Simple solution. Put a second glass of scotch directly behind the first. Reach for it instead of the first one. Sip and enjoy. (The first one.)
Part of the problem is your definition of “reach.” The subatomic particles in you hand will never “touch” the subatomic particles in the glass, even when you’re actually holding it and feeling it. Neither do the particles within your hand or within the glass. At a certain point, you have “reached” the glass.
This is not an answer to your question (it’s already been answered). It’s only another way of thinking about it.
But one feels it should be possible to have motion even in a continuous world, whether or not our own world should happen to be continuous. As Exapno says, the right approach is essentially purely mathematical; considerations of “practicality” or of contingent features of our physical universe can’t be part of proper solutions to the paradox, because the paradox isn’t necessarily about our physical universe. It’s about the concept of continuous motion in the abstract (which may or may not properly model our physical universe), a concept which we strongly feel sensible but which, as the paradox shows, requires us to abandon some superstitious fears of (and unfounded assumptions about) infinity to come to grips with.
That’s fine. However it seems something like something a circle jerk of mathematicians would yell as they orgasm simultaneously. Theoretical physics has revealed a mind-boggling solution to Xeno’s Paradox already. I am not sure why the math department should show up at the same time and declare the problem their own and very difficult to solve. The real world has proved to be so strange as to be more interesting than most imaginary things in its own right. I choose to believe that Xeno really wanted to know how an arrow based in our universe should never reach its target. The fact that theoretical physics does a much better job of answering that than the math department is of no concern.
This is true, but not really relevant to the apparent paradox proposed by the o.p. Regardless of what you consider “touching” you can model the motion toward a position in terms of a sum of fractions of distance and time. Does this result in a problem? No, because the sum of the fractions of distance across a total finite distance is a convergent series; similiarly, for any positive speed, it takes a finite amount of time to get there (not accounting for Lorentz contractions at high relativistic speeds).
Consider this: you have a glass of Glenmorangie that is a distance d away from your hand. So you move your hand at constant velocity v toward it. This gives you a time t to reach the glass, such that d=vt. For any fraction of the distance, say ½d*, the time is ½d/v = ½t. More generally, for any fraction f, fd/v = ft. You can break up d into any number of fractions you like–including an infinite geometric series of increasingly small fractions–as long as they all sum up to d.
Now, these divisions are entirely arbitrary; they have nothing to do with how or if spacetime is quantitized. It’s true that very odd things happen at quantum scales which actually make it impossible to give a discrete answer to the question “What is distance d, and how long will it take us to get there?” down to any degree of precision you select, but this is immaterial to the Gedankenexperiment proposed by Zeno and proffered by the o.p.
For the purposes of coping with quantum mechanics, and particularly relating behavior of fundamental particles to classical systems, stating things in terms of Planck units is useful. This does not mean, however, that space and time are literally quantitized, but rather our ability to measure things on those levels are inhenently limited those graduations by virtue of our measurement devices being themselves quantum systems. By analogy, the tachometer in your car is probably measured in graduations of 100 rpm If you look very closely, you might be able to interpolate between the marks to distinguish differences of 50 rpm or even 25 rpm, but below that you’re not effectively going to be able to discern 2468 rpm from 2473 rpm, even though the rotational speed of the crankshaft is a continuous differentiable function.
It’s possible that spacetime is quantitized, and if so that goes quite a way to allowing for a quantum theory of gravity (and also discombobulating theoriests in General Relativity, but you can’t make an omlette without breaking a few skulls, now can you?) but we can’t at this time prove or disprove this to be true. In any case it has no bearing on the ability of the o.p. to reach his glass of Scotch in a finite time, which converges even in a purely classical framework.
Stranger
You really don’t know what you’re talking about here, and are being needlessly pejorative to boot. Zeno’s (and the o.p.'s) paradox asks the question in terms of arbitrary divisions of distance and time with no reflection upon the behavior of real world system. (Zeno, like is contemporaries, presumably knew nothing of modern quantum mechanics theory and had no reason to believe that the world really came in small, series limiting chunks.) It is a legitimate math question, not a “circle jerk of mathematicians…orgasm[ing] simultaneously,” and it has a very straightforward answer which should be clear to anyone with education in first level calculus. No waving around of Weyl quantizations or Planck units is required to address it, and you don’t need to otherwise limit the division of the denominator to come up with a perfectly rational and useful answer.
