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. — aletheist

So you deny that there's an actual half way point between the start position and the end position?

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

I see how this makes sense with space, but I don't think it makes sense with time. With space it only makes sense to claim that there is a half distance if we can actually identify the real existence of that half distance, to say that the object travels that distance. So if we start with 100m we can mark this, and see that the object travels that spatial unity. We can mark a 50m unit, a 25m unit, and so on, and see the object travel these units. Inevitably there will be a point where we can no longer mark the distance, or observe the object travel it. So it doesn't make sense to speak of space in terms of divisibility like that.

Time however is different. Time is a concept derived from the motions of objects. It relates one motion to another. Because of this, it is not the property of any particular motion. This abstractness provides that it must be inherently divisible in order that we may apply it to ever faster and ever shorter duration of motion. So in the case of time I think we must always allow that even in the shortest identified time period, there is still a possible shorter time period, to provide us the capacity to identify even faster and shorter motions, in the future.

So you deny that there's an actual half way point between the start position and the end position? — Michael

It depends on exactly what you mean by "an actual half way point." As I said, there are no actual points at all, if by that we mean mathematical (i.e., dimensionless) points. There obviously is a location on the continuous line that is equidistant from the start and end positions, but there is only a "point" there if we define and mark it as such for some particular purpose, such as measuring. The object would "pass through" any point that you wish to define and mark on the line - but that act of defining and marking a point does not somehow create a separate, discrete, intermediate step that the object must now take in order to get from one place to the other.

Said another way, the object's motion comes first from a logical standpoint. Drawing a line that traces the object's path, and then defining and marking whatever points on that line serve whatever purpose we may have in doing so, only comes afterward.

There obviously is a location on the continuous line that is equidistant from the start and end positions, but there is only a "point" there if we define and mark it as such for some particular purpose, such as measuring. — aletheist

This seems like you're being unnecessarily pedantic. But fine, I'll use your terminology. There is an infinite series of discrete locations between A and B that an object in continuous motion must actually pass as it moves from A to B – the half-way location, and before that the quarter-way location, and before that the one-eighth location, and so on.

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

The is no paradox if one a treats time and space as indivisible - which is clearly the case. Only those trapped in the works of numbers would agree otherwise. Of course, the is motion and duration always flows, but for some their experiences are not as real as numbers. — Rich

This, incidentally, does not appear far off from what Zeno was arguing for in the first place. Would you consider yourself a Parmenidean? Maybe a Neo-Parmenidean?

Please see my (second) previous response. You can define and mark as many discrete locations between A and B as you like, but this does not in any way affect the continuous motion of the object from A to B, which is logically prior to that mathematical exercise.

So I would set aside the two questions that you formulated - is motion a supertask? and are supertasks (metaphysically?) possible? - as open questions that, prima facie at least, are not incoherent or trivial. Other things that you mention, such as Thompson's lamp, might actually be less problematic than you think, being ultimately language problems rather than problems of metaphysics.

But anyway, if you want to talk about the point, a good way to start would be to give a crisp statement of the alleged paradox. — SophistiCat

I agree that most of the time discussions on this topic tend to descend rather quickly and that’s what I was trying to point out, but you’re quite right. The actual questions raised by the paradox are rarely ever even addressed.

The language problem of Thomson’s lamp: Yes, this is exactly what I’m getting at, that the profundity of Zeno’s paradox (as well as Thomson’s) don’t lie in the realm of mathematics, but in logic/language. This is the point that I feel is often missed.

A form of the paradox that I like is this (from Wikipedia):

• Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
• Supertasks are impossible
• Therefore, motion is impossible

From this, I think it's easy to see that the issues that can be taken with the paradox are issues of logic, not of mathematics and especially not of sums of series.

What does it mean for a motion to be "complete"? Is motion made up of "steps"? These are the core issues that the paradox is getting at.

And before someone brings up physical properties of space and the Planck length, this same argument can be applied to time as well. I find that it is not as contentious a statement to say that time is continuous when compared to space.

I hate to keep stealing my comments from Wikipedia, but there is another interesting version (at least I would call it a version) of Zeno's paradoxes in the form of Bernardete's Paradox of the Gods:

A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods. — J. A. Bernardete

This version takes it out of the physical realm and makes it a pure thought experiment. How would one deal with this version of the paradox?

Depends if you think the sequential distances or steps required to traverse a distance are countably or uncountable infinite. They both have seemingly identical properties, as far as I can tell? They just apply to different sets of numbers.

