Now, Just A Moment, Zeno! (An Arrow Flies By)
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.
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.
Comments (38)
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!
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.
Bergson already solved this paradox adequately in my view.
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?
Quoting emancipate
Thanks. Will look it up.
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.
We are 90% water.
The reality of things is more like an illusion but seems civil and ordered, because of all [b]the nature (of sense) blocking buildings[/B].
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.
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.
Quoting 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.
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.
Quoting javra
No, this is a mistake in the other direction; the theory of relativity assumes that space and time are continuous, rather than discrete.
Quoting 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."
Quoting 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.
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.
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.
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.
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:
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.
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
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).
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.
thats fair.
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.
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.
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?
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:
Quoting aletheist
seems contradictory with what you say here:
Quoting 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.
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?
No, this is conflating reality with existence; I hold that they are not synonymous or coextensive. Reality is that which is as it is regardless of what any individual mind or finite group of minds thinks about it. Existence is reaction with other things in the environment. Everything that exists is real (and discrete), but there are realities (like continuous space and time) that do not exist. Positions and instants are artificial creations, so they only exist after we have deliberately marked them for some purpose, such as description or measurement.
Quoting javra
A discrete position or location is established relative to a coordinate system whose origin, orientation, and unit length are all arbitrary--again, artificial creations.
Quoting javra
No, physical things exist regardless of whether humans ever designate their positions/locations relative to an arbitrary coordinate system.
Quoting javra
Again, being real does not entail existing.
Quoting javra
Yes, but again, the unit by which we measure length or duration is arbitrary. Moreover, both the stick's length and the song's duration are subject to change--we can cut off a portion of the stick, or adjust the tempo of the song.
Got it. Thanks for the clarification. Existence is a very ambiguous term in philosophy: can either imply something along the lines of “that which stands out (perceptually - sometimes, or cognitively - to some observer, some cohort of such, or all coexistent observers … such that, for example, the “points of awareness” which do the observing don’t themselves exist in this sense, for example leading to notions such as the so-called problem of other minds)” or, else, is deemed synonymous to being and, hence, that which is real (in which case, for example, we as “points of awareness” do exist). Hard to tell what gets interpreted by the term existence sometimes.
Quoting aletheist
Arbitrary relative to whom? I ask because you haven't addressed what is to me the difficult question: How does perceptual agreement between all sentient observers that causally interact in regard to the location of physical objects - very much including where they start and where they end - come about?
(While I'm aware of the "god did it" argument, I'm not of this view - nor do I want to debate issues regarding theism in this thread.)
Quoting aletheist
Given your clarification of "existence", how can physical things (note the plurality and, hence, intrinsic quantity to this affirmation - which also entails discrete locations for each as per the law of noncontradiction) exist regardless of whether locations exist - given that the latter are mind-dependent?
Quoting aletheist
We're in agreement to the second sentence in this quote. But, again, length and duration would be arbitrary relative to whom? To me "arbitrariness" loses its meaning when ascribed to that which all coexistent sentient beings do in like manner so as to result in their tacit agreement upon existent physical things that are concretely experienced and interacted with.
So as to not be misinterpreted, I've already given my own perspective in the post to which you've just replied: both continuous spatiotemporal change and discrete aspects of space and time which we quantify are real and existent, though the first is more foundational than the second.
Yes, that it why I offered my definitions of reality and existence--which, by the way, come from Charles Sanders Peirce.
Quoting javra
I am having trouble understanding this question, and I wonder if there is a disconnect between what I mean by "position" and what you mean by "location." Again, what I primarily wish to maintain is that continuous three-dimensional space is not really composed of discrete dimensionless points. Put another way, there are no absolute positions in space, only those that we deliberately mark for some purpose. A physical thing does not occupy a discrete point or collection of discrete points, since it is always in continuous motion. We can only designate its position relative to an arbitrary reference frame, which is also always in continuous motion. We can agree that my computer monitor is consistently three feet in front of me, but we are nevertheless both hurtling through space along a very complex path as the earth rotates about its axis and revolves around the sun, which is revolving around the center of the Milky Way, which is moving toward and away from other galaxies, etc.
Quoting javra
To clarify, I said that the unit of length (e.g., one inch) and the unit of duration (e.g., one second) are arbitrary.
I like some of his takes as well.
Quoting aletheist
That geometric points don’t really hold being other than in our abstract contemplations I too take as a given. To try to better explain my own perspective:
Like the meaning applicable to specific words as percepts - be they written, auditory, or, as is the case with braille, tactile - specific units of length will hold their designation due to communal accord; alternatively stated, they hold an inter-subjective reality ... But not a reality that is solely applicable to one individual (such as would be awareness of some previously experienced, language-less, REM dream), nor one that is universally applicable to all coexistence sentient beings in manners impartial to the wants or needs of any one individual or cohort of such (as is the case with the physical universe).
Then, as with the specific, here visual, percept used to address a concept - as with a word - so too will the use of feet or meters (for example) be inter-subjectively arbitrary. But - here focusing in on the issue of spatial lengths - the spatial length will remain the exact same length regardless of which unit of measurement is used (and even if no unit of measurement is used), thereby making the discrete position at which a given length starts and ends something that is not inter-subjectively arbitrary; instead, these as discrete positions will be universally applicable to all causally interrelated, coexistent beings regardless of a) their individual idiosyncratic properties of body and mind and b) their shared mindsets. Here the discrete positions (what I’ve termed “locations” for brevity) of where the given length starts and ends will not be in any way arbitrary but, by all accounts I can currently think of, objective.
Importantly, I am of this opinion while fully agreeing that there is no objective center to the physical universe, nor any objective top/down, front/back, left/right to it.
Nevertheless, by virtue of there being multiple physical things, there will likewise be multiple discrete positions – such as the start and end of each physical thing’s longest extension (to nitpick, even if this is equally applicable in all directions as is the case with a perfect sphere … which is likely solely conceptual).
To now try to bring this back to the arrow paradox, remember I discussed the point-free topology notion of spatially extended “spots” as an alternative to volumeless geometric points, this in terms of contemplating what space is constituted of. The same conceptual dilemma emerges: if there is a distance - a start-spot and an end-spot to a given length - which has to be traversed, then there will logically be a mid-spot to this distance, this whether or not it is marked by anyone. And, also logically, there will then be an endless quantity of mid-spots getting ever closer to the end-spot but never actualizing a perfectly identical location relative to it.
Hence, in my view, this logic will hold for as long as there is some objective length that needs to be traveled. It doesn’t matter if the length is measure in feet, in meters, some other unit of measurement, or is not measured at all via any inter-subjectively established unit of measurement. The mid-points or, alternatively, mid-spots will be - not because one individual discerns them, nor because of some interpersonally established means of measurement (including those of geometric points and topological spots) - but, it seems to me, these will be just as objectively present as is the very start and end of the given length. And devoid of some start and end to length, width, and height no physical object could itself be - instead, all of physicality would be one center-less and volume-less whole.
Don't know if I expressed myself well enough. At any rate, to me the spatial aspects of the arrow paradox are just an interesting thing to think about at times, this as a distraction of sorts. But I don’t want to beat a proverbial dead horse. I think I get the perspectives you’ve presented, which is what I was interested in. And its clear to me that we both agree on space and time being continuous, with no absolute spatial (or temporal) locations (or durations) to be found in the spatiotemoral universe.
Thanks for the exchange.
Logically, yes; actually, no. Again, what dissolves the paradox is that the arrow need not move to each subsequent midpoint as a discrete step. Even the line itself does not exist until the arrow traverses it, since a gust of wind might alter its actual path.
Quoting javra
Likewise!