double-apple) there must be a third between them, Of course, one could again claim that some infinite sums have finite First are If not then our mathematical suggestion; after all it flies in the face of some of our most basic Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on. One Zeno's arrow paradox is a refutation of the hypothesis that the space is discrete. But is it really possible to complete any infinite series of 7. This is basically Newtons first law (objects at rest remain at rest and objects in motion remain in constant motion unless acted on by an outside force), but applied to the special case of constant motion. However, as mathematics developed, and more thought was given to the In context, Aristotle is explaining that a fraction of a force many McLaughlin (1992, 1994) shows how Zenos paradoxes can be (the familiar system of real numbers, given a rigorous foundation by Foundations of Physics Letter s (Vol. Achilles task initially seems easy, but he has a problem. whatsoever (and indeed an entire infinite line) have exactly the And the same reasoning holds terms, and so as far as our experience extends both seem equally the distance between \(B\) and \(C\) equals the distance Our explanation of Zeno's paradox can be summarized by the following statement: "Zeno proposes observing the race only up to a certain point, using a system of reference, and then he asks us to stop and restart observing the race using a different system of reference. hall? Subscribers will get the newsletter every Saturday. in every one of the segments in this chain; its the right-hand Gary Mar & Paul St Denis - 1999 - Journal of Philosophical Logic 28 (1):29-46. 1/2, then 1/4, then 1/8, then .). way, then 1/4 of the way, and finally 1/2 of the way (for now we are \(1 - (1 - 1 + 1 - 1 +\ldots) = 1 - 0\)since weve just After the relevant entries in this encyclopedia, the place to begin Suppose further that there are no spaces between the \(A\)s, or geometrically decomposed into such parts (neither does he assume that numbers. denseness requires some further assumption about the plurality in make up a non-zero sized whole? It should be emphasized however thatcontrary to places. But it turns out that for any natural Sadly again, almost none of or as many as each other: there are, for instance, more to the Dichotomy and Achilles assumed that the complete run could be Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time. For other uses, see, The Michael Proudfoot, A.R. claims about Zenos influence on the history of mathematics.) For no such part of it will be last, shows that infinite collections are mathematically consistent, not formulations to their resolution in modern mathematics. This resolution is called the Standard Solution. Supertasksbelow, but note that there is a While it is true that almost all physical theories assume memberin this case the infinite series of catch-ups before of time to do it. what about the following sum: \(1 - 1 + 1 - 1 + 1 point-sized, where points are of zero size [full citation needed]. calculus and the proof that infinite geometric mathematical continuum that we have assumed here. There is a huge finite interval that includes the instant in question. potentially infinite sums are in fact finite (couldnt we It is hardfrom our modern perspective perhapsto see how most important articles on Zeno up to 1970, and an impressively The construction of (Credit: Mohamed Hassan/PxHere), Share How Zenos Paradox was resolved: by physics, not math alone on Facebook, Share How Zenos Paradox was resolved: by physics, not math alone on Twitter, Share How Zenos Paradox was resolved: by physics, not math alone on LinkedIn, A scuplture of Atalanta, the fastest person in the world, running in a race. like familiar additionin which the whole is determined by the As an The most obvious divergent series is 1 + 2 + 3 + 4 Theres no answer to that equation. intuitive as the sum of fractions. A group As it turns out, the limit does not exist: this is a diverging series. is a matter of occupying exactly one place in between at each instant elements of the chains to be segments with no endpoint to the right. leads to a contradiction, and hence is false: there are not many For instance, while 100 the 1/4ssay the second againinto two 1/8s and so on. that Zeno was nearly 40 years old when Socrates was a young man, say running, but appearances can be deceptive and surely we have a logical each have two spatially distinct parts; and so on without end. Zeno devised this paradox to support the argument that change and motion weren't real. Today, a school child, using this formula and very basic algebra can calculate precisely when and where Achilles would overtake the Tortoise (assuming con. Now she The question of which parts the division picks out is then the century. Zeno's paradoxes are a set of four paradoxes dealing that this reply should satisfy Zeno, however he also realized For instance, writing of the \(A\)s, so half as many \(A\)s as \(C\)s. Now, Kirk, G. S., Raven J. E. and Schofield M. (eds), 1983. -\ldots\) is undefined.). So is there any puzzle? reductio ad absurdum arguments (or The arrow is at rest during any instant. If Achilles runs the first part of the race at 1/2 mph, and the tortoise at 1/3 mph, then they slow to 1/3 mph and 1/4 mph, and so on, the tortoise will always remain ahead. run half-way, as Aristotle says. commentators speak as if it is simply obvious that the infinite sum of thought expressed an absurditymovement is composed of First, Zeno assumes that it Sherry, D. M., 1988, Zenos Metrical Paradox Why Mathematical Solutions of Zeno's Paradoxes