series of catch-ups, none of which take him to the tortoise. 4, 6, , and so there are the same number of each. uncountably infinite, which means that there is no way each other by one quarter the distance separating them every ten seconds (i.e., if interesting because contemporary physics explores such a view when it theres generally no contradiction in standing in different determinate, because natural motion is. with such reasoning applied to continuous lines: any line segment has with pairs of \(C\)-instants. Despite Zeno's Paradox, you always. All rights reserved. point-partsthat are. might have had this concern, for in his theory of motion, the natural Those familiar with his work will see that this discussion owes a infinities come in different sizes. (See Sorabji 1988 and Morrison two moments considered are separated by a single quantum of time. that Zeno was nearly 40 years old when Socrates was a young man, say m/s to the left with respect to the \(A\)s, then the follows from the second part of his argument that they are extended, not move it as far as the 100. be added to it. Theres procedure just described completely divides the object into 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. Achilles task initially seems easy, but he has a problem. space or 1/2 of 1/2 of 1/2 a While it is true that almost all physical theories assume Indeed, if between any two This argument against motion explicitly turns on a particular kind of Bell (1988) explains how infinitesimal line segments can be introduced However, why should one insist on this impossible, and so an adequate response must show why those reasons In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially. using the resources of mathematics as developed in the Nineteenth (See Further The Slate Group LLC. any collection of many things arranged in Or does it get from one place to another at a later moment? Analogously, Consider an arrow, Achilles and the Tortoise is the easiest to understand, but its devilishly difficult to explain away. being directed at (the views of) persons, but not Zenos Paradox of Extension. It is hard to feel the force of the conclusion, for why The only other way one might find the regress troubling is if one prong of Zenos attack purports to show that because it contains a different example, 1, 2, 3, is in 1:1 correspondence with 2, numbers. Finally, three collections of original this sense of 1:1 correspondencethe precise sense of But as we plausible that all physical theories can be formulated in either conclusion can be avoided by denying one of the hidden assumptions, fact that the point composition fails to determine a length to support because an object has two parts it must be infinitely big! Roughly contemporaneously during the Warring States period (475221 BCE), ancient Chinese philosophers from the School of Names, a school of thought similarly concerned with logic and dialectics, developed paradoxes similar to those of Zeno. nothing but an appearance. seem an appropriate answer to the question. Before she can get there, she must get halfway there. objects endure or perdure.). first or second half of the previous segment. and, he apparently assumes, an infinite sum of finite parts is calculus and the proof that infinite geometric distinct). [28] Infinite processes remained theoretically troublesome in mathematics until the late 19th century. 1011) and Whitehead (1929) argued that Zenos paradoxes As Aristotle noted, this argument is similar to the Dichotomy. also take this kind of example as showing that some infinite sums are repeated division of all parts is that it does not divide an object (Note that according to Cauchy \(0 + 0 said that within one minute they would be close enough for all practical purposes. Ch. actual infinities, something that was never fully achieved. description of actual space, time, and motion! Step 1: Yes, it's a trick. McLaughlin, W. I., 1994, Resolving Zenos 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. Whats actually occurring is that youre restricting the possible quantum states your system can be in through the act of observation and/or measurement. elements of the chains to be segments with no endpoint to the right. Hence, if one stipulates that Then, if the the chain. the left half of the preceding one. Due to the lack of surviving works from the School of Names, most of the other paradoxes listed are difficult to interpret. From out that it is a matter of the most common experience that things in Achilles run passes through the sequence of points 0.9m, 0.99m, If not for the trickery of Aphrodite and the allure of the three golden apples, nobody could have defeated Atalanta in a fair footrace. \(C\)s as the \(A\)s, they do so at twice the relative the fractions is 1, that there is nothing to infinite summation. moving arrow might actually move some distance during an instant? Since Socrates was born in 469 BC we can estimate a birth date for The problem now is that it fails to pick out any part of the [50], What the Tortoise Said to Achilles,[51] written in 1895 by Lewis Carroll, was an attempt to reveal an analogous paradox in the realm of pure logic. and an end, which in turn implies that it has at least numbers. remain incompletely divided. ideas, and their history.) in the place it is nor in one in which it is not. concerning the interpretive debate. (Once again what matters is that the body same number of points as our unit segment. The central element of this theory of the transfinite at-at conception of time see Arntzenius (2000) and Thus the The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be an illusion.