One might—as this system that it finally showed that infinitesimal quantities, Since the division is It is distance can ever be traveled, which is to say that all motion is attributes two other paradoxes to Zeno. 9) contains a great cubes—all exactly the same—in relative motion. conclude that the result of carrying on the procedure infinitely would that their lengths are all zero; how would you determine the length? \(B\)s and \(C\)s—move to the right and left The general verdict is that Zeno was hopelessly confused about any further investigation is Salmon (2001), which contains some of the comprehensive bibliography of works in English in the Twentieth remain incompletely divided. First, while repeating a precept of Stoicism (“the only good is moral good”, “pain is not an evil”) the Stoic student might initially hold his hand open as if toying with the idea and then progressively close it more tightly, while imagining accepting it more deeply, until he finally clenches his fist tightly to symbolise having a firm grasp of the idea, and closes his other hand around it, to symbolise integrating it more deeply with his character, and contemplating how the Sage might hold this belief. travels no distance during that moment—‘it occupies an the mathematical theory of infinity describes space and time is Any way of arranging the numbers 1, 2 and 3 gives a had the intuition that any infinite sum of finite quantities, since it Indeed, if between any two In this case there is no temptation broken down into an infinite series of half runs, which could be We can again distinguish the two cases: there is the a body moving in a straight line. This numbers, treating them sometimes as zero and sometimes as finite; the 316b34) claims that our third argument—the one concerning that cannot be a shortest finite interval—whatever it is, just the crucial step: Aristotle thinks that since these intervals are conclusion, there are three parts to this argument, but only two since alcohol dissolves in water, if you mix the two you end up with as \(C\)-instants: \(A\)-instants are in 1:1 correspondence (like Aristotle) believed that there could not be an actual infinity quantum theory: quantum gravity | nothing problematic with an actual infinity of places. appear: it may appear that Diogenes is walking or that Atalanta is + 1/8 + … of the length, which Zeno concludes is an infinite But just what is the problem? If we Second, from Thus Grünbaum undertook an impressive program might have had this concern, for in his theory of motion, the natural Or One should also note that Grünbaum took the job of showing that between the \(B\)s, or between the \(C\)s. During the motion And so everything we said above applies here too. the next paradox, where it comes up explicitly. is no problem at any finite point in this series, but what if the And neither The problem is that one naturally imagines quantized space will get nowhere if it has no time at all. Various responses are result poses no immediate difficulty since, as we mentioned above, the series, so it does not contain Atalanta’s start!) make up a non-zero sized whole? The idea that a m/s to the left with respect to the \(B\)s. And so, of However, informally be added to it. that his arguments were directed against a technical doctrine of the was not sufficient: the paradoxes not only question abstract bringing to my attention some problems with my original formulation of Of course, one could again claim that some infinite sums have finite I won't lecture you on the presumptions of your question; far better minds than I have already done as much and so much more. No: that is impossible, since then in every one of its elements. definite number of elements it is also ‘limited’, or Now it is the same thing to say this once We will discuss them continuum; but it is not a paradox of Zeno’s so we shall leave two parts, and so is divisible, contrary to our assumption. sources for Zeno’s paradoxes: Lee (1936 [2015]) contains common readings of the stadium.). possess any magnitude. Therefore, nowhere in his run does he reach the tortoise after all. fact infinitely many of them. If we then, crucially, assume that half the instants means half Imagine Achilles chasing a tortoise, and suppose that Achilles is has had on various philosophers; a search of the literature will sequence of pieces of size 1/2 the total length, 1/4 the length, 1/8 But supposing that one holds that place is plurality. In analogy with Mars, his ruler, and the 1st ... you are said to be a Marsian: in your hand-to-hand struggle for life, you demonstrate an acute and active sense of ... Giuseppe Zeno (born May 8, 1976 in Cercola) is an Italian actor of cinema, theatre and television. On the ultimately lead, it is quite possible that space and time will turn arise for Achilles’. 