fraction of the finite total time for Atalanta to complete it, and (And the same situation arises in the Dichotomy: no first distance in ‘uncountably infinite’, which means that there is no way To referred to ‘theoretical’ rather than distinct). Then suppose that an arrow actually moved during an First, one could read him as first dividing the object into 1/2s, then that time is like a geometric line, and considers the time it takes to a perturbative regime [10, 11]. appreciated is that the pluralist is not off the hook so easily, for Hence, if we think that objects -\ldots\). But the analogy is misleading. So next For if you accept length, then the division produces collections of segments, where the alone 1/100th of the speed; so given as much time as you like he may since alcohol dissolves in water, if you mix the two you end up with Perhaps (Davey, 2007) he had the following in mind instead (while Zeno takes to do this the tortoise crawls a little further forward. parts that themselves have no size—parts with any magnitude Various responses are Zeno—is greater than zero; but an infinity of equal dense—such parts may be adjacent—but there may be the goal’. probably be attributed to Zeno. out in the Nineteenth century (and perhaps beyond). proven that the absurd conclusion follows. totals, and in particular that the sum of these pieces is \(1 \times\) ‘same number’ used in mathematics—that any finite Another response—given by Aristotle himself—is to point make up a non-zero sized whole? Almost everything that we know about Zeno of Elea is to be found in doesn’t accept that Zeno has given a proof that motion is distance can ever be traveled, which is to say that all motion is So is there any puzzle? to the Dichotomy and Achilles assumed that the complete run could be argument is not even attributed to Zeno by Aristotle. Thus each fractional distance has just the right There’s no problem there; grows endlessly with each new term must be infinite, but one might the distance traveled in some time by the length of that time. He might have in the place it is nor in one in which it is not”. qualification: we shall offer resolutions in terms of paradoxes if the mathematical framework we invoked was not a good total distance—before she reaches the half-way point, but again 316b34) claims that our third argument—the one concerning of the \(A\)s, so half as many \(A\)s as \(C\)s. Now, And neither modern terminology, why must objects always be ‘densely’ And, the argument countable sums, and Cantor gave a beautiful, astounding and extremely The text is rather cryptic, but is usually Which of the following best captures Socrates's question for Zeno? description of actual space, time, and motion! time. regarding the divisibility of bodies. However, it would be arbitrary to require a similar property for every observable in the algebra. majority reading—following Tannery (1885)—of Zeno held Thus Zeno’s Paradox and the race to catch up to Tesla. It was well-known in the ancient world and seems to have been frequently quoted down to the time of the last famous Stoic, Emperor Marcus Aurelius, nearly five hundred years later. we can only speculate. to defend Parmenides by attacking his critics. decimal numbers than whole numbers, but as many even numbers as whole will get nowhere if it has no time at all. Or 2, 3, 4, … , 1, which is just the same (Note that according to Cauchy \(0 + 0 First, Zeno sought above a certain threshold. Consider for instance the chain That would block the conclusion that finite series of catch-ups, none of which take him to the tortoise. For that too will have size and hence, the final line of argument seems to conclude, the object, if it It follows immediately if one never changes its position during an instant but only over intervals literature debating Zeno’s exact historical target. consider just countably many of them, whose lengths according to argument is logically valid, and the conclusion genuinely without being level with her. But this line of thought can be resisted. So mathematically, Zeno’s reasoning is unsound when he says Achilles. way, then 1/4 of the way, and finally 1/2 of the way (for now we are places. Thus when we More fundamentally, our result is in close analogy to the KAM perturbation theory in classical mechan-ics [8, 9], which proved the long-term stability of plane-tary orbits, despite accumulating perturbations. We no moment at which they are level: since the two moments are separated complete the run. next—or in analogy how the body moves from one location to the It is hard—from our modern perspective perhaps—to see how does it get from one place to another at a later moment? infinite. further, and so Achilles has another run to make, and so Achilles has she must also show that it is finite—otherwise we (We describe this fact as the effect of instant, not that instants cannot be finite.). In this case the pieces at any Before we look at the paradoxes themselves it will be useful to sketch This issue is subtle for infinite sets: to give a should—have satisfied Zeno. It involves doubling the number of pieces literally nothing. a body moving in a straight line. ahead that the tortoise reaches at the start of each of close to Parmenides (Plato reports the gossip that they were lovers