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My paper for phil-201 re: Zenos paradox of motion
Print My paper for phil-201 re: Zenos paradox of motion | |
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richrf
Meaning of Life Blog Usergroup: Members Joined: Jun 06, 2009 Total Topics: 0 Total Posts: 4 |
 #21
I think that Zeno was trying to make the point that there is a conceptual difference between what one might imagine and what actually exists. I personally don't need mathematics to prove that I can catch up to someone else. I do it all the time. However, what makes Zeno interesting, is that I can imagine what he is saying, and cannot understand why what I am imagining does not correspond to reality. My guess is that there is something about time, space, and motion, that has to be reinterpreted in order for everything to make sense. In the same way, Einstein's concept of curved space/time enables us to better understand the nature of gravity. Proving Zeno right or wrong will not remove the underlying paradox of what he has presented. Bravo, Zeno! Rich http://www.MeaningOfLifePhilosophy.com |
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mattmark
Graduate Usergroup: Members Joined: Sep 06, 2009 Total Topics: 1 Total Posts: 39 |
#22
Subject: zeno's pseudo-paradox "I personally don't need mathematics to prove that I can catch up to someone else. I do it all the time." So we know there must be something fishy about an argument purporting to show that we can't. (If only we had this kind of guiding 'clue' handed to us on a silver platter every time we evaluated an argument!) "However, what makes Zeno interesting..." 'Interesting,' or merely curious? Surely it's a sociological curiosity (on a par with the world's believing that the current millennium began in the year 2000) that nominally intelligent individuals can be seduced by a simple logical equivocation into irrelevant head-scratching about Planck constants and the ontological constraints facing infinite divisibility. "...is that I can imagine what he is saying, and cannot understand why what I am imagining does not correspond to reality." I think I can help, and quite easily. "My guess is that there is something about time, space, and motion, that has to be reinterpreted in order for everything to make sense. In the same way, Einstein's concept of curved space/time enables us to better understand the nature of gravity." It's nothing nearly so complicated. It suffices to notice that the evidence cited for the supposed inability of the hare (or, in some versions, Achilles) to overtake the tortoise is irrelevant to establishing this conclusion. Think of what the argument says the hare can't do: it denies that he can set off to overtake the tortoise and arrive at the point where the tortoise was when he (the hare) set off to overtake him (the tortoise), before the tortoise has moved on, for however small a distance. This is perfectly true and will remain true so long as the tortoise is in motion away from the hare, at any speed whatever. But the hare is not required to perform this impossible feat simply to overtake the tortoise. The equivocal concepts at work here are 'H overtakes T,' and, 'H reaches the point (on a line) where T was when H set off to overtake him, before T has moved on--however minimally--from that point.' These concepts are not identical--overtaking the tortoise at some stage, and reaching the tortoise's starting point before he's left it (which would mean reaching it instantaneously), are distinct achievements--but the argument uses the concepts interchangeably. Should it surprise us that such transparent logical equivocation generates a seemingly paradoxical conclusion? "Bravo, Zeno!" Or, perhaps, 'Boo' to society's general inability to detect logical equivocations, even (apparently) in philosophy forums? Edited by mattmark on Sep 8, 2009 - 6:54 PM |
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Jehu
Revealer  Usergroup: Members Joined: Nov 30, 2006 Total Topics: 3 Total Posts: 963 |
#23
bevans15 wrote: In all seriousness however, I believe that I have effectively disproved Zeno’s paradox both mathematically and logically. I do think that, while false, they should continue to be used in order to provoke the proverbial ‘deep-thoughts’ and intellectual enticing conversations that I consider myself lucky to have been a part of this semester. You have made a very common error in approaching Zeno’s Paradox, in that you are arguing that because we can intellectually divide any given portion of a number line without every exhausting our capacity to devise even smaller numbers, that the same is true of actual space or time, but this does not hold. There are indeed infinite series in mathematic (e.g., 1/2+ 1/4+ 1/8+ 1/16+ ... +1/2n+...), and it may rightfully be said that they have boundaries (i.e., 0 and 1), but they cannot have sums, for we cannot sum what is by definition not only incomplete, but incapable of ever being completed. In other words, it is a mistake to hold that just because we can imagine this infinite series to lies within the said boundaries, that it does in fact lie therein. The matter of space or time is quite different, for if we hold that space or time are infinitely divisible, as must be the case if they are continuous, then within any set of spatial or temporal boundaries, there must lie an infinite number of smaller spaces or durations. This is where the logical inconsistencies arise, for to traverse any finite amount of space require a finite amount of time, and so an infinite amount of space would require an infinite amount of time. Thus there can be no movement. However, as modern physics has since abandoned the Newtonian concept of absolute space and time, in favour of Einstein’s relativity, and it was just this sort of notion that Zeno was attempting to refute, it would appear that he may have been right all along. It is not that which the eye can see, but that whereby the eye is able to see, that is the true reality. |
