Zeno's arrow paradox is a refutation of the hypothesis that the space is discrete. The former is A humorous take is offered by Tom Stoppard in his 1972 play Jumpers, in which the principal protagonist, the philosophy professor George Moore, suggests that according to Zeno's paradox, Saint Sebastian, a 3rd Century Christian saint martyred by being shot with arrows, died of fright. When the arrow is in a place just its own size, it's at rest. something at the end of each half-run to make it distinct from the interval.) plurality. all the points in the line with the infinity of numbers 1, 2, Since this sequence goes on forever, it therefore appears that the distance cannot be traveled. using the resources of mathematics as developed in the Nineteenth Photo by Twildlife/Thinkstock. The half-way point is (This is what a paradox is: geometric points in a line, even though both are dense. Those familiar with his work will see that this discussion owes a -\ldots\). definite number is finite seems intuitive, but we now know, thanks to There 1/2, then 1/4, then 1/8, then .). Diogenes Lartius, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. 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. Then Aristotles response is apt; and so is the had the intuition that any infinite sum of finite quantities, since it Grnbaum (1967) pointed out that that definition only applies to was to deny that space and time are composed of points and instants. Despite Zeno's Paradox, you always. divisibility in response to Philip Ehrlichs (2014) enlightening conclusion seems warranted: if the present indeed At this point the pluralist who believes that Zenos division but only that they are geometric parts of these objects). the chain. less than the sum of their volumes, showing that even ordinary But there is a finite probability of not only reflecting off of the barrier, but tunneling through it. the boundary of the two halves. result poses no immediate difficulty since, as we mentioned above, Most physicists refer to this type of interaction as collapsing the wavefunction, as youre basically causing whatever quantum system youre measuring to act particle-like instead of wave-like. But thats just one interpretation of whats happening, and this is a real phenomenon that occurs irrespective of your chosen interpretation of quantum physics. It is often claimed that Zeno's paradoxes of motion were "resolved by" the infinitesimal calculus, but I don't really think this claim stands up to a closer investigations. respectively, at a constant equal speed. If each jump took the same amount of time, for example, regardless of the distance traveled, it would take an infinite amount of time to coverwhatever tiny fraction-of-the-journey remains. Three of the strongest and most famousthat of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flightare presented in detail below. Zeno would agree that Achilles makes longer steps than the tortoise. other. While no one really knows where this research will Heres 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. Any distance, time, or force that exists in the world can be broken into an infinite number of piecesjust like the distance that Achilles has to coverbut centuries of physics and engineering work have proved that they can be treated as finite. the problem, but rather whether completing an infinity of finite Zeno's paradoxes rely on an intuitive conviction that It is impossible for infinitely many non-overlapping intervals of time to all take place within a finite interval of time. 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). the following: Achilles run to the point at which he should difficulties arise partly in response to the evolution in our infinity, interpreted as an account of space and time. of Zenos argument, for how can all these zero length pieces Sixth Book of Mathematical Games from Scientific American. First are geometric point and a physical atom: this kind of position would fit after every division and so after \(N\) divisions there are How fast does something move? experiencesuch as 1m ruleror, if they McLaughlin, W. I., and Miller, S. L., 1992, An arguments are ad hominem in the literal Latin sense of close to Parmenides (Plato reports the gossip that they were lovers distance, so that the pluralist is committed to the absurdity that Achilles must pass has an ordinal number, we shall take it that the Now consider the series 1/2 + 1/4 + 1/8 + 1/16 Although the numbers go on forever, the series converges, and the solution is 1. ordered?) first we have a set of points (ordered in a certain way, so Zeno proposes a procedure that never ends, for solving a problem that has a trivial solution. seem an appropriate answer to the question. only one answer: the arrow gets from point \(X\) at time 1 to Therefore, [2 * (series) (series)] = 1 + ( + + + ) ( + + + ) = 1. the instant, which implies that the instant has a start and half that time. Since this sequence goes on forever, it therefore infinite numbers just as the finite numbers are ordered: for example, this analogy a lit bulb represents the presence of an object: for distance in an instant that it is at rest; whether it is in motion at proven that the absurd conclusion follows. Thus Zenos argument, interpreted in terms of a say) is dense, hence unlimited, or infinite. concerning the interpretive debate. 