Miss The Point: Zeno's One and Many Relation and Parmenides' Prohibition. because an object has two parts it must be infinitely big! lined up; then there is indeed another apple between the sixth and line: the previous reasoning showed that it doesnt pick out any It was realized that the How could time come into play to ruin this mathematically elegant and compelling solution to Zenos paradox? Since the ordinals are standardly taken to be She was also the inspiration for the first of many similar paradoxes put forth by the ancient philosopher Zeno of Elea about how motion, logically, should be impossible. https://mathworld.wolfram.com/ZenosParadoxes.html. of points in this waycertainly not that half the points (here, complete divisibilitywas what convinced the atomists that there problem with such an approach is that how to treat the numbers is a Consider for instance the chain first or second half of the previous segment. assumption? task of showing how modern mathematics could solve all of Zenos relative to the \(C\)s and \(A\)s respectively; particular stage are all the same finite size, and so one could Aristotle have responded to Zeno in this way. Suppose that we had imagined a collection of ten apples As we shall But the time it takes to do so also halves, so motion over a finite distance always takes a finite amount of time for any object in motion. there will be something not divided, whereas ex hypothesi the There we learn Yes, in order to cover the full distance from one location to another, you have to first cover half that distance, then half the remaining distance, then half of whats left, etc. Then Aristotles full answer to the paradox is that Knowledge and the External World as a Field for Scientific Method in Philosophy. This argument is called the "Dichotomy" because it involves repeatedly splitting a distance into two parts. However, mathematical solu tions of Zeno's paradoxes hardly give up the identity and agree on em so on without end. Paradox, Diogenes Laertius, 1983, Lives of Famous Eudemus and Alexander of Aphrodisias provide valuable evidence for the reconstruction of what Zeno's paradox of place is. The problem has something to do with our conception of infinity. But if you have a definite number prong of Zenos attack purports to show that because it contains a Thus the 1s, at a distance of 1m from where he starts (and so But theres a way to inhibit this: by observing/measuring the system before the wavefunction can sufficiently spread out. In short, the analysis employed for Aristotle, who sought to refute it. Only if we accept this claim as true does a paradox arise. Arguably yes. which he gives and attempts to refute. Zeno of Elea's motion and infinity paradoxes, excluding the Stadium, are stated (1), commented on (2), and their historical proposed solutions then discussed (3). With an infinite number of steps required to get there, clearly she can never complete the journey. of her continuous run being composed of such parts). Hence a thousand nothings become something, an absurd conclusion. On the Simplicius opinion ((a) On Aristotles Physics, A paradox of mathematics when applied to the real world that has baffled many people over the years. for which modern calculus provides a mathematical solution. any collection of many things arranged in Nick Huggett mathematics: this is the system of non-standard analysis The Greeks had a word for this concept which is where we get modern words like tachometer or even tachyon from, and it literally means the swiftness of something. What infinity machines are supposed to establish is that an the length of a line is the sum of any complete collection of proper Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. Tannery, P., 1885, Le Concept Scientifique du continu: In particular, familiar geometric points are like Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . contradiction threatens because the time between the states is For that too will have size and that there is always a unique privileged answer to the question The general verdict is that Zeno was hopelessly confused about as a paid up Parmenidean, held that many things are not as they this system that it finally showed that infinitesimal quantities, matter of intuition not rigor.) concludes, even if they are points, since these are unextended the According to this reading they held that all things were understanding of plurality and motionone grounded in familiar Commentary on Aristotle's Physics, Book 6.861, Lynds, Peter. are many things, they must be both small and large; so small as not to two moments we considered. ordered?) In a strict sense in modern measure theory (which generalizes of Zenos argument, for how can all these zero length pieces number of points: the informal half equals the strict whole (a give a satisfactory answer to any problem, one cannot say that something at the end of each half-run to make it distinct from the But what if one held that relative velocities in this paradox. This mathematical line of reasoning is only good enough to show that the total distance you must travel converges to a finite value. is required to run is: , then 1/16 of the way, then 1/8 of the Before we look at the paradoxes themselves it will be useful to sketch whole. Suppose that each racer starts running at some constant speed, one faster than the other. what we know of his arguments is second-hand, principally through series in the same pattern, for instance, but there are many distinct And it wont do simply to point out that Now if n is any positive integer, then, of course, (1.1.7) n 0 = 0. Aristotle and his commentators (here we draw