A supertask is logically impossible. — Metaphysician Undercover

If your statement is true, then the next question is whether motion is a supertask. And if it is, doesn't that mean motion is logically impossible?

Yes, Zeno certainly realized that where there are paradoxes there are flaws in the understanding.

Most of my views and approach parallel those of Bergson.

Yes, this is exactly what I’m getting at, that the profundity of Zeno’s paradox (as well as Thomson’s) don’t lie in the realm of mathematics, but in logic/language. This is the point that I feel is often missed.

A form of the paradox that I like is this (from Wikipedia):

* Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
* Supertasks are impossible
* Therefore, motion is impossible

From this, I think it's easy to see that the issues that can be taken with the paradox are issues of logic, not of mathematics and especially not of sums of series.

What does it mean for a motion to be "complete"? Is motion made up of "steps"? These are the core issues that the paradox is getting at. — Voyeur

These seem to be metaphysical questions, not questions of logic or language. There's nothing logically inconsistent or ambiguous about supertasks (and this is where mathematical treatment of convergence comes in). But one can still ask the questions that you ask as questions of metaphysics (informed by physics).

Thompson's Lamp, on the other hand, as well as a number of other such paradoxes, including the Bernardete paradox that you brought up later, are just logical puzzles. The key to their solution is that their premises are either inconsistent (Bernardete) or incomplete (Thompson). In the former case we can conclude that the premises cannot describe a possible state of affairs, which dissolves the paradox. In the latter case the problem (necessarily) does not have a unique solution, which again renders the seemingly surprising result as inevitable.

These seem to be metaphysical questions, not questions of logic or language. — SophistiCat

True, but in order to progress to a logical analysis, I think metaphysically defining our subject is a worthy cause.

Thompson's Lamp, on the other hand, as well as a number of other such paradoxes, including the Bernardete paradox that you brought up later, are just logical puzzles. The key to their solution is that their premises are either inconsistent (Bernardete) or incomplete (Thompson). — SophistiCat

But in comparison, wouldn't Zeno's paradoxes just be a version of this? Zeno just happened to target physical phenomena as his subject, but isn't his reasoning and the scenario he cooked up just as much a logic puzzle as the other two paradoxes you and I mentioned?

Can't we use the same methods of investigation, discussion, and analysis for all three?

It's interesting because I think it's often overlooked that the point of Zeno's paradoxes isn't to prove that motion is impossible, it's to reduce to absurdity the concept that reality is divisible rather than indivisible.

I've always gained more insight from looking for the hints of things he sort of got right or was on the right track about, than trying to defeat the paradox and move on.

Bergson approached the problem in exactly the same way as Zeno did and for precisely the same reason. No doubt, Bergson was informed by Zeno.

I see how this makes sense with space, but I don't think it makes sense with time. With space it only makes sense to claim that there is a half distance if we can actually identify the real existence of that half distance, to say that the object travels that distance. So if we start with 100m we can mark this, and see that the object travels that spatial unity. We can mark a 50m unit, a 25m unit, and so on, and see the object travel these units. Inevitably there will be a point where we can no longer mark the distance, or observe the object travel it. So it doesn't make sense to speak of space in terms of divisibility like that.

Time however is different. Time is a concept derived from the motions of objects. It relates one motion to another. Because of this, it is not the property of any particular motion. This abstractness provides that it must be inherently divisible in order that we may apply it to ever faster and ever shorter duration of motion. So in the case of time I think we must always allow that even in the shortest identified time period, there is still a possible shorter time period, to provide us the capacity to identify even faster and shorter motions, in the future. — Metaphysician Undercover

The marks in space themselves are also symbolic since nor cannot truly divide space with a mark.

Time, or Duree as Bergson called it to avoid confusion, is not created by motion (this is the scientific time of a repeatable motion in space), but is a feeling that we capture via existing. It comes from consciousness not repeatable movements. I exist and feel my existence flowing as a duration whether or not can see the sun rise and set, or hear a clock. Real time is a psychological feeling of enduring in memory.

If your statement is true, then the next question is whether motion is a supertask. And if it is, doesn't that mean motion is logically impossible? — Voyeur

No, I don't think that motion is a supertask, I think a supertask is an impossibility. I do not believe that motion is impossible though. We observe motion. What I believe is that motion is not well understood. That's what Zeno's paradoxes indicate, that we do not have an adequate understanding of motion, it was this way back then, and it remains so, now. We like to think that human knowledge of motion has greatly advanced with the special and general theories of relativity, and mathematical equations but we really persist in an extremely inadequate understanding of the relationship between space and time. As is evident in modern metaphysics, we have a woefully deficient understanding of what it means to exist, and motion is what existing things do. Until we understand what existence is we will not understand what moving is.