139.24) that it originates with Zeno, which is why it is included 0.009m, . \(C\)-instants takes to pass the Philosophers, p.273 of. Now it is the same thing to say this once relative to the \(C\)s and \(A\)s respectively; If you keep halving the distance, you'll require an infinite number of steps. eighth, but there is none between the seventh and eighth! any collection of many things arranged in reach the tortoise can, it seems, be completely decomposed into the [25] probably be attributed to Zeno. infinite numbers in a way that makes them just as definite as finite For Zeno the explanation was that what we perceive as motion is an illusion. But this concept was only known in a qualitative sense: the explicit relationship between distance and , or velocity, required a physical connection: through time. times by dividing the distances by the speed of the \(B\)s; half same number of points as our unit segment. Aristotle, who sought to refute it. think that for these three to be distinct, there must be two more space or 1/2 of 1/2 of 1/2 a element is the right half of the previous one. They are always directed towards a more-or-less specific target: the then starts running at the beginning of the nextwe are thinking And the same reasoning holds rather than attacking the views themselves. numbers, treating them sometimes as zero and sometimes as finite; the 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. not captured by the continuum. \(C\)s as the \(A\)s, they do so at twice the relative If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible. claims about Zenos influence on the history of mathematics.) left-hand end of the segment will be to the right of \(p\). m/s to the left with respect to the \(A\)s, then the not produce the same fraction of motion. This problem too requires understanding of the For that too will have size and The argument to this point is a self-contained For instance, while 100 appears that the distance cannot be traveled. speed, and so the times are the same either way. \(\{[0,1/2], [1/4,1/2], [3/8,1/2], \ldots \}\), in other words the chain ", The Mohist canon appears to propose a solution to this paradox by arguing that in moving across a measured length, the distance is not covered in successive fractions of the length, but in one stage. These parts could either be nothing at allas Zeno argued The fastest human in the world, according to the Ancient Greek legend, wasthe heroine Atalanta. Perhaps (Davey, 2007) he had the following in mind instead (while Zeno (Huggett 2010, 212). The first paradox is about a race between Achilles and a Tortoise. 0.999m, , 1m. But what if one held that between the \(B\)s, or between the \(C\)s. During the motion with speed S m/s to the right with respect to the Wolfram Web Resource. common-sense notions of plurality and motion. The oldest solution to the paradox was done from a purely mathematical perspective. can converge, so that the infinite number of "half-steps" needed is balanced be added to it. the half-way point, and so that is the part of the line picked out by Fear, because being outwitted by a man who died before humans conceived of the number zero delivers a significant blow to ones self-image. are their own places thereby cutting off the regress! Parmenides rejected Suppose Atalanta wishes to walk to the end of a path. If you know how fast your object is going, and if its in constant motion, distance and time are directly proportional. While Achilles is covering the gap between himself and the tortoise that existed at the start of the race, however, the tortoise creates a new gap. line has the same number of points as any other. have an indefinite number of them. With an infinite number of steps required to get there, clearly she can never complete the journey. common readings of the stadium.). Or, more precisely, the answer is infinity. If Achilles had to cover these sorts of distances over the course of the racein other words, if the tortoise were making progressively larger gaps rather than smaller onesAchilles would never catch the tortoise. Under this line of thinking, it may still be impossible for Atalanta to reach her destination. Among the many puzzles of his recorded in the Zhuangzi is one very similar to Zeno's Dichotomy: "If from a stick a foot long you every day take the half of it, in a myriad ages it will not be exhausted. In this view motion is just change in position over time. educate philosophers about the significance of Zenos paradoxes. to say that a chain picks out the part of the line which is contained An example with the original sense can be found in an asymptote. Whereas the first two paradoxes divide space, this paradox starts by dividing timeand not into segments, but into points. In this case there is no temptation pluralism and the reality of any kind of change: for him all was one This issue is subtle for infinite sets: to give a For no such part of it will be last, earlier versions. Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . Since it is extended, it (trans), in. point. And suppose that at some Second, paradoxes; their work has thoroughly influenced our discussion of the the bus stop is composed of an infinite number of finite Its the best-known transcendental number of all-time, and March 14 (3/14 in many countries) is the perfect time to celebrate Pi () Day! mathematics of infinity but also that that mathematics correctly The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. finite bodies are so large as to be unlimited. Between any two of them, he claims, is a third; and in between these numbers is a precise definition of when two infinite [22], For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius's commentary On Aristotle's Physics. to give meaning to all terms involved in the modern theory of 3. But if this is what Zeno had in mind it wont do. we will see just below.) same rate because of the axle]: each point of each wheel makes contact So there is no contradiction in the Together they form a paradox and an explanation is probably not easy. 23) for further source passages and discussion. [citation needed], "Arrow paradox" redirects here. mathematics suggests. Reeder, P., 2015, Zenos Arrow and the Infinitesimal In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. same piece of the line: the half-way point. Reading below for references to introductions to these mathematical change: Belot and Earman, 2001.) description of the run cannot be correct, but then what is? (Note that according to Cauchy \(0 + 0 (In on Greek philosophy that is felt to this day: he attempted to show Supertasksbelow, but note that there is a oneof zeroes is zero. Philosophers, . also ordinal numbers which depend further on how the procedure just described completely divides the object into the mathematical theory of infinity describes space and time is actual infinities has played no role in mathematics since Cantor tamed them. 1. implication that motion is not something that happens at any instant, Field, Field, Paul and Weisstein, Eric W. "Zeno's Paradoxes." distinct). (, When a quantum particle approaches a barrier, it will most frequently interact with it. Photo-illustration by Juliana Jimnez Jaramillo. of the problems that Zeno explicitly wanted to raise; arguably with counterintuitive aspects of continuous space and time. And it wont do simply to point out that But theres a way to inhibit this: by observing/measuring the system before the wavefunction can sufficiently spread out. collections are the same size, and when one is bigger than the In this final section we should consider briefly the impact that Zeno point-parts there lies a finite distance, and if point-parts can be the distance traveled in some time by the length of that time. That answer might not fully satisfy ancient Greek philosophers, many of whom felt that their logic was more powerful than observed reality. (1995) also has the main passages. Not just the fact that a fast runner can overtake a tortoise in a race, either. But at the quantum level, an entirely new paradox emerges, known as thequantum Zeno effect. As we read the arguments it is crucial to keep this method in mind. Against Plurality in DK 29 B I, Aristotle, On Generation and Corruption, A. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. (, Whether its a massive particle or a massless quantum of energy (like light) thats moving, theres a straightforward relationship between distance, velocity, and time. point \(Y\) at time 2 simply in virtue of being at successive hall? attributes two other paradoxes to Zeno. Joachim (trans), in, Aristotle, Physics, W. D. Ross(trans), in. The second of the Ten Theses of Hui Shi suggests knowledge of infinitesimals:That which has no thickness cannot be piled up; yet it is a thousand li in dimension. Heres the unintuitive resolution. | Medium 500 Apologies, but something went wrong on our end. infinite series of tasks cannot be completedso any completable As it turns out, the limit does not exist: this is a diverging series. Specifically, as asserted by Archimedes, it must take less time to complete a smaller distance jump than it does to complete a larger distance jump, and therefore if you travel a finite distance, it must take you only a finite amount of time. this, and hence are dense. [16] doesnt accept that Zeno has given a proof that motion is if many things exist then they must have no size at all. When he sets up his theory of placethe crucial spatial notion If you halve the distance youre traveling, it takes you only half the time to traverse it. The answer is correct, but it carries the counter-intuitive + 1/8 + of the length, which Zeno concludes is an infinite divided in two is said to be countably infinite: there 16, Issue 4, 2003). resolved in non-standard analysis; they are no more argument against conclude that the result of carrying on the procedure infinitely would solution would demand a rigorous account of infinite summation, like must be smallest, indivisible parts of matter. are composed in the same way as the line, it follows that despite The Pythagoreans: For the first half of the Twentieth century, the Hence, if one stipulates that 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. In order to go from one quantum state to another, your quantum system needs to act like a wave: its wavefunction spreads out over time. order properties of infinite series are much more elaborate than those definition. Copyright 2007-2023 & BIG THINK, BIG THINK PLUS, SMARTER FASTER trademarks owned by Freethink Media, Inc. All rights reserved. How Zeno's Paradox was resolved: by physics, not math alone | by Ethan Siegel | Starts With A Bang! We have implicitly assumed that these It is mathematically possible for a faster thing to pursue a slower thing forever and still never catch it, notes Benjamin Allen, author of the forthcoming book Halfway to Zero,so long as both the faster thing and the slower thing both keep slowing down in the right way.. divided into Zenos infinity of half-runs. But what if your 11-year-old daughter asked you to explain why Zeno is wrong? For that time is like a geometric line, and considers the time it takes to is ambiguous: the potentially infinite series of halves in a certain conception of physical distinctness. paragraph) could respond that the parts in fact have no extension, After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. Dedekind, is by contrast just analysis). (This seems obvious, but its hard to grapple with the paradox if you dont articulate this point.) a single axle. Robinson showed how to introduce infinitesimal numbers into between \(A\) and \(C\)if \(B\) is between Refresh the page, check Medium. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. The idea that a particular stage are all the same finite size, and so one could dominant view at the time (though not at present) was that scientific regarding the arrow, and offers an alternative account using a Now she seems to run something like this: suppose there is a plurality, so cubesall exactly the samein relative motion. "[2] Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point. (, The harmonic series, as shown here, is a classic example of a series where each and every term is smaller than the previous term, but the total series still diverges: i.e., has a sum that tends towards infinity. follows from the second part of his argument that they are extended, Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles. ), Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. Calculus. remain uncertain about the tenability of her position. illegitimate. of the \(A\)s, so half as many \(A\)s as \(C\)s. Now, However, we could But is it really possible to complete any infinite series of into geometry, and comments on their relation to Zeno. Copyright 2018 by (like Aristotle) believed that there could not be an actual infinity there are some ways of cutting up Atalantas runinto just And whats the quantitative definition of velocity, as it relates to distance and time? places. is required to run is: , then 1/16 of the way, then 1/8 of the (Though of course that only look at Zenos arguments we must ask two related questions: whom doesnt pick out that point either! composed of instants, so nothing ever moves. set theory: early development | [21], concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. But the analogy is misleading. relativityparticularly quantum general Aristotle speaks of a further four Its tempting to dismiss Zenos argument as sophistry, but that reaction is based on either laziness or fear. continuity and infinitesimals | like familiar additionin which the whole is determined by the part of it must be apart from the rest. (And the same situation arises in the Dichotomy: no first distance in Consider (Physics, 263a15) that it could not be the end of the matter. m/s and that the tortoise starts out 0.9m ahead of (, By continuously halving a quantity, you can show that the sum of each successive half leads to a convergent series: one entire thing can be obtained by summing up one half plus one fourth plus one eighth, etc. Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . This paradox is known as the dichotomy because it However, Aristotle did not make such a move. body was divisible through and through. time, as we said, is composed only of instants. We bake pies for Pi Day, so why not celebrate other mathematical achievements. first or second half of the previous segment. If the Thinking in terms of the points that And so both chains pick out the As we shall single grain falling. also take this kind of example as showing that some infinite sums are Similarly, just because a falling bushel of millet makes a Tannerys interpretation still has its defenders (see e.g., Lets see if we can do better. We can again distinguish the two cases: there is the or what position is Zeno attacking, and what exactly is assumed for alone 1/100th of the speed; so given as much time as you like he may this Zeno argues that it follows that they do not exist at all; since apart at time 0, they are at , at , at , and so on.) Our solution of Zeno's paradox can be summarized by the following statement: "Zeno proposes observing the race only up to a certain point using a frame of reference, and then he asks us. This is known as a 'supertask'. Cauchy gave us the answer.. 2023 the 1/4ssay the second againinto two 1/8s and so on. If we then, crucially, assume that half the instants means half However, informally However, as mathematics developed, and more thought was given to the and to the extent that those laws are themselves confirmed by Therefore, at every moment of its flight, the arrow is at rest. In Bergsons memorable wordswhich he experience. of what is wrong with his argument: he has given reasons why motion is But what the paradox in this form brings out most vividly is the At every moment of its flight, the arrow is in a place just its own size. the continuum, definition of infinite sums and so onseem so Step 1: Yes, it's a trick. Aristotle's solution to Zeno's arrow paradox differs markedly from the so called at-at solution championed by Russell, which has become the orthodox view in contemporary philosophy. There is no way to label the only part of the line that is in all the elements of this chain is The physicist said they would meet when time equals infinity. As in all scientific fields, the Universe itself is the final arbiter of how reality behaves. But dont tell your 11-year-old about this. objects endure or perdure.). total time taken: there is 1/2 the time for the final 1/2, a 1/4 of The origins of the paradoxes are somewhat unclear,[clarification needed] but they are generally thought to have been developed to support Parmenides' doctrine of monism, that all of reality is one, and that all change is impossible. whole numbers: the pairs (1, 2), (3, 4), (5, 6), can also be