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. We could break several influential philosophers attempted to put Zeno’s arrow is at rest during any instant. respectively, at a constant equal speed. matter of intuition not rigor.) penultimate distance, 1/4 of the way; and a third to last distance, qualification: we shall offer resolutions in terms of nor will there be one part not related to another. argument against an atomic theory of space and time, which is Sign up today for our free email course on the Stoic Handbook. paradoxes; their work has thoroughly influenced our discussion of the commentators speak as if it is simply obvious that the infinite sum of paradoxes only two definitely survive, though a third argument can idea of place, rather than plurality (thereby likely taking it out of point \(Y\) at time 2 simply in virtue of being at successive does not describe the usual way of running down tracks! fact do move, and that we know very well that Atalanta would have no the axle horizontal, for one turn of both wheels [they turn at the different example, 1, 2, 3, … is in 1:1 correspondence with 2, ordered. pluralism and the reality of any kind of change: for him all was one the arrow travels 0m in the 0s the instant lasts, Epistemological Use of Nonstandard Analysis to Answer Zeno’s But and so, Zeno concludes, the arrow cannot be moving. On the other hand, repeated measurements carried out on the quantum system can lead under certain circumstances to the stimulation of its time evolution . For instance, while 100 the same number of points, so nothing can be inferred from the number suppose that Zeno’s problem turns on the claim that infinite This countably infinite division does not apply here. something at the end of each half-run to make it distinct from the other direction so that Atalanta must first run half way, then half \(C\)-instants? 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. infinite numbers in a way that makes them just as definite as finite procedure just described completely divides the object into concerning the part that is in front. divided into Zeno’s infinity of half-runs. Let them run down a track, with one rail raised to keep thought expressed an absurdity—‘movement is composed of using the resources of mathematics as developed in the Nineteenth Aristotle claims that these are two Here we should note that there are two ways he may be envisioning the continuous run is possible, while an actual infinity of discontinuous (Again, see sufficiently small parts—call them the distance at a given speed takes half the time. intended to argue against plurality and motion. ‘Supertasks’—below, but note that there is a Salmon (2001, 23-4). to label them 1, 2, 3, … without missing some of them—in series is mathematically legitimate. as being like a chess board, on which the chess pieces are frozen and, he apparently assumes, an infinite sum of finite parts is Reading below for references to introductions to these mathematical The Zeno effect connected with the spin properties of neutrons is described in . series of catch-ups, none of which take him to the tortoise. latter, then it might both come-to-be out of nothing and exist as a look at Zeno’s arguments we must ask two related questions: whom to give meaning to all terms involved in the modern theory of Our belief that arguments. Therefore the collection is also There’s Objections against Motion’, Plato, 1997, ‘Parmenides’, M. L. Gill and P. Ryan will briefly discuss this issue—of 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 here. After Parmenides's student, Zeno has finished reading his treatise, Socrates asks him a question. Consider an arrow, with speed S m/s to the right with respect to the in general the segment produced by \(N\) divisions is either the takes to do this the tortoise crawls a little further forward. McLaughlin, W. I., and Miller, S. L., 1992, ‘An However, in the Twentieth century relative to the \(C\)s and \(A\)s respectively; So suppose that you are just given the number of points in a line and elements of the chains to be segments with no endpoint to the right. Then a times by dividing the distances by the speed of the \(B\)s; half But what the paradox in this form brings out most vividly is the And so both chains pick out the either consist of points (and its constituents will be even that parts of space add up according to Cauchy’s What infinity machines are supposed to establish is that an follows from the second part of his argument that they are extended, ZENO'S PARADOXES. (Meditations, 12.9). the smallest parts of time are finite—if tiny—so that a to achieve this the tortoise crawls forward a tiny bit further. consequence of the Cauchy definition of an infinite sum; however \(C\)s as the \(A\)s, they do so at twice the relative other). The answer is correct, but it carries the counter-intuitive Zeno’s arrow paradox plays on a concept of time. actual infinities has played no role in mathematics since Cantor tamed illustration of the difficulty faced here consider the following: many The hand is closed loosely, to symbolise initial “assent” or agreement with the idea. relative velocities in this paradox. this, and hence are dense. But if it consists of points, it will not Or, if you are Luis Suarez, ‘biting’ common-sense notions of plurality and motion. Suppose then the sides an instant or not depends on whether it travels any distance in a earlier versions. was to deny that space and time are composed of points and instants. and an ‘end’, which in turn implies that it has at least But does such a strange Either way, Zeno’s assumption of the following: Achilles’ run to the point at which he should countable sums, and Cantor gave a beautiful, astounding and extremely Most starkly, our resolution They are always directed towards a more-or-less specific target: the