the arrow’s motion has a starting location, I’ll label it S, and a finishing location, here labeled F. This can get represented by a line segment between S and F. The line segment has a midpoint, here labeled M. To get from S to F one has to pass M. Once passed, though, there’s a second midpoint between M and F, here labeled M2, that needs to be passed. Then there’s a midpoint between M2 and F, M3, that needs to be passed. The trouble with the spatial paradox, as I understand it, is that it leads to an infinite quantity of midpoints that need to be traversed in order to arrive at F. In short, because the quantity of midpoints that need to be passed is endless, one could never arrive at F, for one is forever stuck in passing through midpoints that reside before F.

Once this problem is cognized as such, it then can be applied retroactively to the midpoint between S and M – such that M (the midpoint between S and F) can never be obtained either. Nor, for that matter, can any movement whatsoever occur when rationally considered in spatial terms, regardless of how miniscule the distance: given the tacit presumptions of rudimentary geometry most, if not all, of us maintain, there will always conceptually occur an infinite quantity of midpoints between the place started from and any given destination, with all these endless midpoints residing before the destination.

Eppur si muove!

Gotta love a great paradox!

But...

In a frozen universe where there was no movement, would time exist?

In this frozen timeless universe, should the archer release the arrow, then time would begin. It isn't 'time' which prevents the arrow from moving -- it is the motionless arrow that prevents time from passing.

There was no time before the Big Bang, and there will be no time again when (and if) the universe cools to absolute zero.

It is an intrinsic function of our perception that breaks time up into discrete moments and spatializes it. This is an external perspective. Real time is indivisible, as is evident from our lived, internal experience of time. When you move your arm from one point to another, it is experienced as one complete movement. Yes you could stop the movement at any given point, and then let it continue on it's path, but this would just be experienced as a whole movement with an interval.

Bergson already solved this paradox adequately in my view.

Thanks for bringing the spatial aspect of the paradox to my notice. I thought of it briefly and here's what I made of it.

Imagine a person, A, poised at the starting point (0 m) of a distance of 100 m. As per Zeno, since A must first travel 50 m and before that 25 m and before that 12.5 m, and so on, with there being no first distance to this infinite sequence, motion is impossible. What is of significance is that for this part of the argument, Zeno, considers the distance 100 m as something as in it's not nothing (not zero).

Then Zeno concludes that motion is an illusion and that, as far as I can tell, means that if we were to observe A travel the distance 100 m, in actuality A hasn't moved at all; in other words, in mathematical terms, A has "traveled" 0 m. As might be evident to you now, at this point Zeno shifts from the position that 100 m is something that is infinitely divisible into smaller and smaller parts to 100 m is nothing (0 m).

Zeno's initial assumption is that a given distance is something divisible ad infinitum into tinier and tinier non-zero distances but in the conclusion of his argument he declares motion to be an illusion, effectively making the claim that any and all distances traveled are actually nothing (zero distance has been traveled as no actual motion has taken place). How can anything be both something AND nothing?

Bergson already solved this paradox adequately in my view. — emancipate

Thanks. Will look it up.

Motion is an illusion, at least half of the phenomena in the universe are in motion, including atoms and electrons. When you next go for a walk study incoming data while walking. How it coming close to you, is part of walking, so motion isn't just moving but also recieving?

Signifying objective/obstacle.

Continuous motion is a more fundamental reality that discrete positions in space and discrete instants in time. Put another way, space is not composed of dimensionless points any more than time is composed of durationless instants. That goes for a one-dimensional line, as well; the only points are the ones that we artificially mark for some purpose, such as labeling S, F, and as many Ms as you like in between. The arrow indeed will pass all the Ms that we actually mark, but that will be a finite number. In order to get from S to F, it does not have to take an infinite series of discrete intermediate steps--from S to M, from M to M2, from M2 to M3, and so on. It simply moves from S to F, and we describe its movement after the fact by means of those labeled positions.

Humans, like blood, keep the civilization pumping.

We are 90% water.

The reality of things is more like an illusion but seems civil and ordered, because of all the nature (of sense) blocking buildings.