Stranger
To be fair, the math department proffered its solution centuries before the physics community decided the world really was quantized. And the math department’s solution really is better, applying in full generality (maybe space is quantized, but some other quantity isn’t; the math department’s solution applies equally to all types of quantities in our physical universe, plus any theoretical types of quantities we might apply towards any purposes, physical or not). The math department’s solution stands firm, incapable of being shot down by any further experiment. The math department’s solution really gets at the core of the problem, rather than sidestepping it by saying “Oh, continuous motion really isn’t possible, you’re right, but discrete motion is cool, so I guess it’s a good thing the real world turns out discrete, eh?”
You reflect on Zeno’s paradox, and you start thinking “Is continuous change actually possible?”. Solving that is the accomplishment of the math department; what we’ve since discovered about quantization of the physical universe is laudable for its own purposes, but not particularly relevant towards this target.
It’s basically the question, can any supertasks be carried out in finite time? The usual reason to believe no supertasks can be finitely carried out is because it seems an infinite number of tasks each taking positive time must take infinitely long to carry out. The accomplishment of the mathematicians was to point out that this was an unfounded assumption. They didn’t need physics to do this, and, indeed, physics didn’t answer this, it only showed that one particular sort of thing might not actually be a supertask after all. That’s much weaker than the response the mathematicians could give.
I should clarify, it isn’t really about continuous vs. discrete; Zeno’s paradox would confront us even in a discrete (discretish?) realm where all quantities are rational numbers, for example. It’s really about supertasks, which is to say it’s really about infinite divisibility within a finite interval. It’s really about “Can A and B be finitely far apart, and yet have infinitely many things inbetween?”, or, specifically, “Can A and B be finitely far apart, with B > B_1 > B_2 > B_3 > … > A?” (To which the mathematicians can say “Of course that’s possible; just look at, e.g., the rationals. The only reason you think it’s not possible is because you have misguided assumptions about how infinite series act. But if you’ll look here, at the very situation you are thinking about, you’ll see that infinite series don’t always act the way you might naively think they do.”)
So it’s not really about continuous vs. discrete. But the main reason ordinary people would ever worry about it is because of the concept of continuous change, which is a very natural and intuitively sensible concept, so, in that sense, it’s tied to continuity.
As others have said, this statement totally misses Zeno’s point.
He was, by definition, interested in the mathematical solution to the real-world problem. His assumption was that the world was infinitely divisible. This distinction is crucial, because it applies not just to this single problem but to the range of all his “paradoxes.” He was concerned with infinite sums in a finite world.
Yes, he phrased it as a real-world problem, because he could not imagine a mathematics that was different from physics. (He couldn’t really imagine physics at all, but to the extent of his understanding they were identical.) There can only be a mathematical answer because he asked only a mathematical question. To look at it in any other way is both an insult to the perceptiveness of his question and to the generations of mathematicians who have rewritten our understanding of math based on these questions.
When we try to redefine the question, then you may indeed get a different solution. There are times for which doing so in interesting and important. Whether the physical universe is quantized is an interesting and important question in and of itself.
But it is not Zeno’s question. It doesn’t read to me as if it were the OP’s question. As such, insisting on answering it doesn’t shine light on the question; it obscures the core that has made the question live for 2500 years. You can choose to believe Zeno’s question was something totally other that what everybody else in the universe understands it to be, but you shouldn’t inflict that false belief upon others.
And that doesn’t even get to the point that no physicist can say for sure that the universe is actually quantized at the Planck level. This is a matter of huge debate and uncertainty. Unless and until the next grander more unifying theory comes about, that point is up in the air. So you’re disrupting the solidest of mathematics in favor of a physical supposition. Your “solution” fails on every level. Sorry.
No, it’s not real in a mathematical sense. The paradox comes from the fact that Zeno didn’t have a good understanding of infinite series. In that sense my calculus 2 students are in good company.
My very first sig line:
Tell Zeno I’m willing to meet him halfway.
You’ve been drinking scotch since your teens? Damn, that’s some cool parents you had.
That post is a massive hijack but I will refer people to the endless discussion on that question and suggest any further posts on it be in a new thread.
OK, so there was a mathematician and an engineer. They were put at one end of a room across from a beautiful woman. They were told, “You may move halfway across the room. After you have reached the halfway mark, you may again move half the remaining distance. You may repeat this as many times as you want.”
The mathematician said, “But that’s Zeno’s paradox! I can never actually reach her!”
The engineer said, “Well, I can get close enough for all practical purposes.”