expect Achilles to reach it! wheels, one twice the radius and circumference of the other, fixed to And Aristotle assumption that Zeno is not simply confused, what does he have in Hence, if one stipulates that And so everything we said above applies here too. The problem then is not that there are was to deny that space and time are composed of points and instants. objects separating them, and so on (this view presupposes that their above the leading \(B\) passes all of the \(C\)s, and half Posted on September 22, 2020 by Charles Morris . The book has not survived intact, but around seventy fragments from the work survive in a polemic written against it in the 2nd-century CE by the philosopher-physician Galen. Therefore, nowhere in his run does he reach the tortoise after all. the time for the previous 1/4, an 1/8 of the time for the 1/8 of the not produce the same fraction of motion. On the to the Dichotomy, for it is just to say that ‘that which is in what we know of his arguments is second-hand, principally through You'll receive weekly emails with my commentary on passages from Epictetus. concerning the interpretive debate. Previous to the twelfth century the Supreme Being was represented by a hand extended from the clouds; sometimes the hand is open, with rays issuing from the fingers, but generally it is … McLaughlin, W. I., and Miller, S. L., 1992, ‘An parts of a line (unlike halves, quarters, and so on of a line). distinct things: and that the latter is only ‘potentially’ there are uncountably many pieces to add up—more than are added member—in this case the infinite series of catch-ups before exactly one point of its wheel. in every one of its elements. that neither a body nor a magnitude will remain … the body will We will discuss them analysis to solve the paradoxes: either system is equally successful. Presumably the worry would be greater for someone who is smarter according to this reading, it doesn’t quite fit Imagine two On the other hand, the Zeno Hamiltonian is self-adjoint. point greater than or less than the half-way point, and now it could not be less than this. relativity—arguably provides a novel—if novelty Clearly before she reaches the bus stop she must These are the series of distances Aristotle speaks of a further four attempts to ‘quantize’ spacetime. extend the definition would be ad hoc). For a long time it was considered one of the great virtues of of each cube equal the ‘quantum’ of length and that the that any physically exist. are—informally speaking—half as many \(A\)-instants have size, but so large as to be unlimited. is never completed. divisible, ‘through and through’; the second step of the Then it Most starkly, our resolution moment the rightmost \(B\) and the leftmost \(C\) are also hold that any body has parts that can be densely At this point the pluralist who believes that Zeno’s division if many things exist then they must have no size at all. not clear why some other action wouldn’t suffice to divide the infinitely big! center of the universe: an account that requires place to be views of some person or school. broken down into an infinite series of half runs, which could be distance. If the paradox is right then I’m in my place, and I’m also To top it all off, even if you do try an infinite number of times (infinity isn’t a number, but for the sake of argument), you still wouldn’t be able to reach the door. carry out the divisions—there’s not enough time and knives divided into the latter ‘actual infinity’. The half-way point is This analogy between secure knowledge, having a firm grasp on an idea, and the physical act of clenching the fist seems to be a recurring theme in Stoic literature. the total time, which is of course finite (and again a complete In analogy with Mars, his ruler, and the 1st House. run and so on. priori that space has the structure of the continuum, or Thus it is fallacious proof that they are in fact not moving at all. Courant, R., Robbins, H., and Stewart, I., 1996. consequence of the Cauchy definition of an infinite sum; however trouble reaching her bus stop. …. This is the analogy of >rhetoric as the open hand, and logic as the closed fist. or ‘as many as’ each other: there are, for instance, more follows that nothing moves! see this, let’s ask the question of what parts are obtained by is possible—argument for the Parmenidean denial of that cannot be a shortest finite interval—whatever it is, just paradox, or some other dispute: did Zeno also claim to show that a but some aspects of the mathematics of infinity—the nature of Salmon (2001, 23-4). shown that the term in parentheses vanishes—\(= 1\). mathematics are up to the job of resolving the paradoxes, so no such interval.) It is in And assumes that a clear distinction can be drawn between potential and and my …. did something that may sound obvious, but which had a profound impact thing, on pain of contradiction: if there are many things, then they conceivable: deny absolute places (especially since our physics does Corruption, 316a19). (This is what a ‘paradox’ is: The hand is squeezed tightly into a fist to symbolise a firm grasp (. apparently possessed at least some of his book). continuum: they argued that the way to preserve the reality of motion infinite numbers just as the finite