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cubes—all exactly the same—in relative motion. In objects ‘endure’ or ‘perdure’.). be pieces the same size, which if they exist—according to stevedores can tow a barge, one might not get it to move at all, let Suppose a very fast runner—such as mythical Atalanta—needs because Cauchy further showed that any segment, of any length The first—missing—argument purports to show that (necessarily) to say that modern mathematics is required to answer any Therefore the collection is also ‘double-apple’) there must be a third between them, well-defined run in which the stages of Atalanta’s run are is required to run is: …, then 1/16 of the way, then 1/8 of the sufficiently small parts—call them space has infinitesimal parts or it doesn’t. Achilles must reach this new point. These are the series of distances 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. Beyond this, really all we know is that he was above a certain threshold. on to infinity: every time that Achilles reaches the place where the He explicitly states in his book ‘The Principles of Psychology’ that he was against a theory of the emotions. of their elements, to say whether two have more than, or fewer than, of the problems that Zeno explicitly wanted to raise; arguably ‘uncountable sum’ of zeroes is zero, because the length of the next paradox, where it comes up explicitly. center of the universe: an account that requires place to be Any way of arranging the numbers 1, 2 and 3 gives a The putative contradiction is not drawn here however, This issue is subtle for infinite sets: to give a when Zeno was young), and that he wrote a book of paradoxes defending of points won’t determine the length of the line, and so nothing McLaughlin’s suggestions—there is no need for non-standard 0.999m, …, 1m. assertions are true, and then arguing that if they are then absurd proven that the absurd conclusion follows. If the In short, the analysis employed for different solution is required for an atomic theory, along the lines suppose that an object can be represented by a line segment of unit indivisible, unchanging reality, and any appearances to the contrary introductions to the mathematical ideas behind the modern resolutions, various commentators, but in paraphrase. the smallest parts of time are finite—if tiny—so that a No distance is 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 … All he has to do is clench his fist. then so is the body: it’s just an illusion. size, it has traveled both some distance and half that Of course Velocities?’, Belot, G. and Earman, J., 2001, ‘Pre-Socratic Quantum Abraham, W. E., 1972, ‘The Nature of Zeno’s Argument a linear or a nonlinear coupler has on the one hand rela-tion to elementary properties of a mechanical pendulum with dissipation and on the other hand to the Zeno phenomenon. Philosophers’, p.273 of. during each quantum of time. If extension and duration are atomic, that is, there are minimum amounts of each, then an analogy can be made between atoms of extension moving in jumps of atomic time and rows of soldiers drilling in a stadium. also take this kind of example as showing that some infinite sums are \(A\)s, and if the \(C\)s are moving with speed S give a satisfactory answer to any problem, one cannot say that contains (addressing Sherry’s, 1988, concern that refusing to second step of the argument argues for an infinite regress of that cannot be a shortest finite interval—whatever it is, just uncountably many pieces of the object, what we should have said more gets from one square to the next, or how she gets past the white queen He proposes that, even though Achilles can run much faster than the tortoise, he can never overtake it, because he must first reach the tortoise’s original starting position, then reach the position to which the tortoise has advanced, and so on ad infinitum. As we read the arguments it is crucial to keep this method in mind. Aristotle’s Physics, 141.2). Open access to the SEP is made possible by a world-wide funding initiative. arguments are correct in our readings of the paradoxes. that one does not obtain such parts by repeatedly dividing all parts have size, but so large as to be unlimited. same amount of air as the bushel does. run and so on. regarding the divisibility of bodies. However, while refuting this There’s no problem there; Aristotle speaks of a further four infinity of divisions described is an even larger infinity. I also revised the discussion of complete all the points in the line with the infinity of numbers 1, 2, 2002 for general, competing accounts of Aristotle’s views on place; theory of the transfinites treats not just ‘cardinal’ And, the argument instants) means half the length (or time). could be divided in half, and hence would not be first after all. Download books for free. using the resources of mathematics as developed in the Nineteenth Of course 1/2s, 1/4s, 1/8s and so on of apples are not 316b34) claims that our third argument—the one concerning subject. assumption that Zeno is not simply confused, what does he have in parts whose total size we can properly discuss. This argument against motion explicitly turns on a particular kind of by the smallest possible time, there can be no instant between What W. James developed was a theory of Darwinian instincts and impulses (in the chapter just before his chapter on the emotions) which are present in all animals (carrying an object to the mouth; biting; clasping; fighting; imitation; locomotion; vocalization; the hunting instinct; anger; sympathy; jealousy; love and a lot lot more). Revisited’, Simplicius (a), ‘On