I just found this while looking through old files on my computer. It’s something I wrote way back when, during my very first quarter as a grad student in Philosophy five years ago. It was a take-home midterm or something like that.
I don’t vouch for the quality of the thing. But it does serve as a nice overview of the different ways Zeno’s paradox has been discussed by contemporary writers.
I notice that, in the essay, I took issue with the standard view that Russell has handily dispensed with the paradox. For one thing, apparently, I claimed Russell’s argument was different than what people on this thread have been claiming, and his argument didn’t work. For another, apparently, I argued that the argument people have given on Russell’s behalf also doesn’t work. I am typing this paragraph to register formally that I have no idea whether I was on the right track about all that when I wrote the essay. Also, I wish to note that I know the Russell of the essay is not persuasively written, because it presumes the reader has read the Russell. That was okay at the time, given the nature of my assignment. But with this post, I am not trying to persuade, and do not expect to do so. I’m just pasting the essay in to serve as an overview and as a catalyst for thought.
-FrL-
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The fundamental fact which gives Zeno’s paradoxes their punch is that he seems to have put a finger on a disconnect between our intuitions regarding space and motion on the one hand, and what’s really happening (or what could logically possibly be really happening) “out there,” without regard to our observations or intuitions. Intuitively, it seems to us that, when an object moves, it goes through a series of points. It starts at one point. At the next moment, it is at another point. Then it is at another, then another, and so on infinitely until it reaches an end point. What Zeno succeeds in doing is to show that this understanding of motion will not do - it leads to absurdity. What Zeno wrongly thinks he has done is to show that we must therefore embrace monism. His implicit argument seems to be that, since he has shown that our intuitions about motion can’t be right, it must be the case that there is no such thing as motion. If there is no motion, there is no space. And if there is no space, there can be no distinction between objects. Therefore, there is only one object. Zeno is wrong here because it does not follow from the fact that our intuitions about something are incorrect to the idea that that something itself does not exist. But I will be focusing on what Zeno succeeded in doing, by recounting how philosophers have tried to come to grips with Zeno’s insight.
An explanation of one of Zeno’s paradoxes will give the tenor, very generally, of all of the paradoxes - sufficiently so for the purposes of this paper, in any case. So I will only specifically recount one of them - the Achilles paradox. Suppose Achilles is running a race against a tortoise. Now, Achilles knows he can run much faster than the tortoise, so to make a real challenge out of it, he agrees to give the tortoise a head start. Zeno’s argument is that, under these conditions, it is impossible for Achilles to ever catch up with the tortoise. While Achilles runs up to the point from which the tortoise started, the tortoise will have run some small distance ahead to another point. Then, while Achilles continues to run up to that point, the tortoise will himself have continued to run, again, to a point some small distance ahead of his previous position. And again, while Achilles runs to catch up to that point, the tortoise will have run a small distance to yet another point. This can continue forever - no matter how many times Achilles “catches up” to where the tortoise has been, it seems he can never reach the point where the tortoise is, and hence can never actually pass the tortoise and will never win the race. (Zeno’s Paradoxes, pp 8-9)
A first instinct which many have when presented with this setup is simply to point out that it seems silly to think Achilles couldn’t catch up with a tortoise. When one runner is faster than another, then even given that there is a head start involved, nevertheless the distance between the two runners is certain to shrink steadily until it reaches zero and at that point the faster runner will pass the slower one. However, this does not answer Zeno. Zeno can easily agree that it seems silly to think Achilles couldn’t catch up with a tortoise - especially given the impeccable mathematical logic of the objector. But Zeno also seems to have some impeccable mathematical logic, and according to his logic, it seems equally silly to think Achilles ever will catch up! Both possibilities - Achilles’ catching up and his not catching up - are shown to be impossible on (what seem to be) firmly logical grounds, and Zeno urges that the only way out of this dilemma is simply to deny that Achilles or the tortoise is ever moving at all. No matter what our senses are telling us, the logic of the situation shows that they must be mistaken - there can be no motion taking place because accepting the reality of that motion forces an absurdity on us.
Bertrand Russell objects that the logic of Zeno’s paradox is not so impeccable. He characterizes Zeno as having said that Achilles must pass through an infinite number of instants before reaching the tortoise - and that it would therefore take Achilles an infinite amount of time. (p. 50) Russell rightly points out that an infinite number of instances do not add up to an infinite amount of time - but he was wrong to characterize Zeno in this way in the first place. Zeno does not concern himself with instances and amounts of time - rather he concerns himself with distances. According to Zeno, Achilles must traverse an infinite number of distances in order to reach the tortoise, and this seems to Zeno to be impossible.