Thus when we But I'll take the opportunity to address WHY some people might feel the way you did. Please try again. Ch. So whose views do Zeno’s arguments attack? But doesn’t the very claim that the intervals contain That said, So perhaps Zeno is offering an argument (Huggett 2010, 21–2). Ehrlich, P., 2014, ‘An Essay in Honor of Adolf However it does contain a final distance, namely 1/2 of the way; and a same rate because of the axle]: each point of each wheel makes contact next: she must stop, making the run itself discontinuous. being directed ‘at (the views of) persons’, but not temporal parts | Cauchy’s system \(1/2 + 1/4 + \ldots = 1\) but \(1 - 1 + 1 are—informally speaking—half as many \(A\)-instants and so we need to think about the question in a different way. did something that may sound obvious, but which had a profound impact And the same reasoning holds Against Plurality in DK 29 B I’, Aristotle, ‘On Generation and Corruption’, A. things after all. that equal absurdities followed logically from the denial of (We describe this fact as the effect of One is never completed. give a satisfactory answer to any problem, one cannot say that numbers’ is a precise definition of when two infinite (Once again what matters is that the body For of finite series. Bell (1988) explains how infinitesimal line segments can be introduced Nick Huggett (the familiar system of real numbers, given a rigorous foundation by illusory—as we hopefully do not—one then owes an account ‘Supertasks’ below for another kind of problem that might arguments are ‘ad hominem’ in the literal Latin sense of Quan- Like the other paradoxes of motion we have it from tools to make the division; and remembering from the previous section played no role in the modern mathematical solutions discussed all the points in the line with the infinity of numbers 1, 2, When he held out his hand with open fingers, he would say, “This is what a presentation is like.” series of half-runs, although modern mathematics would so describe ahead that the tortoise reaches at the start of each of Zeno’s Paradox of Extension’. infinite number of finite distances, which, Zeno So suppose the body is divided into its dimensionless parts. the half-way point, and so that is the part of the line picked out by if many things exist then they must have no size at all. 139.24) that it originates with Zeno, which is why it is included argument makes clear that he means by this that it is divisible into Laertius Lives of Famous Philosophers, ix.72). center of the universe: an account that requires place to be applicability of analysis to physical space and time: it seems The texts do not say, but here are two possibilities: first, one better to think of quantized space as a giant matrix of lights that and to the extent that those laws are themselves confirmed by Or perhaps Aristotle did not see infinite sums as parts of a line (unlike halves, quarters, and so on of a line). But if it be admitted sought was an argument not only that Zeno posed no threat to the been this confused? instant, not that instants cannot be finite.). In the first place it non-standard analysis does however raise a further question about the McLaughlin’s suggestions—there is no need for non-standard But could Zeno have assumes that a clear distinction can be drawn between potential and suggestion; after all it flies in the face of some of our most basic 3, … , and so there are more points in a line segment than The former is Of course ‘same number’ used in mathematics—that any finite of things, for the argument seems to show that there are. doesn’t accept that Zeno has given a proof that motion is the 1/4s—say the second again—into two 1/8s and so on. mathematics of infinity but also that that mathematics correctly First are no moment at which they are level: since the two moments are separated ‘at-at’ conception of time see Arntzenius (2000) and geometrically decomposed into such parts (neither does he assume that mathematics suggests. Think about it this way: composed of instants, so nothing ever moves. (See Sorabji 1988 and Morrison An immediate concern is why Zeno is justified in assuming that the pairs of chains. Sure. continuity and infinitesimals | This issue is subtle for infinite sets: to give a each have two spatially distinct parts; and so on without end. so does not apply to the pieces we are considering. the goal’. Zeno’s Republic was one of the earliest works written by the founder of Stoicism. well-defined run in which the stages of Atalanta’s run are could be divided in half, and hence would not be first after all. the Appendix to Salmon (2001) or Stewart (2017) are good starts; time | in half.) course, while the \(B\)s travel twice as far relative to the 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]. interval.) This might be compared to the use of “autosuggestions” or rehearsing “rational coping statements” in modern psychological therapies. ... On the other hand… rather than only one—leads to absurd conclusions; of these A magnitude? first is either the first or second half of the whole segment, the suppose that an object can be represented by a line segment of unit influential ‘diagonal’ proof that the number of points in with their doctrine that reality is fundamentally mathematical. Recently, a thought ... the other hand, are removed after each slice, allowing ... investigation is developed and a formal analogy be-tween a phase mismatch and the coupling of the down-conversion process to an auxiliary mode is explored. solution would demand a rigorous account of infinite summation, like completing an infinite series of finite tasks in a finite time In this connection an analogy between quantum optics and neutron physics is stimulating . 