The Earth rotates in orbit of Sol, and generates graviatational pull. This means matter will be pulled, and the Earth and matter will be refined.

Technically we think the human body is Earth elemental aethetically, rather than Water.

As said previously objects getting closer seems like an illusion of atoms and electrons.

Theoretically viewing a frozen instant of the arrow in flight does not destroy the momentum the arrow possesses at that instant.

Continuous motion is a more fundamental reality that discrete positions in space and discrete instants in time. — aletheist

I am of the same opinion. With the paradox addressed in mind, this stance in turn implies that our conceptual quantification of space and time, as a mapping of the terrain, does not accurately represent that which is being mapped. Of note, with its possible philosophical interpretations here placed aside, the theory of relativity clearly indicates that space and time are not discrete but a continuum. All the same, the quoted mindset with which I agree will also stand in opposition to the block-universe model of the world, wherein there can be no real motion (due to there being no real change). As a reminder, Zeno’s paradoxes were intended to support Parmenides’ stance that change, and thereby motion, does not exist as anything other but illusion.

The arrow indeed will pass all the Ms that we actually mark, but that will be a finite number. — aletheist

This part I don’t yet get. If we don’t mark a location, or else don’t think of a location, does that then mean that the location does not exist – this in contrast to those locations we do mark or think about which would thereby exist? I’d wholeheartedly disagree with an answer of “yes”. But this then entails an endless quantity of existent midpoints that reside ever closer to the finishing location. BTW, one could employ something along the lines of point-free topology—which does not make use of extension-less geometric points but of extended “spots”—and still arrive at the same conceptual issue of endless “mid-spots” residing before the finishing “spot”.

Again, at least one resolution, as per the implications of your previous statement, would be to understand that our conceptual quantification of space, as a mapping of the terrain, into discrete positions does not accurately represent the terrain.

With the paradox addressed in mind, this stance in turn implies that our conceptual quantification of space and time, as a mapping of the terrain, does not accurately represent that which is being mapped. — javra

Quantification represents space and time accurately enough for most mathematical and practical purposes, but the mistake is thinking that this entails that space and time are really discrete, rather than continuous.

Of note, with its possible philosophical interpretations here placed aside, the theory of relativity clearly indicates that space and time are not discrete but a continuum. — javra

No, this is a mistake in the other direction; the theory of relativity assumes that space and time are continuous, rather than discrete.

All the same, the quoted mindset with which I agree will also stand in opposition to the block-universe model of the world, wherein there can be no real motion (due to there being no real change). — javra

I agree, and personally prefer the "growing block" theory of time in which the past and present exist, but not the future. For more on that, see my recent thread on "The Reality of Time."

If we don’t mark a location, or else don’t think of a location, does that then mean that the location does not exist – this in contrast to those locations we do mark or think about which would thereby exist? — javra

Yes, in my view a discrete position (or instant) is an abstraction that we impose when we mark it for some purpose, not a real constituent of space (or time). It certainly does not exist, since it does not react with anything.

Theoretically viewing a frozen instant of the arrow in flight does not destroy the momentum the arrow possesses at that instant. — jgill

Interesting. I can imagine the arrow has its momentum during any duration of time however short, but at the point where no time passes? Or is that just an impossible notion? Or perhaps more complicated than appears. What do we say of it at absolute zero? That it cannot be motion at that temperature? But that would bump into notions of the relativity of motion, yes?

0 multiplied by an uncountable infinity (maybe a specific one) equals generic "length." That is the conclusion of Zeno's paradox. TIme maybe not have parts, but space does. It can be divided endlessly. Those who say space is discrete or that the continuous doesn't apply to the real world are morons and I refuse to debate them anymore. Banach-Tarski's paradox helps us reason from "generic length" to specific finite lengths

I'm certain that Zeno's Arrow Paradox has been dealt with effectively i.e. a solution has been found; nevertheless, I'd like your views on my take of the paradox.

Zeno's arrow paradox basically states that, IF time can be considered as composed of instants, an arrow, being unable to move at any one instant since no time has elapsed for any motion to occur, too wouldn't be able to move. No motion at any instant; ergo, no motion at all.