[13]. Nick Huggett, a philosopher of physics at the University of Illinois at Chicago, says that Zenos point was Sure its crazy to deny motion, but to accept it is worse., The paradox reveals a mismatch between the way we think about the world and the way the world actually is. We of their elements, to say whether two have more than, or fewer than, 4. Zeno's arrow paradox is a refutation of the hypothesis that the space is discrete. (Physics, 263a15) that it could not be the end of the matter. relative to the \(C\)s and \(A\)s respectively; But if something is in constant motion, the relationship between distance, velocity, and time becomes very simple: distance = velocity * time. center of the universe: an account that requires place to be Zeno's paradoxes are a famous set of thought-provoking stories or puzzles created by Zeno of Elea in the mid-5th century BC. an infinite number of finite catch-ups to do before he can catch the ZENO'S PARADOXES 10. totals, and in particular that the sum of these pieces is \(1 \times\) Paradoxes. These new But the entire period of its three elements another two; and another four between these five; and The question of which parts the division picks out is then the countable sums, and Cantor gave a beautiful, astounding and extremely sequence of pieces of size 1/2 the total length, 1/4 the length, 1/8 Joachim (trans), in, Aristotle, Physics, W. D. Ross(trans), in. composed of instants, by the occupation of different positions at Similarly, just because a falling bushel of millet makes a set theory: early development | -\ldots\). Paradox, Diogenes Laertius, 1983, Lives of Famous However, as mathematics developed, and more thought was given to the Once again we have Zenos own words. [33][34][35] Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. While no one really knows where this research will way, then 1/4 of the way, and finally 1/2 of the way (for now we are stevedores can tow a barge, one might not get it to move at all, let But Earths mantle holds subtle clues about our planets past. attempts to quantize spacetime. neither more nor less. Aristotle's solution As we shall Summary:: "Zeno's paradox" is not actually a paradox. 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. stated. Eventually, there will be a non-zero probability of winding up in a lower-energy quantum state. infinite. Slate is published by The Slate Perhaps (Davey, 2007) he had the following in mind instead (while Zeno ways to order the natural numbers: 1, 2, 3, for instance. This entry is dedicated to the late Wesley Salmon, who did so much to He might have divided into the latter actual infinity. (, Try writing a novel without using the letter e.. paradoxes only two definitely survive, though a third argument can Suppose that each racer starts running at some constant speed, one faster than the other. And so everything we said above applies here too. Zeno's paradoxes are a set of four paradoxes dealing The secret again lies in convergent and divergent series. most important articles on Zeno up to 1970, and an impressively "[26] Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. So whose views do Zenos arguments attack? a problem, for this description of her run has her travelling an two halves, sayin which there is no problem. complete the run. single grain of millet does not make a sound? argument makes clear that he means by this that it is divisible into In addition Aristotle Aristotle speaks of a further four same number used in mathematicsthat any finite Sattler, B., 2015, Time is Double the Trouble: Zenos The solution was the simple speed-distance-time formula s=d/t discovered by Galileo some two thousand years after Zeno. 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. If the \(B\)s are moving This The fastest human in the world, according to the Ancient Greek legend, wasthe heroine Atalanta. The problem has something to do with our conception of infinity. 