,

The marks in space themselves are also symbolic since nor cannot truly divide space with a mark. — Rich

There is always substance though, a surface which we mark, or a ruler, or some such thing. So we measure space by referring to material substance, but we can only go so small with material substance, that is the point.

Time, or Duree as Bergson called it to avoid confusion, is not created by motion (this is the scientific time of a repeatable motion in space), but is a feeling that we capture via existing. It comes from consciousness not repeatable movements. I exist and feel my existence flowing as a duration whether or not can see the sun rise and set, or hear a clock. Real time is a psychological feeling of enduring in memory. — Rich

I do not believe that we get a sense of time simply from existing. I think we get the experience of time from sensing things. If you go to a quiet place and meditate, you can lose track of time. I do believe that you always know that time is passing, regardless of any sensing, but this is not "time" as we know it, i.e. time as a measure of duration, and maybe this is why you are distinguishing it from "scientific time". I do believe that we must recognize two distinct concepts, but time without measure needs to be better defined if we are going to call this "time".

There is always substance though, a surface which we mark, or a ruler, or some such thing. So we measure space by referring to material substance, but we can only go so small with material substance, that is the point. — Metaphysician Undercover

But if you examine it closely, you are not cutting space. The mark simply dissolves into space as more precision is required. There is no materiality but there a continuum of substantial or density of the underlying field. The is simply no way to create units within continuity and if one tries to, out pops Zeno. There is no way to say it other than mathematical ideas and symbols do not carry any underlying ontological weight. So what do you use instead? Bergson said intuition in the sense of conscious penetrating consciousness. Actually, this is the heart of meditation. One just needs to practice the art of peeling away the layers of the onion. A totally unique skill.

I do not believe that we get a sense of time simply from existing. — Metaphysician Undercover

When I examine time, all I sense is a feeling of flow memories. I don't feel and units of measurment. Time sometimes feel like it is passing slowly and sometimes quickly and sometimes it seems to disappear into something else when I am dreaming or call unconscious, this last experience being particularly interesting.

But if you examine it closely, you are not cutting space. The mark simply dissolves into space as more precision is required. There is no materiality but there a continuum of substantial or density of the underlying field. — Rich

The problem though is that we actually are marking things, with a ruler and other forms of measurement. But when we think about space in our minds, we think about dividing it, making geometrical figures, and whatever, this is an imaginary space. The imaginary space, which we can divide infinitely is not consistent with the space full of substance which we work with, which we cannot divide infinitely. In other words our concept of space is inadequate, because we don't know how space is really divided. We only actually divide space by dividing up substance, and substance divides quite differently from the way that we divide space conceptually.

The is simply no way to create units within continuity and if one tries to, out pops Zeno. — Rich

But don't you have this reversed? What exists is units, objects, but we want to talk about space in terms of a continuity. So it's not like we're trying to create units within a continuity, what exists is units and we are trying to make these units into a continuity. That's what Zeno shows us. It's not that motion is continuous, and we are trying to understand it as units, it's that it is not continuous, but we are trying to model it as being continuous. And this creates the paradox.

When I examine time, all I sense is a feeling of flow memories. I don't feel and units of measurment. Time sometimes feel like it is passing slowly and sometimes quickly and sometimes it seems to disappear into something else when I am dreaming or call unconscious, this last experience being particularly interesting. — Rich

Do you think that you can sense a feeling of time when you are unconscious? I don't think so. Do you think you sense a feeling of time when you are dreaming?

Do you think that you can sense a feeling of time when you are unconscious? I don't think so. Do you think you sense a feeling of time when you are dreaming? — Metaphysician Undercover

If one insists on a discontinuous space and time then Zeno will always be there along with Achilles not ever reaching the finish line and arrows that are forever stopping in mid-air and restarting itself. The only way out is to challenge the assumptions and the methods used. If mathematics as a tool to reveal is too precious to give up then so be it. As for me, the idea of symbols actually being used as a placeholder for nature has passed. A piano n teacher one taught me to disregard the notes when you actually play music. The notes are inadequate.