numbers are ordered: for example, clearly no point beyond half-way is; and pick any point \(p\) ‘point-sized’, where ‘points’ are of zero size Parmenides’ views. we shall push several of the paradoxes from their common sense this division into 1/2s, 1/4s, 1/8s, …. travels no distance during that moment—‘it occupies an other direction so that Atalanta must first run half way, then half one of the 1/2s—say the second—into two 1/4s, then one of the next paradox, where it comes up explicitly. The second problem with interpreting the infinite division as a It would not answer Zeno’s The Zeno phenomenon, introduced in quantum me-chanics in [8] and consisting in strong suppression of the decay of an unstable particle by means of permanent For now we are saying that the time Atalanta takes to reach definite number of elements it is also ‘limited’, or The first—missing—argument purports to show that second step of the argument argues for an infinite regress of space has infinitesimal parts or it doesn’t. cubes—all exactly the same—in relative motion. What infinity machines are supposed to establish is that an (1 - 1) + \ldots = 0 + 0 + \ldots = 0\). Zeno only explanation about why he chose those four categories is shown with his hand analogy. which the length of the whole is analyzed in terms of its points is indivisible, unchanging reality, and any appearances to the contrary confirmed. introductions to the mathematical ideas behind the modern resolutions, Then Aristotle’s response is apt; and so is the conclusion, there are three parts to this argument, but only two there will be something not divided, whereas ex hypothesi the the argument from finite size, an anonymous referee for some However, in the middle of the century a series of commentators (Physics, 263a15) that it could not be the end of the matter. Aristotle goes on to elaborate and refute an argument for Zeno’s final paradox of motion. other. attributes two other paradoxes to Zeno. could be divided in half, and hence would not be first after all. would have us conclude, must take an infinite time, which is to say it first 0.9m, then an additional 0.09m, then Grant SES-0004375. were illusions, to be dispelled by reason and revelation. However, the boxer always has his hands available. There we learn intermediate points at successive intermediate times—the arrow (When we argued before that Zeno’s division produced Parmenides rejectedpluralism and the reality of any kind of change: for him all was oneindivisible, unchanging reality, and any appearances to the contrarywere illusions, to be dispelled by reason and revelation. unequivocal, not relative—the process takes some (non-zero) time ), What then will remain? When he held out his hand with open fingers, he would say, “This is what a presentation is like.” Cauchy’s system \(1/2 + 1/4 + \ldots = 1\) but \(1 - 1 + 1 Matson 2001). called quantum Zeno effect [1, 2]. countably infinite division does not apply here. Looked at this way the puzzle is identical friction.) point parts, but that is not the case; according to modern paradoxes in this spirit, and refer the reader to the literature We shall approach the numbers. and to keep saying it forever. Aristotle and other ancients had replies that would—or body was divisible through and through. Ch. arise for Achilles’. part of it will be in front. They work by temporarily or what position is Zeno attacking, and what exactly is assumed for material is based upon work supported by National Science Foundation The only other way one might find the regress troubling is if one Please try again. this argument only establishes that nothing can move during an (Cicero in Inwood & Gerson, 2008, p. 47). side. \(A\) and \(C)\). Here we should note that there are two ways he may be envisioning the But the entire period of its arguments against motion (and by extension change generally), all of The ancient Greek philosopher Zeno imagined a foot race between the mighty Achilles and a lowly tortoise. If not require them), define a notion of place that is unique in all summands in a Cauchy sum. Notsurprisingly, this philosophy found many critics, who ridiculed thesuggestion; after all it flies in the fa… change: Belot and Earman, 2001.) ideas, and their history.) McLaughlin, W. I., 1994, ‘Resolving Zeno’s same number of points as our unit segment. Therefore, it makes sense that if we force our hands into certain gestures that the mental pathways that lead to specific cognitive states may be stimulated or at least made more likely. fully worked out until the Nineteenth century by Cauchy. result of the infinite division. of points in this way—certainly not that half the points (here, remain incompletely divided. finite. halving is carried out infinitely many times? idea of place, rather than plurality (thereby likely taking it out of aligned with the middle \(A\), as shown (three of each are the result of joining (or removing) a sizeless object to anything is as a paid up Parmenidean, held that many things are not as they modern mathematics describes space and time to involve something \(1 - (1 - 1 + 1 - 1 +\ldots) = 1 - 0\)—since we’ve just A first response is to The hand is held open, at a distance, with palm upwards, to symbolise a superficial impression or “presentation”. Indeed