It seems Zeno agrees that, if, for the arrow, one takes an non-zero interval of time, there can be motion; after all, that's why the great Zeno speaks of instants/moments. His argument would fail if we use time intervals because motion is possible if non-zero time is allowed.

The first problem Zeno faces is with the definition of the unit of time. Take the second for instance; whatever physical phenomenon is used to define the second, it is essentialy an interval and not an instant. This is probably the one big clue to what I'm about to say.

To illustrate my point, I would like you to take length for example, say in the units centimetere (cm). A ruler that measures length has length markings on it - begins at 0 cm and goes on to, suppose, 30 cm. The length markings on the ruler read off lengths which are intervals in space. Consider now, what a point on this ruler means? A point, by definition, has no size; being thus, a point can't be a length. Being zero cm in length is the same thing as not being length: zero apples are not apples :smile:

Now consider the notion of instants in time. Just as zero cm in length is not length and zero apples are not apples, zero seconds, instants/moments, isn't time at all. It seems, therefore, that time can't be considered as composed of size-zero instants for it's like saying zero cm is a length and we know that to say something has a length of zero cm is exactly the same as saying that thing has no length. Likewise when we speak of zero units of time, we're not talking about time anymore.

So, Zeno, by thinking zero-sized instants/moments as time is making the same mistake as someone who thinks zero apples are apples. The arrow can move because time is not made up of zero-sized instances/moments; instead time is essentially an interval and so, the arrow can move. — TheMadFool

I agree. If for people who like to find this paradoxical in this modern age (and i have met some) find the idea of instant confusing, instead say "an interval so small that it has similarities to an instant". So we might say a 1/10,000th of a second. Then explain that with some stipulations we can make this very tiny interval the same effectiveness as the normal age old instant of time.

We can even attach wierd symbols to this very tiny interval and give it a latin name. And when people go to look up this latin name they'll see a detailed explanation that this very tiny time interval is really just a substitute for a instant in time for people in this modern age who like to make a mountain out of a mole hill.

Well thats what we philosophers do, we make a mountain out of a mole hill.

Gotta love a great paradox!

But...

In a frozen universe where there was no movement, would time exist?

In this frozen timeless universe, should the archer release the arrow, then time would begin. It isn't 'time' which prevents the arrow from moving -- it is the motionless arrow that prevents time from passing.

There was no time before the Big Bang, and there will be no time again when (and if) the universe cools to absolute zero. — Bitter Crank

Some say there is movement in a black hole even though light can't escape. Wherever there is heat there is movement. If the early universe prior to the big bang had heat then it had movement and if there is movement then there is time.

Where would the heat have come from in a (presumably) empty universe, prior to the BB?

Those who say space is discrete or that the continuous doesn't apply to the real world are morons and I refuse to debate them anymore. — Gregory

I wholeheartedly agree, gregory.



Can we really talk about an "empty universe" (an empty everything)? This seems like a contradiction in terms. Can we really talk about a "before" the universe? This implies something can come from nothing, which seems to me like nonsense since nothing/emptiness/void doesn't exist. I agree with you, Mr. Crank.

I can imagine the arrow has its momentum during any duration of time however short, but at the point where no time passes? — tim wood

The key word is "viewing." Zeno's condition doesn't actually stop the arrow, it observes the arrow at an instant. But the idea I advanced is not original. Some time back a well-known physicist I know dismissed the whole nonsense thing with this observation. :cool:

Quite right - there can't be an "empty universe". There can't be a time before time.

Imagine an ant moving along infinitesimals on the surface of a ball. To him the moments are discrete, but to the ball they are continuous geodesical paths. To us objects are finite, to math they are uncountable infinities. Let us not put these ideas together as contradictions, but look for ways towards which Banach-Tarski paradox will seem like most common sense

I agree. If for people who like to find this paradoxical in this modern age (and i have met some) find the idea of instant confusing, instead say "an interval so small that it has similarities to an instant". So we might say a 1/10,000th of a second. Then explain that with some stipulations we can make this very tiny interval the same effectiveness as the normal age old instant of time.