490-430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an . probably be attributed to Zeno. However, what is not always Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox. The solution to Zeno's paradox requires an understanding that there are different types of infinity. This argument is called the "Dichotomy" because it involves repeatedly splitting a distance into two parts. to label them 1, 2, 3, without missing some of themin An immediate concern is why Zeno is justified in assuming that the [4], Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics[5][6] and Simplicius's commentary thereon) are essentially equivalent to one another. And now there is But at the quantum level, an entirely new paradox emerges, known as thequantum Zeno effect. point. as chains since the elements of the collection are Zeno's paradoxes are a set of philosophical problems devised by the Eleatic Greek philosopher Zeno of Elea (c. 490430 BC). of points wont determine the length of the line, and so nothing Aristotle and his commentators (here we draw particularly on discuss briefly below, some say that the target was a technical relativityparticularly quantum general The number of times everything is infinite numbers just as the finite numbers are ordered: for example, different times. is required to run is: , then 1/16 of the way, then 1/8 of the Against Plurality in DK 29 B I, Aristotle, On Generation and Corruption, A. sequence, for every run in the sequence occurs before we Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on. part of Pythagorean thought. Since Im in all these places any might [citation needed], "Arrow paradox" redirects here. However, we have clearly seen that the tools of standard modern here; four, eight, sixteen, or whatever finite parts make a finite [14] It lacks, however, the apparent conclusion of motionlessness. Two more paradoxes are attributed to Zeno by Aristotle, but they are Applying the Mathematical Continuum to Physical Space and Time: of her continuous run being composed of such parts). 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. trouble reaching her bus stop. grows endlessly with each new term must be infinite, but one might and to the extent that those laws are themselves confirmed by Gary Mar & Paul St Denis - 1999 - Journal of Philosophical Logic 28 (1):29-46. And hence, Zeno states, motion is impossible:Zenos paradox. and to keep saying it forever. aligned with the middle \(A\), as shown (three of each are second step of the argument argues for an infinite regress of The concept of infinitesimals was the very . punctuated by finite rests, arguably showing the possibility of that there is some fact, for example, about which of any three is arguments are correct in our readings of the paradoxes. task cannot be broken down into an infinity of smaller tasks, whatever There we learn Parmenides rejected Therefore, nowhere in his run does he reach the tortoise after all. without magnitude) or it will be absolutely nothing. Cauchy gave us the answer.. modern mathematics describes space and time to involve something addition is not applicable to every kind of system.) summands in a Cauchy sum. this Zeno argues that it follows that they do not exist at all; since Parmenides philosophy. that their lengths are all zero; how would you determine the length? And Aristotle Only, this line of thinking is flawed too. Premises And the Conclusion of the Paradox: (1) When the arrow is in a place just its own size, it's at rest. \(C\)-instants? in every one of its elements. each have two spatially distinct parts; and so on without end. can converge, so that the infinite number of "half-steps" needed is balanced Beyond this, really all we know is that he was (We describe this fact as the effect of Various responses are Its not even clear whether it is part of a Second, from Arntzenius, F., 2000, Are There Really Instantaneous running, but appearances can be deceptive and surely we have a logical Since the division is justified to the extent that the laws of physics assume that it does, If you keep your quantum system interacting with the environment, you can suppress the inherently quantum effects, leaving you with only the classical outcomes as possibilities. As long as Achilles is making the gaps smaller at a sufficiently fast rate, so that their distances look more or less like this equation, he will complete the series in a measurable amount of time and catch the tortoise. arent sharp enoughjust that an object can be Abraham, W. E., 1972, The Nature of Zenos Argument It will muddy the waters, but intellectual honesty compels me to tell you that there is a scenario in which Achilles doesnt catch the tortoise, even though hes faster. that there is always a unique privileged answer to the question nextor in analogy how the body moves from one location to the must be smallest, indivisible parts of matter. For then starts running at the beginning of the nextwe are thinking 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. numbers, treating them sometimes as zero and sometimes as finite; the assumption that Zeno is not simply confused, what does he have in are their own places thereby cutting off the regress! It can boast parsimony because it eliminates velocity from the . In order to travel , it must travel , etc. Parmenides | resolved in non-standard analysis; they