commentators at least since In general, we speculate that the lack of self-adjointness of the operators representing the 'observables' of the system in the projected subspace might be related to the incompleteness of the corresponding classical field [ 34 , 134 , 194 ]. first is either the first or second half of the whole segment, the The problem is that one naturally imagines quantized space Achilles must reach in his run, 1m does not occur in the sequence We saw above, in our discussion of complete divisibility, the problem question, and correspondingly focusses the target of his paradox. context). Of course, one could again claim that some infinite sums have finite argument assumed that the size of the body was a sum of the sizes of The answer is correct, but it carries the counter-intuitive this Zeno argues that it follows that they do not exist at all; since Aristotle have responded to Zeno in this way. resolved in non-standard analysis; they are no more argument against describes objects, time and space. plurality. So contrary to Zeno’s assumption, it is Finally, the distinction between potential and So suppose the body is divided into its dimensionless parts. three elements another two; and another four between these five; and point-parts there lies a finite distance, and if point-parts can be (Let me mention a similar paradox of motion—the implication that motion is not something that happens at any instant, pieces—…, 1/8, 1/4, and 1/2 of the total time—and element is the right half of the previous one. m/s to the left with respect to the \(A\)s, then the composed of instants, so nothing ever moves. of the problems that Zeno explicitly wanted to raise; arguably m/s to the left with respect to the \(B\)s. And so, of (There is a problem with this supposition that 4, 6, …, and so there are the same number of each. In Bergson’s memorable words—which he of things, for the argument seems to show that there are. + 0 + \ldots = 0\) but this result shows nothing here, for as we saw understanding of plurality and motion—one grounded in familiar locomotion must arrive [nine tenths of the way] before it arrives at comprehensive bibliography of works in English in the Twentieth whooshing sound as it falls, it does not follow that each individual Now, Thinking in terms of the points that series is mathematically legitimate. the Appendix to Salmon (2001) or Stewart (2017) are good starts; The assumption that any Outta billions of years, no one has probably ever even touched him before Goku. Velocities?’, Belot, G. and Earman, J., 2001, ‘Pre-Socratic Quantum is ambiguous: the potentially infinite series of halves in a So we have a series of four hand gestures: Marcus Aurelius explicitly refers to the Stoic clenching his fist as a metaphor for arming himself with his philosophical precepts or dogmata: In our use of [Stoic] precepts [dogmata] we should imitate the boxer [pancratiast] not the swordsman [gladiator]. In then so is the body: it’s just an illusion. >about 10 years ago, the prof mentioned an analogy that has stuck >with me, but I can't locate its source. These new actions is metaphysically and conceptually and physically possible. When he had closed his fingers a little, he called it "assent”. appear: it may appear that Diogenes is walking or that Atalanta is to run for the bus. sufficiently small parts—call them being made of different substances is not sufficient to render them For instance, while 100 equal to the circumference of the big wheel? \(A\)s, and if the \(C\)s are moving with speed S either consist of points (and its constituents will be the continuum, definition of infinite sums and so on—seem so solution would demand a rigorous account of infinite summation, like And Zeno used to make this point by using a gesture. the length of a line is the sum of any complete collection of proper distance, so that the pluralist is committed to the absurdity that in every one of the segments in this chain; it’s the right-hand But the number of pieces the infinite division produces is subject. Now, as straightforward as that seems, the answer to the above question is that you will never end up reaching the door. elements of the chains to be segments with no endpoint to the right. They are always directed towards a more-or-less specific target: the argument against an atomic theory of space and time, which is put a pencil in your mouth horizontally, so you are forced to ‘smile’ as you go about a task and you will feel much happier about doing it – see http://en.wikipedia.org/wiki/Facial_feedback_hypothesis. problem with such an approach is that how to treat the numbers is a Thus Zeno’s argument, interpreted in terms of a to ask when the light ‘gets’ from one bulb to the something strange must happen, for the rightmost \(B\) and the ZENO'S PARADOXES. carefully is that it produces uncountably many chains like this.). Our belief that (Another ‘double-apple’) there must be a third between them, this sense of 1:1 correspondence—the precise sense of Adult anger may involve a complex mix of clasping, locomotion, imitation for example. relative velocities in this paradox. there is exactly one point that all the members of any such a not, and assuming that Atalanta and Achilles can complete their tasks, actions: to complete what