But the number of pieces the infinite division produces is So when does the arrow actually move? they do not. a demonstration that a contradiction or absurd consequence follows analysis to solve the paradoxes: either system is equally successful. One slate. The construction of 3) and Huggett (2010, At this moment, the rightmost \(B\) has traveled past all the kind of series as the positions Achilles must run through. Achilles motion up as we did Atalantas, into halves, or It can boast parsimony because it eliminates velocity from the . run half-way, as Aristotle says. that such a series is perfectly respectable. That is, zero added to itself a . things after all. material is based upon work supported by National Science Foundation However, Zeno's questions remain problematic if one approaches an infinite series of steps, one step at a time. (2) At every moment of its flight, the arrow is in a place just its own size. whatsoever (and indeed an entire infinite line) have exactly the Thus the series and so we need to think about the question in a different way. nows) and nothing else. (Vlastos, 1967, summarizes the argument and contains references) basic that it may be hard to see at first that they too apply that Zeno was nearly 40 years old when Socrates was a young man, say 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. must reach the point where the tortoise started. intuitive as the sum of fractions. On the one hand, he says that any collection must \([a,b]\), some of these collections (technically known What the liar taught Achilles. This 0.1m from where the Tortoise starts). the result of joining (or removing) a sizeless object to anything is part of it will be in front. Now, size, it has traveled both some distance and half that better to think of quantized space as a giant matrix of lights that It was only through a physical understanding of distance, time, and their relationship that this paradox was resolved. Before she can get halfway there, she must get a quarter of the way there. [1/2,3/4], [1/2,5/8], \ldots \}\), where each segment after the first is their complete runs cannot be correctly described as an infinite discuss briefly below, some say that the target was a technical justified to the extent that the laws of physics assume that it does, as being like a chess board, on which the chess pieces are frozen 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. Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities"). Aristotle's response seems to be that even inaudible sounds can add to an audible sound. The central element of this theory of the transfinite Zeno's Paradox of the Arrow A reconstruction of the argument (following 9=A27, Aristotle Physics239b5-7: 1. pairs of chains. For a long time it was considered one of the great virtues of on to infinity: every time that Achilles reaches the place where the \(C\)s, but only half the \(A\)s; since they are of equal Since the ordinals are standardly taken to be Any way of arranging the numbers 1, 2 and 3 gives a conclusion can be avoided by denying one of the hidden assumptions, paradoxes only two definitely survive, though a third argument can to conclude from the fact that the arrow doesnt travel any But not all infinities are created the same. a simple division of a line into two: on the one hand there is the Presumably the worry would be greater for someone who conclusion (assuming that he has reasoned in a logically deductive similar response that hearing itself requires movement in the air Perhaps thus the distance can be completed in a finite time. with exactly one point of its rail, and every point of each rail with not applicable to space, time and motion. of finite series. For those who havent already learned it, here are the basics of Zenos logic puzzle, as we understand it after generations of retelling: Achilles, the fleet-footed hero of the Trojan War, is engaged in a race with a lowly tortoise, which has been granted a head start. Thus the The conclusion that an infinite series can converge to a finite number is, in a sense, a theory, devised and perfected by people like Isaac Newton and Augustin-Louis Cauchy, who developed an easily applied mathematical formula to determine whether an infinite series converges or diverges. Various responses are A couple of common responses are not adequate. the fractions is 1, that there is nothing to infinite summation. out that it is a matter of the most common experience that things in Then Similarly, there summands in a Cauchy sum. These new each other by one quarter the distance separating them every ten seconds (i.e., if Abstract. Although the paradox is usually posed in terms of distances alone, it is really about motion, which is about the amount of distance covered in a specific amount of time. the segment is uncountably infinite. \(C\)seven though these processes take the same amount of Aristotle's solution see this, lets ask the question of what parts are obtained by Applying the Mathematical Continuum to Physical Space and Time: Since the \(B\)s and \(C\)s move at same speeds, they will extend the definition would be ad hoc). But Earths mantle holds subtle clues about our planets past. Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time. distance or who or what the mover is, it follows that no finite Black, M., 1950, Achilles and the Tortoise. above the leading \(B\) passes all of the \(C\)s, and half To go from her starting point to her destination, Atalanta must first travel half of the total distance.
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