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mattmark
Graduate Usergroup: Members Joined: Sep 06, 2009 Total Topics: 1 Total Posts: 39 |
#24
”…you are arguing that because we can intellectually divide any given portion of a number line without ever exhausting our capacity to devise even smaller numbers, that the same is true of actual space or time, but this does not hold.” I don’t know if the post to which you’re replying actually argues this, but there’s no need to argue it. Zeno hasn’t bequeathed us a mathematical or ontological paradox but a logical one that owes whatever persuasiveness it has to conceptual ambiguity. I’m sure you would agree that the straight line distance between any two non-coincident points can be infinitely subdivided--and this suffices for the purpose. Straight lines are abstractions, remember, and so are their constituent, dimensionless points (they lack extension or ‘area’). It makes no difference that ‘actual space or time’ may consist of ‘smallest possible’ units; we can indefinitely subdivide both length and duration without having to advance claims about the ontological status of any (purely abstract) ‘infinitely smaller’ units that may arise from the process. The ‘actuality’ of such units is irrelevant. “…within any set of spatial or temporal boundaries, there must lie an infinite number of smaller spaces or durations.” It couldn’t be otherwise, given that there are an infinite number of points in any line segment. “This is where the logical inconsistencies arise, for to traverse any finite amount of space require a finite amount of time, and so an infinite amount of space would require an infinite amount of time.” But it’s clearly equivocal to jump from ‘infinite number of smaller spaces or durations’ to ‘infinite amount of space or time.’ Granted, we can indefinitely subdivide any space, no matter how minuscule (it’s this realization that gives rise to the seeming paradox, yes?), thus yielding, in principle, an infinitely large number. But the ‘amount’ of space so subdivided remains just as finite as it was before we started dividing. “Thus there can be no movement.” If this conclusion followed from the considerations discussed above, it would similarly follow that there could be nothing stationary either. By definition, all line segments--no matter how short--contain an infinite number of points. Shouldn’t we be as concerned about the ontological status of these infinitely numerous points as about the (supposedly dubious) ontological status of infinitely small subdivisions of line segments? But we aren’t. “However, as modern physics has since abandoned the Newtonian concept of absolute space and time, in favour of Einstein’s relativity, and it was just this sort of notion that Zeno was attempting to refute, it would appear that he may have been right all along.” It’s doubtful that Newton, Einstein or physics have significant roles to play in helping us come to grips with Zeno’s paradox. What’s crucial to realize is that there’s a logical equivocation at the heart of the paradox (please see my earlier post, above); once it is discerned, the problem disappears. Perhaps the conceptual opacity at the root of this equivocation can be addressed by means of a simple thought-experiment. Instead of asking ourselves how the hare is able to traverse an infinitely divisible distance between himself and the tortoise, why not ask how he can traverse the equally infinitely divisible distance between himself and some fixed point (on the imaginary line on which he and the tortoise are situated) that is located well beyond the tortoise? Now the question becomes not, ‘Can the hare overtake the tortoise?’, but, ‘Can the tortoise and the hare both reach a fixed point that is well beyond them and further away for the hare?’ If we imagine the pursuing hare and the tortoise to be already very close together, with the hare moving at much the greater rate of speed, is there any conceptual impediment to imagining not only that both succeed in reaching this distant point but that the hare might do so before the tortoise? And if we can conceive of the hare arriving first, why not of his also arriving first at an intermediate point which is slightly less distant--and so backward along the line at various intermediate points, nearer and nearer to the respective starting positions, until the hare’s arrival at some very proximate point barely precedes the tortoise’s or is even coincident with it? What does it matter that all of the line segments defined by the hare’s linear position(s) and any of these other arbitrarily selected points can be infinitely subdivided? What relevance does this have for the concept of ‘overtaking,’ or of motion in general? Even if it had the least relevance, what difference could it make where the defining points of the line segments happened to be located in relation to the tortoise? Why should we regard the line segments defined by his position, either in front of, coincident with or behind the hare, as special cases rather than as red herrings? |
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Jehu
Revealer Usergroup: Members Joined: Nov 30, 2006 Total Topics: 3 Total Posts: 963 |