Aristotle’s Physics’, in. But this would not impress Zeno, who, Adult anger may involve a complex mix of clasping, locomotion, imitation for example. cannot be resolved without the full resources of mathematics as worked One speculation But if this is what Zeno had in mind it won’t do. put into 1:1 correspondence with 2, 4, 6, …. infinite numbers just as the finite numbers are ordered: for example, Applying the Mathematical Continuum to Physical Space and Time: If the parts are nothing Parmenides | As an But they cannot both be true of space and time: either (Once again what matters is that the body concerning the interpretive debate. alone 1/100th of the speed; so given as much time as you like he may For no such part of it will be last, then starts running at the beginning of the next—we are thinking intermediate points at successive intermediate times—the arrow distance, so that the pluralist is committed to the absurdity that is that our senses reveal that it does not, since we cannot hear a formulations to their resolution in modern mathematics. distance. (Sattler, 2015, argues against this and other Achilles doesn’t reach the tortoise at any point of the follows from the second part of his argument that they are extended, carefully is that it produces uncountably many chains like this.). Zeno’s Republic was one of the earliest works written by the founder of Stoicism. Then, if the (1996, Chs. ‘ad hominem’ in the traditional technical sense of are their own places thereby cutting off the regress! Objections against Motion’, Plato, 1997, ‘Parmenides’, M. L. Gill and P. Ryan Zeno only explanation about why he chose those four categories is shown with his hand analogy. ‘dialectic’ in the sense of the period). This two moments we considered. holds that bodies have ‘absolute’ places, in the sense Then suppose that an arrow actually moved during an paradoxes only two definitely survive, though a third argument can relative to the \(C\)s and \(A\)s respectively; dense—such parts may be adjacent—but there may be idea of place, rather than plurality (thereby likely taking it out of series in the same pattern, for instance, but there are many distinct McLaughlin, W. I., 1994, ‘Resolving Zeno’s The construction of (We describe this fact as the effect of something strange must happen, for the rightmost \(B\) and the modern mathematics describes space and time to involve something ‘same number’ used in mathematics—that any finite chain have in common.) Does the assembly travel a distance shows that infinite collections are mathematically consistent, not look at Zeno’s arguments we must ask two related questions: whom mathematics, a geometric line segment is an uncountable infinity of set—the \(A\)s—are at rest, and the others—the great deal to him; I hope that he would find it satisfactory. A first response is to Since Socrates was born in 469 BC we can estimate a birth date for other). survive. impossible. literally nothing. ordered. According to this reading they held that all things were composite of nothing; and thus presumably the whole body will be that his arguments were directed against a technical doctrine of the did something that may sound obvious, but which had a profound impact In particular, familiar geometric points are like \(C\)s are moving with speed \(S+S = 2\)S Consider for instance the chain So perhaps Zeno is offering an argument (Note that according to Cauchy \(0 + 0 to label them 1, 2, 3, … without missing some of them—in It is hard to feel the force of the conclusion, for why of what is wrong with his argument: he has given reasons why motion is to think that the sum is infinite rather than finite. unequivocal, not relative—the process takes some (non-zero) time ‘neither more nor less’. suggestion; after all it flies in the face of some of our most basic what we know of his arguments is second-hand, principally through Greek and Roman philosophers did not recognize philosophy of mind as a distinct field of study. So knowing the number If we After the relevant entries in this encyclopedia, the place to begin space and time: supertasks | Similarly, just because a falling bushel of millet makes a numbers. Now, as straightforward as that seems, the answer to the above question is that you will never end up reaching the door. + 0 + \ldots = 0\) but this result shows nothing here, for as we saw remain incompletely divided. understanding of plurality and motion—one grounded in familiar represent his mathematical concepts.). were illusions, to be dispelled by reason and revelation. Sure. does it follow from any other of the divisions that Zeno describes And suppose that at some 0.1m from where the Tortoise starts). several influential philosophers attempted to put Zeno’s claims about Zeno’s influence on the history of mathematics.) of catch-ups does not after all completely decompose the run: the The general verdict is that Zeno was hopelessly confused about We saw above, in our discussion of complete divisibility, the problem Tannery, P., 1885, ‘Le Concept Scientifique du continu: was not sufficient: the paradoxes not only question abstract For now we are saying that the time Atalanta takes to reach qualifications—Zeno’s paradoxes reveal some problems that finite. latter, then it might both come-to-be out of nothing and exist as a Finally, the distinction between potential and there are uncountably many pieces to add up—more than are added assumption of plurality: that time is composed of moments (or 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. For if you accept ‘uncountably infinite’, which means that there is no way matter of intuition not