We can even attach wierd symbols to this very tiny interval and give it a latin name. And when people go to look up this latin name they'll see a detailed explanation that this very tiny time interval is really just a substitute for a instant in time for people in this modern age who like to make a mountain out of a mole hill.

Well thats what we philosophers do, we make a mountain out of a mole hill. — christian2017

I wouldn't say that the arrow paradox is something philosophers are making a mountain out of a molehill of. If Zeno is right, motion would be impossible and all that we see around us would be an illusion. Isn't that something to worry about?

As for infinitesimal calculus, I think it's a clever way around the problem of instantaneous velocity.

I wouldn't say that the arrow paradox is something philosophers are making a mountain out of a molehill of. If Zeno is right, motion would be impossible and all that we see around us would be an illusion. Isn't that something to worry about?

As for infinitesimal calculus, I think it's a clever way around the problem of instantaneous velocity. — TheMadFool

The problem is not specifically motion, although that is how Zeno phrased it. The difficulty is space with having no final term while having a spatial limit. In other words, being infinite and finite. There can be motion because there is a limit and no motion because there is no final term. If it can be figured out how space can exist in this state in the first place, motion can be explained

Time is not composed of instances. Time is relative change. Movement is relative.

An interval is simply a string of instances - each a particular snapshot of time in the mind. Instances and intervals only exist in minds. Time exists everywhere there is relative change. The mind breaks up time into instances, just like it breaks processes into objects. The mind is converting the analog signal of the world into binary bits - objects of thought (instances in time and objects in space).

Where would the heat have come from in a (presumably) empty universe, prior to the BB? — Bitter Crank

Not all Physicists agree. If there is matter (very dense matter) there is heat, and the thus the universe is not empty but the universe is very small. If there is heat or pressure, then there is movment & if there is movement then there is time. Just like a black hole has movement inside of the black hole. A black hole is like a really dense universe but is not as dense as the early universe.

I agree. If for people who like to find this paradoxical in this modern age (and i have met some) find the idea of instant confusing, instead say "an interval so small that it has similarities to an instant". So we might say a 1/10,000th of a second. Then explain that with some stipulations we can make this very tiny interval the same effectiveness as the normal age old instant of time.

We can even attach wierd symbols to this very tiny interval and give it a latin name. And when people go to look up this latin name they'll see a detailed explanation that this very tiny time interval is really just a substitute for a instant in time for people in this modern age who like to make a mountain out of a mole hill.

Well thats what we philosophers do, we make a mountain out of a mole hill.
— christian2017

I wouldn't say that the arrow paradox is something philosophers are making a mountain out of a molehill of. If Zeno is right, motion would be impossible and all that we see around us would be an illusion. Isn't that something to worry about?

As for infinitesimal calculus, I think it's a clever way around the problem of instantaneous velocity. — TheMadFool

thats fair.

Time is not composed of instances. Time is relative change. Movement is relative.

An interval is simply a string of instances - each a particular snapshot of time in the mind. Instances and intervals only exist in minds. Time exists everywhere there is relative change. The mind breaks up time into instances, just like it breaks processes into objects. The mind is converting the analog signal of the world into binary bits - objects of thought (instances in time and objects in space). — Harry Hindu

What do you mean by movement is relative? Time is relative in that is really an iteration of events, and is hard to accurately measure unless it is a small subset of the universe (special relativity). But two objects can pass through and gauge their velocity based on the same point in 3d space. Next thing you are going to tell me is two objects can't pass through the same "point" in 3d space.

What do you mean by movement is relative? Time is relative in that is really an iteration of events, and is hard to accurately measure unless it is a small subset of the universe (special relativity). But two objects can pass through and gauge their velocity based on the same point in 3d space. Next thing you are going to tell me is two objects can't pass through the same "point" in 3d space. — christian2017

Looks like movement is relative, whether relative to another object, or a point in space (a point in space seems to qualify as an object in space). The mind has this habit of quantifying (or objectifying) space/change.