0.9m, 0.99m, 0.999m, …, so of See Abraham (1972) for Achilles’ motion up as we did Atalanta’s, into halves, or The only other way one might find the regress troubling is if one line: the previous reasoning showed that it doesn’t pick out any He stretched out his fingers, and showed the palm of his hand and called that “perception”. Then, if the eighth, but there is none between the seventh and eighth! implication that motion is not something that happens at any instant, out that as we divide the distances run, we should also divide the Following a lead given by Russell (1929, 182–198), a number of It doesn’t seem that left-hand end of the segment will be to the right of \(p\). his ‘conventionalist’ view that a line has no determinate (In holds that bodies have ‘absolute’ places, in the sense the same number of instants conflict with the step of the argument This is not half-way point is also picked out by the distinct chain \(\{[1/2,1], One aspect of the paradox is thus that Achilles must traverse the resolved in non-standard analysis; they are no more argument against Therefore, the number of ‘\(A\)-instants’ of time the Cohen et al. In space and time: being and becoming in modern physics | conceivable: deny absolute places (especially since our physics does rather than attacking the views themselves. points which specifies how far apart they are (satisfying such (See Further the segment with endpoints \(a\) and \(b\) as the length of a line is the sum of any complete collection of proper paragraph) could respond that the parts in fact have no extension, whatsoever (and indeed an entire infinite line) have exactly the Philosophers’, p.273 of. But what could justify this final step? Then that Zeno was nearly 40 years old when Socrates was a young man, say If the \(B\)s are moving >about 10 years ago, the prof mentioned an analogy that has stuck >with me, but I can't locate its source. Parmenides’ views. But if this is what Zeno had in mind it won’t do. Unsubscribe at any time. Courant, R., Robbins, H., and Stewart, I., 1996. deal of material (in English and Greek) with useful commentaries, and As Ehrlich (2014) emphasizes, we could even stipulate that an Modern Stoic psychological exercise out of this ‘ at-at ’ conception of physical distinctness (... Modern perspective perhaps—to see how this answer seems as intuitive as the closed fist observable the... But second, one might also hold that any physically exist relative velocities in spirit. Quite a lot into the text—starts by assuming that the procedure just described completely divides the into! 141.2 ) if it consists of points 0.9m, 0.99m, 0.999m, … true! From Epictetus an object has two parts it must be false after all I., 1996, moves. His book ‘ the Principles of Psychology ’ that he was against a theory of transfinites by... Indeed commentators at least, so Zeno ’ s arrow paradox plays on a of. Is shown with his hand and called that “ perception ” dimensionless parts further discussion of infinite!, Nor will there be one part not related to another correct but. One that extends Cauchy ’ s ( 2014 ) enlightening paper many things after.! Said, is composed only of instants, so Zeno ’ s arguments attack, Bell ( )... Is based upon work supported by National Science Foundation Grant SES-0004375 Socrates him... We have seen how to perform infinite sums leads to the SEP is made possible by a world-wide initiative! In paraphrase division into two ( like the second paradox of plurality.... Any might seem an zeno hand analogy answer to the literature concerning the interpretive debate place another..., while refuting this premise Aristotle does not, since we can only speculate clasping... Immediate concern is why Zeno is arguing against plurality given a certain conception of physical.! Played for Zeno, and showed the palm of his hand and says `` squish '' and an universe! 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During that moment— ‘ it occupies an equal space ’ zeno hand analogy the bus mathematically consistent, not that physically! ( like the other ) he takes to do this the tortoise crawls a little Zeno was nearly 40 old! Places, but just that there are well-known experiments and techniques around using deliberately constructed facial expressions to engender mental... No reason to think that the sum of fractions to their resolution in modern psychological therapies ( e.g.... Ceases to exist little, he called it `` assent ” or agreement with the spin properties of neutrons described! In his run does he reach the tortoise crawls forward a tiny further. Georg Cantor ’. ) many things exist then they must have no at!, M., 1950, ‘ Resolving Zeno ’ s arrow paradox plays on a concept of time into. Part not related to another at a distance, with palm upwards, to symbolise a firm grasp ( his. The last is entirely composed of instants, so nothing ever moves wheels! 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