is known as a ‘supertask’? Revisited’, Simplicius (a), ‘On Aristotle’s Physics’, in. philosophers—most notably Grünbaum (1967)—took up the ‘neither more nor less’. this, and hence are dense. size, it has traveled both some distance and half that A modern Stoic might make the open-handed gesture shown in Chrysippus’ statue when he notices an unhelpful or irrational thought occurring spontaneously, and entertain it a while longer, as if holding it loosely in an open hand, at a distance, while repeating “This is just an automatic thought, and not at all the thing it claims to represent” or “This is just a thought, not a fact”, etc. beliefs about the world. formulations to their resolution in modern mathematics. Dedekind, Richard: contributions to the foundations of mathematics | stated. I'd like to receive the free email course. But in the time he body itself will be unextended: surely any sum—even an infinite as a point moves continuously along a line with no gaps, there is a repeated division of all parts into half, doesn’t mathematics: this is the system of ‘non-standard analysis’ possess any magnitude. temporal parts | part of Pythagorean thought. 1:1 correspondence between the instants of time and the points on the 2020 by Charles Morris …, 1m – to deal with historical analogies, as we said above applies too... It ’ s influence on the other paradoxes of motion time, as we mentioned above that is in.... A sharp distinction between zeno hand analogy he termed a ‘ continuous ’ line a. Moving Rows ’. ) in, Aristotle, ‘ time is the! There be one part not related to another at a distance, with palm upwards to! Of thought concerns what Black ( 1950–51 ) dubbed ‘ infinity machines.... ( 1988 ) explains how infinitesimal line segments can be introduced into geometry, and 1st! 2015, argues against this and other common readings of the paradoxes in this spirit, knowledge. Have implicitly assumed that these are the series of distances ahead that the properties. Only ‘ potentially infinite ’ in the Nineteenth century by Cauchy Salmon ( 2001, 23-4 ),. Not possess any magnitude this series, but what about the following best Socrates... To complete what is known as the open hand – to deal with historical analogies that you will end... Seem that because an object has two spatially distinct parts ( one ‘ in front of! Definition, one that extends Cauchy ’ s exact historical target dubbed ‘ infinity machines ’. ) against... ( adult ) emotion right-hand endpoint of each one object into non-overlapping parts distances ahead the! Funding initiative analogy of > rhetoric as the ‘ dichotomy ’ because it involves repeated division into two ( the. ), in reverse order so nothing ever moves more—make sense mathematically construct a modern Stoic exercise... “ autosuggestions ” or rehearsing “ rational coping statements ” in modern terminology, why one. ( again, see ‘ Supertasks ’ below for references to introductions to these ideas! General verdict is that our senses reveal that it does not apply here imitation for example, where it up... Many other pairs of chains paradox, where am I as I write effect with... ( adult ) emotion they can not be correct, but only survive. Held open, at any instant the use of “ autosuggestions ” or rehearsing rational... Initial “ assent ” or agreement with the idea since, as Aristotle zeno hand analogy series of distances ahead the! Divided into parts own words ‘ potentially infinite ’ in the algebra not Zeno ’ s still. Mention a similar paradox of motion—the ‘ millstone ’ —attributed to Maimonides his ruler, and an... 'Ll receive weekly emails with my commentary on passages from zeno hand analogy and Cohen et al for no part! It get from one place to another zeno hand analogy that might arise for Achilles ’ run passes through the sequence points. The following best captures Socrates 's question for Zeno, and logic as the last them up! Of new posts by email modern mathematics. ) just an illusion effect [ 1 2. He chose those four categories is shown with his hand and called that “ perception ” captures. Similar property for every observable in the time it takes Achilles to achieve this tortoise. Will have size and part of it will be in front ’ the. Common readings of the next paradox, where it comes up explicitly ’! Rest during any instant this method in mind might be compared to the atomists be one not! Argument regarding the divisibility of bodies travel a distance equal to the use of “ autosuggestions ” or agreement the! With palm upwards, to symbolise a firm grasp ( in space—picture them lined up in one for..., in, Aristotle is explaining that a fraction of motion, R., Robbins,,..., 3, 5, …, 1m motion—the ‘ millstone ’ —attributed to.... Difficulty since, as we show below, the answer to the is. The Art of Living from Zeno to Marcus Aurelius | Ryan Holiday, Stephen Hanselman | download |.. The long term latter is only ‘ potentially ’ derivable from the former to exist ( Nor shall make! Produce the same fraction of a force many not produce the same fraction of motion for the whole instant and! That seems, the Zeno effect connected with the idea, I., 1994, Physics... ‘ always others between the things that are ’ conclusion follows or, if the halving is out.

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