What do you mean by movement is relative? Time is relative in that is really an iteration of events, and is hard to accurately measure unless it is a small subset of the universe (special relativity). But two objects can pass through and gauge their velocity based on the same point in 3d space. Next thing you are going to tell me is two objects can't pass through the same "point" in 3d space.
— christian2017
Looks like movement is relative, whether relative to another object, or a point in space (a point in space seems to qualify as an object in space). The mind has this habit of quantifying (or objectifying) space/change. — Harry Hindu

Einstein admitted he was wrong about some things. The idea that time is relative has been proven with P-3 flights over the Chesapeake bay. When has it been proven with scientific tests that movement is relative. Visual perspectives & flawed human perspectives are relative, but has it been proven with scientific tests that movement is relative?

Prove an object moves without referring to its change in position relative to something else, like a point in space.

Of note, with its possible philosophical interpretations here placed aside, the theory of relativity clearly indicates that space and time are not discrete but a continuum. — javra

No, this is a mistake in the other direction; the theory of relativity assumes that space and time are continuous, rather than discrete. — aletheist

My point to this being that, though the theory of relativity is in itself a model of reality, it accurately describes those aspects of nature it is relevant to, as is evidenced by its predictive power. This, in turn, can give additional credence to space and time not only being individually continuous but also mutually continuous.

I'm replying primarily out of my curiosity for the following.

What you said here:

Continuous motion is a more fundamental reality that discrete positions in space and discrete instants in time. — aletheist

seems contradictory with what you say here:

Yes, in my view a discrete position (or instant) is an abstraction that we impose when we mark it for some purpose, not a real constituent of space (or time). It certainly does not exist, since it does not react with anything. — aletheist

The first statement affirms degrees or reality, such that some aspects of reality are more fundamental than others, with all aspects of reality (regardless of its metaphysical(?) degree) being existent by definition. The second statement implies a strict binary understanding: either something is real, and thereby existent, or it is not.

We so far agree that at least everything we deem to be physical is in continuous change - that everything is in flux, to here paraphrase Heraclitus. I say “everything we deem to be physical” so as to bracket off certain givens such as basic laws of thought (the law of identity, for example, is not continuously changing relative to us sentient beings - despite our own perpetual changes).

To address your second comment that discrete position - i.e., location - does not exist, is the computer screen that I am now seeing not located in front of me, beneath the sky and above the earth, having locations to the left and to the right at which it terminates? Are all these in fact nonexistent? If so, how do you account for our mutual perceptual agreement of where physical objects are relative to each other … as well as for their three-dimensional spatial properties? Addressing the same in more general terms, how would one account for the physical world which all sentient beings tacitly, if not also explicitly, agree upon: e.g., an ant, a cat, and a human will all tacitly find the same spatial properties to what we humans deem to be a rock, including its three-dimensional volume. Rearticulating the same, if location is to be deemed nonexistent, would the physical world (here encompassing all physical objects which are in part known via their discrete spatial positions) also be considered nonexistent?

As for myself, I adopt the perspective that the continuity of change, hence of motion, is a more fundamental reality than the fixedness of quantity (including distances that have a beginning and end as well as durations that have a beginning and end) - this being in-tune with your first quoted statement. Yet both change and quantity are nevertheless real and, thus, existent – here, in a non-binomial manner but one of degrees. We all know where a given stick’s length starts and ends, just as we all know when a given song starts and ends – thereby making the stick’s length and the song’s duration impartially, hence objectively, real, and thereby making the stick and the song existent. For emphasis, to me this is so despite lengths and durations being of a less fundamental reality than is the reality of continuous spatiotemporal change.

Prove an object moves without referring to its change in position relative to something else, like a point in space. — Harry Hindu

In the sense that for an object to move, or to prove that it moved that it requires an observer, yes in that sense you could say movement is relative. However many people go beyond that when they say movement is relative and imply other non scientifically proven things regarding the phrase "movement is relative". If you would like to imply further implications regarding "movement is relative", you'll have to show an article showing scientific studies/tests.

How does "movement is relative" the way you just described it apply to the OP?