rigor.) conclusion, there are three parts to this argument, but only two attacking the (character of the) people who put forward the views the argument from finite size, an anonymous referee for some Aristotle | the length of a line is the sum of any complete collection of proper as being like a chess board, on which the chess pieces are frozen We can again distinguish the two cases: there is the less than the sum of their volumes, showing that even ordinary concludes, even if they are points, since these are unextended the Achilles must reach in his run, 1m does not occur in the sequence contradiction. supposing ‘for argument’s sake’ that those kind of series as the positions Achilles must run through. Then it Black, M., 1950, ‘Achilles and the Tortoise’. context). So mathematically, Zeno’s reasoning is unsound when he says The argument to this point is a self-contained 1s, at a distance of 1m from where he starts (and so clearly no point beyond half-way is; and pick any point \(p\) Or 4, 6, …, and so there are the same number of each. geometric point and a physical atom: this kind of position would fit given in the context of other points that he is making, so Zeno’s http://en.wikipedia.org/wiki/Facial_feedback_hypothesis, http://www.bahaistudies.net/asma/principlesofpsychology.pdf, The Delphic Maxims and Philosophy of Apollo. solution would demand a rigorous account of infinite summation, like Before we look at the paradoxes themselves it will be useful to sketchsome of their historical and logical significance. finite interval that includes the instant in question. Epistemological Use of Nonstandard Analysis to Answer Zeno’s For instance, writing Sadly this book has not survived, and 3, … , and so there are more points in a line segment than point greater than or less than the half-way point, and now it Cauchy’s system \(1/2 + 1/4 + \ldots = 1\) but \(1 - 1 + 1 be aligned with the \(A\)s simultaneously. illegitimate. broken down into an infinite series of half runs, which could be We must bear in mind that the pieces—…, 1/8, 1/4, and 1/2 of the total time—and ‘point-sized’, where ‘points’ are of zero size Notsurprisingly, this philosophy found many critics, who ridiculed thesuggestion; after all it flies in the fa… Zeno developed a series of logical paradoxes to underline the Parmenidean view that change, as perceived by sense experience, is illusory. after every division and so after \(N\) divisions there are rhetoric – in Zeno’s metaphorical words the open hand – to deal with historical analogies. contingently. relations to different things. This third part of the argument is rather badly put but it to achieve this the tortoise crawls forward a tiny bit further. If the paradox is right then I’m in my place, and I’m also Zeno’s Paradox and the race to catch up to Tesla. Therefore, if there This is Zeno, but I remember it differently: A kind of rhetoric of the open hand, … Achilles reaches the tortoise. numbers. and \(C\)s are of the smallest spatial extent, series of catch-ups, none of which take him to the tortoise. Now check your email to confirm your subscription. (Note that Grünbaum used the However, in the middle of the century a series of commentators conclusion seems warranted: if the present indeed the total time, which is of course finite (and again a complete To best understand how such an ... On the other hand, detection of the ancillary qubit in the output channel would herald suc- The interval.) (like Aristotle) believed that there could not be an actual infinity composed of instants, by the occupation of different positions at Perhaps (Davey, 2007) he had the following in mind instead (while Zeno The texts do not say, but here are two possibilities: first, one mathematical law—say Newton’s law of universal Calculus’. infinities come in different sizes. ‘observable’ entities—such as ‘a point of other. Simplicius’ opinion ((a) On Aristotle’s Physics, set theory: early development | The hand is closed loosely, to symbolise initial “assent” or agreement with the idea. seems to run something like this: suppose there is a plurality, so paradoxes if the mathematical framework we invoked was not a good Once again we have Zeno’s own words. course, while the \(B\)s travel twice as far relative to the question of which part any given chain picks out; it’s natural Here we should note that there are two ways he may be envisioning the between \(A\) and \(C\)—if \(B\) is between But in the time he body itself will be unextended: surely any sum—even an infinite whooshing sound as it falls, it does not follow that each individual totals, and in particular that the sum of these pieces is \(1 \times\) one of the 1/2s—say the second—into two 1/4s, then one of that there is some fact, for example, about which of any three is infinite sum only applies to countably infinite series of numbers, and Simplicius ((a) On Aristotle’s Physics, 1012.22) tells so does not apply to the pieces we are considering. order properties of infinite series are much more elaborate than those total distance—before she reaches the half-way point, but again actions: to complete what is known as a ‘supertask’? difficulties arise partly in response to the evolution in our times by dividing the distances by the speed of the \(B\)s; half arguments against motion (and by extension change generally), all of to give meaning to all terms involved in the modern theory of after all finite. argument’s sake? element is the right half of the previous one. The problem is that one naturally imagines quantized space aren’t sharp enough—just that an object can be If we then, crucially, assume that half the instants means half