#25
mattmark wrote: I don’t know if the post to which you’re replying actually argues this, but there’s no need to argue it. Zeno hasn’t bequeathed us a mathematical or ontological paradox but a logical one that owes whatever persuasiveness it has to conceptual ambiguity. I’m sure you would agree that the straight line distance between any two non-coincident points can be infinitely subdivided--and this suffices for the purpose. Straight lines are abstractions, remember, and so are their constituent, dimensionless points (they lack extension or ‘area’). It makes no difference that ‘actual space or time’ may consist of ‘smallest possible’ units; we can indefinitely subdivide both length and duration without having to advance claims about the ontological status of any (purely abstract) ‘infinitely smaller’ units that may arise from the process. The ‘actuality’ of such units is irrelevant. Zeno was not concerned with the abstractions of mathematics, but with whether or not the prevailing metaphysical concepts of his day logically held when applied to the actual world. As an idealist, Zeno was attempting to draw attention to the logical inconsistencies that the notion of real and extended objects moving through a continuous, and therefore infinitely divisible, space and time entailed. Further, he was not the first to point out these logical inconsistencies, for the great Buddhist logistician Nagarjuna tried to call attention to the problem of time and motion when he wrote, and I paraphrase, on the path there is only the road already taken and the road not yet taken, and since the past no longer exists, and the future does not yet exist, where then exists there any time for movement. While the abstract concept of a line may entail the notion of a sequence of infinitesimal points, an actual line does not; and it is a categorical mistake to think that the abstract spaces of mathematics are identical with actual/physical space. It is not that which the eye can see, but that whereby the eye is able to see, that is the true reality. |
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mattmark
Graduate Usergroup: Members Joined: Sep 06, 2009 Total Topics: 1 Total Posts: 39 |
#26
"Zeno was not concerned with the abstractions of mathematics, but with whether or not the prevailing metaphysical concepts of his day logically held when applied to the actual world." As I'm unable to distill meanings from two conjunctions of terms in this sentence I'm at a loss to understand what you take Zeno's concerns to have been. If you could perhaps explain (ideally by citing an instance) what it means for a concept (as opposed to an argument) to 'logically hold,' and what kind of 'application' to the 'actual world' you have in mind that bears on such holding, I'd have a better chance of grasping the connection between your observation and the 'paradox.' "As an idealist, Zeno was attempting to draw attention to the logical inconsistencies that the notion of real and extended objects moving through a continuous, and therefore infinitely divisible, space and time entailed." Can you specify what these inconsistencies are? If they're 'entailed' by anything it should be an easy and convincing demo. The only 'inconsistency' I've been able to discern is a merely apparent one, arising from the equivocal use of non-equivalent concepts. "...on the path there is only the road already taken and the road not yet taken, and since the past no longer exists, and the future does not yet exist, where then exists there any time for movement." Metaphorical analogy, to be instructive, must be genuinely analogical and--just as importantly--coherent. If Nagarjuna is unwilling to grant some sort of existence to past and future, it's clear that he can no longer speak of a 'path' either. Expunging past and future eradicates the possibility of linear metaphor. "While the abstract concept of a line may entail the notion of a sequence of infinitesimal points, an actual line does not; and it is a categorical mistake to think that the abstract spaces of mathematics are identical with actual/physical space." Yes, we agree: the map is not the territory. But what connection does this have with the equivocation that gives rise to the so-called paradox? It remains equivocal irrespective of the relation between abstraction and the realm of the 'actual/physical.' Edited by mattmark on Sep 11, 2009 - 6:07 AM |
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Jehu
Revealer Usergroup: Members Joined: Nov 30, 2006 Total Topics: 3 Total Posts: 963 |
#27
mattmark[quote wrote: As I'm unable to distill meaning from the conjunction of terms in this sentence I'm at a loss to understand what you take Zeno's concerns to have been. If you could perhaps explain (ideally by citing an instance) what it means for a concept (as opposed to an argument) to 'logically hold,' and what kind of 'application' to the 'actual world' you have in mind that bears on such holding, I'd have a better chance of grasping the connection between your observation and the 'paradox.' Let me restate it then, in terms that perhaps you might better understand: when Zeno thought deeply upon the metaphysical views of other philosophical schools of his period, he discovered that a logical analysis of those views gave rise to such absurd conclusions as that there can be no motion. Now, Zeno did not believe that there was in fact no motion, but as it was clear that his reasoning was valid, it could only mean that one or more of the underlying premises was faulty. Can you specify what these inconsistencies are? If