As for what happens to time once I am unconscious or in a state of sleep is a question I have been long exploring and I come back to it now and then. As far as I can tell, it is a state where time is at its slowest, where there is some feeling of existing (via dreams) but it seems as though time isn't passing at all. Possibly an analog for the life-death cycle.

It's not that motion is continuous, and we are trying to understand it as units, it's that it is not continuous, but we are trying to model it as being continuous. — Metaphysician Undercover

You still have it exactly backwards. Space, time, and motion are all continuous; we only model them as being discrete.

Prompted by some of the discussion in my spin-off thread on "Continuity and Mathematics," I have been reading up on category theory and one of its outcomes, synthetic differential geometry, also known as smooth infinitesimal analysis. I just came across this very pertinent passage on page 277 of John L. Bell's 2005 book, The Continuous and the Infinitesimal in Mathematics and Philosophy (emphases added).

The Principle of Microstarightness yields an intuitively satisfying account of motion. For it entails that infinitesimal parts of (the curve representing a) motion are not points at which, as Aristotle observed, no motion is detectable - or, indeed, even possible. Rather, infinitesimal parts of the motion are nondegenerate [i.e., non-zero] spatial segments just large enough for motion through each to be discernible. On this reckoning a state of motion is to be accorded an intrinsic status, and not merely identified with its result - the successive occupation of a series of distinct positions. Rather, a state of motion is represented by the smoothly varying straight microsegment, the infinitesimal tangent vector, of its associated curve. This straight microsegment may be thought of as an infinitesimal “rigid rod”, just long enough to have a slope - and so, like a speedometer needle, to indicate the presence of motion - but too short to bend, and so too short to indicate a rate of change of motion.

This analysis may also be applied to the mathematical representation of time. Classically, time is represented as a succession of discrete instants, isolated “nows” at which time has, as it were, stopped. The principle of microstraightness, however, suggests that time be instead regarded as a plurality of smoothly overlapping timelets each of which may be held to represent a “now” or “specious present” and over which time is, so to speak, still passing. This conception of the nature of time is similar to that proposed by Aristotle to refute Zeno’s paradox of the arrow; it is also closely related to Peirce’s ideas on time.

As I said before, continuous motion is the most fundamental concept here. It is logically prior to any series of discrete locations - including an infinite one - through which an object passes while traveling from one place to another. In fact, the object's actual path that includes those identified "points" only exists as the result of the motion.

You still have it exactly backwards. Space, time, and motion are all continuous; we only model them as being discrete. — aletheist

We measure space, time and motion as discrete, because that's the only way we can apply the numbers. But we tend to believe that these are continuous. It is this false belief, that space and time are continuous, which give rise to Zeno's paradoxes. So long as you hold this belief, that space, time and motion are continuous you will have paradoxes.

The concept of "infinitesimal points" is incompatible with continuous motion, it is only compatible with discrete motion. An infinitesimal point must be separate from another infinitesimal point or else it is not a point, and this negates any possibility of continuity. A series of "timelets" is a description of something discrete. Your quote from John Bell has provided a description of discrete motion, not continuous motion. He has perhaps recognized that our belief in continuous motion must be adapted to be represented as discrete.

We measure space, time and motion as discrete, because that's the only way we can apply the numbers. — Metaphysician Undercover

I have no problem with measuring continuous things using discrete models; as I have acknowledged previously, they are very useful for that purpose.

So long as you hold this belief, that space, time and motion are continuous you will have paradoxes. — Metaphysician Undercover

No, I have explained how Zeno's paradox dissolves when continuous motion is properly understood as more basic than discrete locations. Besides, a paradox by definition is only an apparent contradiction, not an actual contradiction; beliefs that are paradoxical are not necessarily false.

The concept of "infinitesimal points" is incompatible with continuous motion, it is only compatible with discrete motion. An infinitesimal point must be separate from another infinitesimal point or else it is not a point, and this negates any possibility of continuity. — Metaphysician Undercover

No one is talking about "infinitesimal points" except you. Infinitesimals are not separate dimensionless points, they are lines of extremely small but non-zero length that smoothly blend together so as to be indistinct. A continuum is that which has parts, all of which have parts of the same kind . A one-dimensional line cannot be divided into zero-dimensional points, only shorter and shorter one-dimensional lines.

A series of "timelets" is a description of something discrete. Your quote from John Bell has provided a description of discrete motion, not continuous motion. — Metaphysician Undercover

That would be news to him. I guess you missed the part about the timelets "smoothly overlapping" such that "time is, so to speak, still passing" within each of them, rather than being frozen in a discrete instant.