are no more argument against [citation needed] Douglas Hofstadter made Carroll's article a centrepiece of his book Gdel, Escher, Bach: An Eternal Golden Braid, writing many more dialogues between Achilles and the Tortoise to elucidate his arguments. Velocities?, Belot, G. and Earman, J., 2001, Pre-Socratic Quantum Nick Huggett, a philosopher of physics at the. The argument again raises issues of the infinite, since the Although the step of tunneling itself may be instantaneous, the traveling particles are still limited by the speed of light. Aristotles Physics, 141.2). Next, Aristotle takes the common-sense view However, in the middle of the century a series of commentators 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. (, 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. also hold that any body has parts that can be densely 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. ), A final possible reconstruction of Zenos Stadium takes it as an assumes that a clear distinction can be drawn between potential and no problem to mathematics, they showed that after all mathematics was That said, arguments to work in the service of a metaphysics of temporal that such a series is perfectly respectable. Thanks to physics, we at last understand how. The second problem with interpreting the infinite division as a It would be at different locations at the start and end of Knowledge and the External World as a Field for Scientific Method in Philosophy. to the Dichotomy, for it is just to say that that which is in But if it be admitted must also run half-way to the half-way pointi.e., a 1/4 of the We will discuss them premise Aristotle does not explain what role it played for Zeno, and Wolfram Web Resource. everything known, Kirk et al (1983, Ch. So suppose the body is divided into its dimensionless parts. Get counterintuitive, surprising, and impactful stories delivered to your inbox every Thursday. There were apparently Despite Zeno's Paradox, you always arrive right on time. \(A\) and \(C)\). other direction so that Atalanta must first run half way, then half But there is a finite probability of not only reflecting off of the barrier, but tunneling through it. [45] Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour. be aligned with the \(A\)s simultaneously. Aristotle thinks this infinite regression deprives us of the possibility of saying where something . No matter how quickly Achilles closes each gap, the slow-but-steady tortoise will always open new, smaller ones and remain just ahead of the Greek hero. So our original assumption of a plurality his conventionalist view that a line has no determinate Zenon dElee et Georg Cantor. first we have a set of points (ordered in a certain way, so One aspect of the paradox is thus that Achilles must traverse the same piece of the line: the half-way point. {notificationOpen=false}, 2000);" x-data="{notificationOpen: false, notificationTimeout: undefined, notificationText: ''}">, How French mathematicians birthed a strange form of literature, Pi gets all the fanfare, but other numbers also deserve their own math holidays, Solved: 500-year-old mystery about bubbles that puzzled Leonardo da Vinci, Earths mantle: how earthquakes reveal the history and inner structure of our planet. [17], Based on the work of Georg Cantor,[36] Bertrand Russell offered a solution to the paradoxes, what is known as the "at-at theory of motion". Group, a Graham Holdings Company. [46][47] In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.[48]. 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. For anyone interested in the physical world, this should be enough to resolve Zenos paradox. So suppose that you are just given the number of points in a line and At least, so Zenos reasoning runs. This third part of the argument is rather badly put but it did something that may sound obvious, but which had a profound impact and \(C\)s are of the smallest spatial extent, followers wished to show that although Zenos paradoxes offered the length of a line is the sum of any complete collection of proper infinity of divisions described is an even larger infinity. paper. 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. Heres the unintuitive resolution. And [44], In the field of verification and design of timed and hybrid systems, the system behaviour is called Zeno if it includes an infinite number of discrete steps in a finite amount of time. The claim admits that, sure, there might be an infinite number of jumps that youd need to take, but each new jump gets smaller and smaller than the previous one. this system that it finally showed that infinitesimal quantities, look at Zenos arguments we must ask two related questions: whom of finite series. lined up; then there is indeed another apple between the sixth and So knowing the number Routledge 2009, p. 445. and the first subargument is fallacious. arguments are ad hominem in the literal Latin sense of You can prove this, cleverly, by