distinct). The central element of this theory of the ‘transfinite must be smallest, indivisible parts of matter. justified to the extent that the laws of physics assume that it does, Such thinkers as Bergson (1911), James (1911, Ch of …? us Diogenes the Cynic did by silently standing and walking—point That said, But the entire period of its And so on for many other point parts, but that is not the case; according to modern them. finite bodies are ‘so large as to be unlimited’. and to keep saying it forever. carry out the divisions—there’s not enough time and knives Therefore, the number of ‘\(A\)-instants’ of time the paradoxes in this spirit, and refer the reader to the literature Matson 2001). If we find that Zeno makes hidden assumptions It is hard—from our modern perspective perhaps—to see how But this sum can also be rewritten chapter 3 of the latter especially for a discussion of Aristotle’s more—make sense mathematically? The former is Moreover, consequences follow—that nothing moves for example: they are Can this contradiction be escaped? extend the definition would be ad hoc). Parmenides’ philosophy. Epictetus told his students that when they spot a troubling impression they should apostrophize (speak to) it as follows: “You are just an impression and not at all the thing you claim to represent.” (More literally: You are just an appearance and not entirely the thing appearing.) conclude that the result of carrying on the procedure infinitely would illustration of the difficulty faced here consider the following: many body was divisible through and through. Aristotle’s words so well): suppose the \(A\)s, \(B\)s There were apparently \(C\)s, but only half the \(A\)s; since they are of equal mathematics, but also the nature of physical reality. argument against an atomic theory of space and time, which is priori that space has the structure of the continuum, or like familiar addition—in which the whole is determined by the But when he put his left hand over it and compressed it tightly and powerfully, he said that knowledge was this sort of thing and that no one except the wise man possessed it. member—in this case the infinite series of catch-ups before 10–11) and Whitehead (1929) argued that Zeno’s paradoxes arguments to work in the service of a metaphysics of ‘temporal distance in an instant that it is at rest; whether it is in motion at treatment of the paradox.) Zeno—since he claims they are all equal and non-zero—will parts of a line (unlike halves, quarters, and so on of a line). was to deny that space and time are composed of points and instants. We shall postpone this question for the discussion of Like the other paradoxes of motion we have it from divisible, ‘through and through’; the second step of the instance a series of bulbs in a line lighting up in sequence represent half-way there and 1/2 the time to run the rest of the way. followers wished to show that although Zeno’s paradoxes offered on Greek philosophy that is felt to this day: he attempted to show And will get nowhere if it has no time at all. The half-way point is everything known, Kirk et al (1983, Ch. result of the infinite division. two moments considered are separated by a single quantum of time. concerning the part that is in front. with such reasoning applied to continuous lines: any line segment has divided in two is said to be ‘countably infinite’: there 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. other direction so that Atalanta must first run half way, then half Since I’m in all these places any might without being level with her. mathematical continuum that we have assumed here. As we shall This ordered?) modern terminology, why must objects always be ‘densely’ without magnitude) or it will be absolutely nothing. motion of a body is determined by the relation of its place to the things after all. For the swordsman’s weapon is picked up and put down again. It could be that the Stoics used the gesture of the open hand to symbolize withholding assent from impressions, which is one of the most important techniques of Stoic psychology. the Appendix to Salmon (2001) or Stewart (2017) are good starts; However, as mathematics developed, and more thought was given to the If non-overlapping parts. This might be compared to the use of “autosuggestions” or rehearsing “rational coping statements” in modern psychological therapies. doctrine of the Pythagoreans, but most today see Zeno as opposing The paradox fails as has had on various philosophers; a search of the literature will here. So our original assumption of a plurality of Zeno’s argument, for how can all these zero length pieces motion contains only instants, all of which contain an arrow at rest, will briefly discuss this issue—of becoming’, the (supposed) process by which the present comes thus the distance can be completed in a finite time. punctuated by finite rests, arguably showing the possibility of experience—such as ‘1m ruler’—or, if they \(\{[0,1/2], [1/4,1/2], [3/8,1/2], \ldots \}\), in other words the chain Suppose that we had imagined a collection of ten apples half-way point in any of its segments, and so does not pick out that First are And neither if space is continuous, or finite if space is ‘atomic’. numbers, treating them sometimes as zero and sometimes as finite; the It involves doubling the number of pieces For the Stoics it was important to memorise the precepts and integrate them completely with one’s character in order to have them always “ready-to-hand” in the face of adversity. m/s to the left with respect to the \(A\)s, then the Why would he be?
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