No one is talking about "infinitesimal points" except you. Infinitesimals are not separate dimensionless points, they are lines of extremely small but non-zero length that smoothly blend together so as to be indistinct. A continuum is that which has parts, all of which have parts of the same kind . A one-dimensional line cannot be divided into zero-dimensional points, only shorter and shorter one-dimensional lines. — aletheist

It doesn't matter how you lay the infinitesimal out, as a point, or as a line, there is still the assumed separation between it and other infinitesimals, and therefore it is necessarily a discreteness. A continuum cannot have parts, or else it is by virtue of those parts, not continuous, it is discrete.

That would be news to him. I guess you missed the part about the timelets "smoothly overlapping" such that "time is, so to speak, still passing" within each of them, rather than being frozen in a discrete instant. — aletheist

By saying "smoothly overlapping" you are speaking in terms of discreteness. You have identified separate parts which overlap. This is not continuity.

It doesn't matter how you lay the infinitesimal out, as a point, or as a line, there is still the assumed separation between it and other infinitesimals, and therefore it is necessarily a discreteness. — Metaphysician Undercover

Wrong. There is no separation (assumed or otherwise) between infinitesimals. Neighboring infinitesimals are indistinct; the principle of excluded middle does not apply to them.

A continuum cannot have parts, or else it is by virtue of those parts, not continuous, it is discrete. — Metaphysician Undercover

Wrong. A continuum - by definition - is that which has parts, all of which have parts of the same kind. What a continuum cannot have are indivisible parts, like points.

By saying "smoothly overlapping" you are speaking in terms of discreteness. You have identified separate parts which overlap. — Metaphysician Undercover

Wrong. Discreteness requires separation and distinction; infinitesimals, as defined by synthetic differential geometry (a.k.a. smooth infinitesimal analysis), are neither separate nor distinct.

Wrong. There is no separation (assumed or otherwise) between infinitesimals. Neighboring infinitesimals are indistinct; the principle of excluded middle does not apply to them. — aletheist

Then they are not infinitesimals, are they? They are united as one large continuum and it is false to refer to them as separate infinitesimals.

Wrong. A continuum - by definition - is that which has parts, all of which have parts of the same kind. What a continuum cannot have are indivisible parts, like points. — aletheist

A continuum is a continuity. It is the desire to model the continuum as a serious of parts, like we do in a mathematical model, which negates the essence of the continuity, rendering it as a series of discrete units. To say that a continuum has parts is contradictory. By saying it consists of parts, you no longer describe it as continuous.

Wrong. Discreteness requires separation and distinction; infinitesimals, as defined by synthetic differential geometry (a.k.a. smooth infinitesimal analysis), are neither separate nor distinct. — aletheist

I agree that discreteness requires separation, but what you seem to be failing to recognize is that "part" also requires separation. That is why it is contradictory to say that a continuity consists of parts. The true continuum must be indivisible, that's why it cannot be modeled mathematically.

They are united as one large continuum and it is false to refer to them as separate infinitesimals. — Metaphysician Undercover

One more time: By definition, infinitesimals are not separate.

To say that a continuum has parts is contradictory. — Metaphysician Undercover

One more time: By definition, a continuum has parts, all of which have parts of the same kind.

... what you seem to be failing to recognize is that "part" also requires separation. — Metaphysician Undercover

No, it does not. Once again, you are rejecting the commonly accepted definitions of terms, and imposing your own idiosyncratic ones.

The true continuum must be indivisible, that's why it cannot be modeled mathematically. — Metaphysician Undercover

No, it must be infinitely divisible - i.e., there cannot be any indivisible parts - and smooth infinitesimal analysis does model this mathematically, whether you recognize it or not.

One more time: By definition, a continuum has parts, all of which have parts of the same kind. — aletheist

Clearly I do not accept your contradictory definitions. "Part" implies of necessity, a separation, and this negates any claim of continuity, which is a lack of such separation. You may proceed with your deluded metaphysics if you so desire.

"Part" implies of necessity, a separation, and this negates any claim of continuity, which is a lack of such separation. — Metaphysician Undercover

Unless you can demonstrate that the concept of "part" necessarily involves separation, rather than just baldly asserting this over and over again as your own idiosyncratic definition, I have no reason to take it seriously. It blatantly begs the question to insist that anything with parts of any kind must be classified as "discrete," rather than "continuous."