Some have taken Zeno’s argument to rest on a mathematical fallacy. They argue that Zeno does not realize that the sum of an infinite series can, in some cases, be finite. The series .1 + .01 + .001 + .0001 + …, for example, though possessing an infinite number of terms, nevertheless has a finite sum - 1/9. (p. 68) Though Achilles’ run can be characterized as being composed of an infinite number of ever decreasing distances, nevertheless, this does not mean he never reaches the end - the sum of all those distances is finite, after all. But this seems to miss the point. In one sense, this is simply a repetition of the simplistic objection given above, along the lines of saying “Of course Achilles catches up with the tortoise. The distance gets smaller and smaller until it reaches zero - then he passes him.” We know that, and Zeno knows that. But he is pointing out that for us to affirm this seems to require us to also affirm that somehow, Achilles, while traversing only a finite distance, nevertheless seems to be accomplishing an infinite number of tasks in the finite time it takes him to do so - as pointed out by Max Black. (p. 71) This seems problematic, on the face of it.
Some have suggested that, in fact, there are not an infinite number of tasks involved in Achilles’ run because there are not in fact an infinite number of points between his start point and the point at which he catches up with the tortoise. This is the atomistic solution - relying for its force on an assumption that space is made up of discrete units, so that between any two points there is not an infinite but a hugely finite number of these units. Achilles only needs to accomplish a number of “tasks” equal to the number of these units between the two points involved. There are some serious problems with this view however. For example, if there is in fact a smallest unit of space, then that unit must have some kind of shape (or it can not actually take up space). But if this shape is one which smoothly tiles, then it is necessarily non-spherical, and if it is non-spherical, then we are forced to conclude that space has an inherent directionality to it. Motion in some directions will require traveling through fewer units than motion in other directions. But if the units of space do not smoothly tile, then there would be “space” in between them. This is of course silly since the units we are talking about are what space is. There can be no space “in between” space. (Weyl makes an observation along these lines, as well.) (p. 175.)
But this leaves us with the notion that an actually infinite number of tasks are accomplished by Achilles in a finite amount of time. Several of the articles in the book try to deal with this by showing how it can be possible, however unintuitive it may seem. Some argue that the infinity involved is merely conceptual - a product of mathematics and not of observation - that physically speaking we need only concern ourselves with the finite task of running, performed by a finite agent. (p. 81) Others argue that the fact that we can say there are an infinite number of sub-tasks involved in running the race simply has nothing to do with whether we can say the whole task of running the race can be accomplished or not. (Space does not permit me to elaborate on this but much of Benaceraff’s article is concerned with just this point.)
All of these discussions were in every sense interesting and fruitful, but in the end, even if it is logically shown that an infinite number of subtasks can be performed in the performance of a whole task, we are still left with the force of Zeno’s paradox - that this logically proven idea nevertheless does not seem to make sense in the context of our intuition that the finite can not actually be divided into an infinite number of parts. A finite racecourse, when considered as an infinite number of smaller racecourses, suddenly seems to become (to our intuition) an infinite racecourse, impossible to traverse.
Zeno took the fact that our intuition leads to such absurdity to mean that there was something fundamentally wrong with it and that therefore we should reject the intuition - namely, the very idea of motion. I would agree that there is something wrong with our intuition, but I think this simply calls for a redefinition of the concept of motion without undue regard for our intuition. I think the fundamental problem with the intuition is that it thinks of a moving object as occupying first one point, then next another point, and next another point, and so on. This makes the number of points on the racecourse a countable infinity, which makes it seem paradoxically infinitely large while at the same time as being of only finite size. But if we understand that the number of points on that racecourse are not countably infinite - that given any point on it, there is no next point, then we see that the points do not come in a series. Rather than thinking of motion as traveling through a series of points, we can think of motion as simply the fact that, at one time, an object occupies one point, and at another, it occupies a different point, and that there is a sensible relation between the points occupied and the times at which they are occupied. Thinking of motion this way prevents it from seeming to be the case that the racecourse is both infinitely large and finite at the same time - because in thinking of motion this way we never conceive of all the points on the race course being laid out “in a row” as it were. We simply think of them as a set of points which we can point out individually at our leisure. There is no pressure for Achilles to traverse all the points “one after another.” He can simply point out that at given times, he does in fact occupy the proper points, and therefore he is moving at the proper speed, and will catch up and pass the tortoise as he should.