they're 'entailed' by anything it should be a very easy and convincing demo. The only 'inconsistency' I've been able to discern is a merely apparent one, arising from the equivocal use of non-equivalent concepts. I think that Zeno did an exemplary job of outlining the logical inconsistencies, for what does the term ‘paradox’ means, if not that which is logically inconsistent? To be sure, I am incapable of stating the case any better than he. Metaphorical analogy, to be instructive, must be genuinely analogical and--just as importantly--coherent. If Nagarjuna is unwilling to grant some sort of existence to past and future, it's clear that he cannot legitimately speak of a 'path' either. No past and future, no linear metaphor. True, but missed the point again. Nagarjuna, like Zeno, is an idealist, and is questioning the same sort of materialistic/physicalistic views within the Buddhist community of his day. For an idealist, such a Nagarjuna, the five elements of the physical world: space, time, matter, energy and motion are all illusory (like in a dream), and so his view was in complete accord with his logical conclusions. It is not that which the eye can see, but that whereby the eye is able to see, that is the true reality. |
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mattmark
Graduate Usergroup: Members Joined: Sep 06, 2009 Total Topics: 1 Total Posts: 39 |
#28
"Let me restate it then, in terms that perhaps you might better understand:" My understanding would benefit best from your answering my question. What does it mean for a concept to 'logically hold?' Do you simply mean a non-contradictory concept? "…when Zeno thought deeply upon the metaphysical views of other philosophical schools of his period, he discovered that a logical analysis of those views gave rise to such absurd conclusions as that there can be no motion." Whether Zeno thought deeply or shallowly isn't at issue here. If there's a specific absurdity that follows from specific premises in the paradox we're considering, what is it? The only absurdity I'm able to discern doesn't 'follow' from anything. It's simply the sort of confusion one expects to encounter in any tale that's mired in equivocation. "I think that Zeno did an exemplary job of outlining the logical inconsistencies…" Shouldn’t that make it easy to specify one? "…for what does the term ‘paradox’ means, if not that which is logically inconsistent? To be sure, I am incapable of stating the case any better than he." This is evasive. I'm not asking you to recapitulate Zeno. I'm trying to discover what particular logical inconsistency you discern in this paradox, apart from the one arising from its equivocal terms. "True, but missed the point again. Nagarjuna, like Zeno, is an idealist, and is questioning the same sort of materialistic/physicalistic views within the Buddhist community of his day. For an idealist, such a Nagarjuna, the five elements of the physical world: space, time, matter, energy and motion are all illusory (like in a dream), and so his view was in complete accord with his logical conclusions." Whatever you hoped to achieve with this sort of historical generalization, I'm afraid it hasn't done much to clarify your initial rejoinder. Zeno’s and Nagarjuna’s motives and historical circumstances are, of course, of considerable interest to us, but this alone can’t confer on them any particular relevance for assessing the logical status of Zeno’s paradox. If we’re to be persuaded that a genuine paradox exists here, the supposedly ‘paradoxical’ element must manifest itself on its own, as formulated, and withstand our critical scrutiny. |
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Jehu
Revealer Usergroup: Members Joined: Nov 30, 2006 Total Topics: 3 Total Posts: 963 |
#29
mattmark wrote: If there's a specific absurdity that follows from specific premises in the paradox we're considering, what is it? When a rigorous deductive enquiry begins from premises such as that space and time are continuous and therefore infinitely divisible, and ends in such an obviously absurd conclusion as that there can be no motion at all, this is called a ‘paradox’. Further, paradoxes are not permitted in a rational universe, and as there is no fault to be found with Zeno’s reasoning – or Nagarjuna’s for that matter, it follows that there must be something amiss with one or more of the founding premises. Now, many have claimed to have resolved the paradox, but this has been chiefly by some sort of mathematical slight of hand (such as in the OP), however, Zeno was not speaking of mathematics, nor of formal logic, but of actual things moving in actual space and time. It is not that which the eye can see, but that whereby the eye is able to see, that is the true reality. |
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ragus
Tenured Poster  Usergroup: Members Joined: Apr 23, 2006 Total Topics: 16 Total Posts: 2541 |
#30
mattmark wrote we can indefinitely subdivide both length and duration without having to advance claims about the ontological status of any (purely abstract) ‘infinitely smaller’ units that may arise from the process. Does this mean that we can ignore the computational aspect of this? We might think that we can produce strings representing subdivisions but as these strings get longer the reality of their formulation becomes problematic. If the logician waves this aside isn't this another way of calling on the infinitely fast to do the job? Our awareness of truth and falsehood gives us the capacity, but not the reason, to lie. |
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