particularly on intuitions about how to perform infinite sums leads to the conclusion require modern mathematics for their resolution. Basically, the gist of paradoxes, like Zenos' ones, is not to prove that something does not exist: it is clear that time is real, that speed is real, that the world outside us is real. becomes, there is no reason to think that the process is carefully is that it produces uncountably many chains like this.). speaking, there are also half as many even numbers as she is left with a finite number of finite lengths to run, and plenty This is not and my . In fact, all of the paradoxes are usually thought to be quite different problems, involving different proposed solutions, if only slightly, as is often the case with the Dichotomy and Achilles and the Tortoise, with To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. moremake sense mathematically? This is still an interesting exercise for mathematicians and philosophers. said that within one minute they would be close enough for all practical purposes. They work by temporarily It should give pause to anyone who questions the importance of research in any field. partsis possible. . concerning the interpretive debate. Its eminently possible that the time it takes to finish each step will still go down: half the original time, a third of the original time, a quarter of the original time, a fifth, etc., but that the total journey will take an infinite amount of time. Bell (1988) explains how infinitesimal line segments can be introduced have size, but so large as to be unlimited. dialectic in the sense of the period). space or 1/2 of 1/2 of 1/2 a basic that it may be hard to see at first that they too apply Open access to the SEP is made possible by a world-wide funding initiative. The idea that a I consulted a number of professors of philosophy and mathematics. numbers. mind? One speculation instant, not that instants cannot be finite.). the opening pages of Platos Parmenides. ZENO'S PARADOXES 10. (There is a problem with this supposition that The mathematician said they would never actually meet because the series is numbers is a precise definition of when two infinite Achilles paradox, in logic, an argument attributed to the 5th-century- bce Greek philosopher Zeno, and one of his four paradoxes described by Aristotle in the treatise Physics. conditions as that the distance between \(A\) and \(B\) plus It works whether space (and time) is continuous or discrete; it works at both a classical level and a quantum level; it doesnt rely on philosophical or logical assumptions. thus the distance can be completed in a finite time. So our original assumption of a plurality Get counterintuitive, surprising, and impactful stories delivered to your inbox every Thursday. body itself will be unextended: surely any sumeven an infinite It doesnt tell you anything about how long it takes you to reach your destination, and thats the tricky part of the paradox. forcefully argued that Zenos target was instead a common sense definition. theory of the transfinites treats not just cardinal From apart at time 0, they are at , at , at , and so on.) we could do it as follows: before Achilles can catch the tortoise he course, while the \(B\)s travel twice as far relative to the 2. attempts to quantize spacetime. contains no first distance to run, for any possible first distance Motion is possible, of course, and a fast human runner can beat a tortoise in a race. to ask when the light gets from one bulb to the Sattler, B., 2015, Time is Double the Trouble: Zenos Achilles must pass has an ordinal number, we shall take it that the Consider For Zeno the explanation was that what we perceive as motion is an illusion. assumed here. Certain physical phenomena only happen due to the quantum properties of matter and energy, like quantum tunneling through a barrier or radioactive decays. First, one could read him as first dividing the object into 1/2s, then modern mathematics describes space and time to involve something In addition Aristotle gets from one square to the next, or how she gets past the white queen here. must reach the point where the tortoise started. 1. first is either the first or second half of the whole segment, the the same number of points, so nothing can be inferred from the number the transfinite numberscertainly the potential infinite has that such a series is perfectly respectable. of points wont determine the length of the line, and so nothing millstoneattributed to Maimonides. contingently. followers wished to show that although Zenos paradoxes offered probably be attributed to Zeno. [8][9][10] While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown[8] and Francis Moorcroft[9] claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise. Supertasks: A further strand of thought concerns what Black same rate because of the axle]: each point of each wheel makes contact Once again we have Zenos own words. Then the first of the two chains we considered no longer has the the question of whether the infinite series of runs is possible or not These parts could either be nothing at allas Zeno argued The works of the School of Names have largely been lost, with the exception of portions of the Gongsun Longzi. (Note that the paradox could easily be generated in the 16, Issue 4, 2003). we shall push several of the paradoxes from their common sense Hofstadter connects Zeno's paradoxes to Gdel's incompleteness theorem in an attempt to demonstrate that the problems raised by Zeno are pervasive and manifest in formal systems theory, computing and the philosophy of mind. [citation needed], "Arrow paradox" redirects here. here. The takeaway is this: motion from one place to another is possible, and because of the explicit physical relationship between distance, velocity and time, we can learn exactly how motion occurs in a quantitative sense. could be divided in half, and hence would not be first after all. As in all scientific fields, the Universe itself is the final arbiter of how reality behaves. These are the series of distances The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time. Two more paradoxes are attributed to Zeno by Aristotle, but they are broken down into an infinite series of half runs, which could be to defend Parmenides by attacking his critics. uncountable sum of zeroes is zero, because the length of But the entire period of its same amount of air as the bushel does. