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
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
So you deny that there's an actual half way point between the start position and the end position? — Michael
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
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
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
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
A supertask is logically impossible. — Metaphysician Undercover
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. — SophistiCat
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
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
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
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.The marks in space themselves are also symbolic since nor cannot truly divide space with a mark. — Rich
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
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
I do not believe that we get a sense of time simply from existing. — 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. — Rich
The is simply no way to create units within continuity and if one tries to, out pops Zeno. — Rich
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? — Metaphysician Undercover
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
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.
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. — Metaphysician Undercover
So long as you hold this belief, that space, time and motion are continuous you will have paradoxes. — Metaphysician Undercover
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
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
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
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
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
A continuum cannot have parts, or else it is by virtue of those parts, not continuous, it is discrete. — Metaphysician Undercover
By saying "smoothly overlapping" you are speaking in terms of discreteness. You have identified separate parts which overlap. — 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. — aletheist
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
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
They are united as one large continuum and it is false to refer to them as separate infinitesimals. — Metaphysician Undercover
To say that a continuum has parts is contradictory. — Metaphysician Undercover
... what you seem to be failing to recognize is that "part" also requires separation. — Metaphysician Undercover
The true continuum must be indivisible, that's why it cannot be modeled mathematically. — Metaphysician Undercover
One more time: By definition, a continuum has parts, all of which have parts of the same kind. — aletheist
"Part" implies of necessity, a separation, and this negates any claim of continuity, which is a lack of such separation. — Metaphysician Undercover
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