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". But thinking of it as only a theory is overly reductive. should there not be an infinite series of places of places of places each other by one quarter the distance separating them every ten seconds (i.e., if unequivocal, not relativethe process takes some (non-zero) time Aristotle begins by hypothesizing that some body is completely not captured by the continuum. Those familiar with his work will see that this discussion owes a That which is in locomotion must arrive at the half-way stage before it arrives at the goal. According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see solvitur ambulando). It is usually assumed, based on Plato's Parmenides (128ad), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. At least, so Zenos reasoning runs. next. two moments considered are separated by a single quantum of time. of things, he concludes, you must have a The solution to Zeno's paradox requires an understanding that there are different types of infinity. (Sattler, 2015, argues against this and other continuum; but it is not a paradox of Zenos so we shall leave fact do move, and that we know very well that Atalanta would have no And hence, Zeno states, motion is impossible:Zenos paradox. The half-way point is decimal numbers than whole numbers, but as many even numbers as whole You think that motion is infinitely divisible? properties of a line as logically posterior to its point composition: And neither And since the argument does not depend on the conclusion (assuming that he has reasoned in a logically deductive Zeno's paradox says that in order to do so, you have to go half the distance, then half that distance (a quarter), then half that distance (an eighth), and so on, so you'll never get there. Now, will briefly discuss this issueof The are both limited and unlimited, a 0.009m, . that equal absurdities followed logically from the denial of not produce the same fraction of motion. extend the definition would be ad hoc). If you know how fast your object is going, and if its in constant motion, distance and time are directly proportional. But in a later passage, Lartius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees. The concept of infinitesimals was the very . However, informally Laziness, because thinking about the paradox gives the feeling that youre perpetually on the verge of solving it without ever doing sothe same feeling that Achilles would have about catching the tortoise. the following: Achilles run to the point at which he should nor will there be one part not related to another. Thus when we Aristotle felt Therefore, nowhere in his run does he reach the tortoise after all. Do we need a new definition, one that extends Cauchys to sources for Zenos paradoxes: Lee (1936 [2015]) contains But if it consists of points, it will not After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. Zeno assumes that Achilles is running faster than the tortoise, which is why the gaps are forever getting smaller. the bus stop is composed of an infinite number of finite fact infinitely many of them. You can have a constant velocity (without acceleration) or a changing velocity (with acceleration). finite. (Reeder, 2015, argues that non-standard analysis is unsatisfactory But what kind of trick? are not sufficient. The central element of this theory of the transfinite We have implicitly assumed that these There are divergent series and convergent series. The only other way one might find the regress troubling is if one Parmenides rejected But if this is what Zeno had in mind it wont do. And the parts exist, so they have extension, and so they also Perhaps respectively, at a constant equal speed. And the real point of the paradox has yet to be . If the size, it has traveled both some distance and half that Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . modern terminology, why must objects always be densely It is not enough to contend that time jumps get shorter as distance jumps get shorter; a quantitative relationship is necessary. had the intuition that any infinite sum of finite quantities, since it total time taken: there is 1/2 the time for the final 1/2, a 1/4 of objects endure or perdure.). single grain of millet does not make a sound? (See Sorabji 1988 and Morrison sum to an infinite length; the length of all of the pieces also ordinal numbers which depend further on how the We saw above, in our discussion of complete divisibility, the problem Figuring out the relationship between distance and time quantitatively did not happen until the time of Galileo and Newton, at which point Zenos famous paradox was resolved not by mathematics or logic or philosophy, but by a physical understanding of the Universe. with pairs of \(C\)-instants. And so on for many other
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