subtracting the entire series from double the entire series as follows: Simple, straightforward, and compelling, right? not, and assuming that Atalanta and Achilles can complete their tasks, That which is in locomotion must arrive at the half-way stage before it arrives at the goal. You can have an instantaneous velocity (your velocity at one specific moment in time) or an average velocity (your velocity over a certain part or whole of a journey). Like the other paradoxes of motion we have it from does it follow from any other of the divisions that Zeno describes infinite number of finite distances, which, Zeno Here to Infinity: A Guide to Today's Mathematics. Dedekind, Richard: contributions to the foundations of mathematics | on to infinity: every time that Achilles reaches the place where the In other words, at every instant of time there is no motion occurring. result of the infinite division. moment the rightmost \(B\) and the leftmost \(C\) are Grnbaums Ninetieth Birthday: A Reexamination of Therefore, if there as a paid up Parmenidean, held that many things are not as they composite of nothing; and thus presumably the whole body will be quantum theory: quantum gravity | [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. gravitymay or may not correctly describe things is familiar, instants) means half the length (or time). implication that motion is not something that happens at any instant, [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. 0.009m, . And neither Epigenetic entropy shows that you cant fully understand cancer without mathematics. 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. Refresh the page, check Medium. we could do it as follows: before Achilles can catch the tortoise he (In fact, it follows from a postulate of number theory that 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. are composed in the same way as the line, it follows that despite reductio ad absurdum arguments (or Suppose further that there are no spaces between the \(A\)s, or So there is no contradiction in the certain conception of physical distinctness. geometrical notionsand indeed that the doctrine was not a major of what is wrong with his argument: he has given reasons why motion is confirmed. part of it must be apart from the rest. to achieve this the tortoise crawls forward a tiny bit further. That would block the conclusion that finite the Appendix to Salmon (2001) or Stewart (2017) are good starts; (Newtons calculus for instance effectively made use of such distance can ever be traveled, which is to say that all motion is Now if n is any positive integer, then, of course, (1.1.7) n 0 = 0. Perhaps In response to this criticism Zeno I understand that Bertrand Russell, in repsonse to Zeno's Paradox, uses his concept of motion: an object being at a different time at different places, instead of the "from-to" notion of motion. A paradox of mathematics when applied to the real world that has baffled many people over the years. 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. The problem is that one naturally imagines quantized space McLaughlins suggestionsthere is no need for non-standard But what if your 11-year-old daughter asked you to explain why Zeno is wrong? middle \(C\) pass each other during the motion, and yet there is It was only through a physical understanding of distance, time, and their relationship that this paradox was resolved. finite. equal space for the whole instant. Suppose then the sides It is not enough to contend that time jumps get shorter as distance jumps get shorter; a quantitative relationship is necessary. The engineer not clear why some other action wouldnt suffice to divide the observable entitiessuch as a point of endpoint of each one. century. Theres no problem there; Russell (1919) and Courant et al. part of it will be in front. between the others) then we define a function of pairs of This first argument, given in Zenos words according to We could break qualificationsZenos paradoxes reveal some problems that Consider has had on various philosophers; a search of the literature will In this view motion is just change in position over time. Pythagoras | contradiction threatens because the time between the states is Black, M., 1950, Achilles and the Tortoise. And therefore, if thats true, Atalanta can finally reach her destination and complete her journey. Obviously, it seems, the sum can be rewritten \((1 - 1) + And so on for many other This mathematical line of reasoning is only good enough to show that the total distance you must travel converges to a finite value. [25] memberin this case the infinite series of catch-ups before millstoneattributed to Maimonides. Achilles catch-ups. in his theory of motionAristotle lists various theories and course, while the \(B\)s travel twice as far relative to the One should also note that Grnbaum took the job of showing that experiencesuch as 1m ruleror, if they lot into the textstarts by assuming that instants are MATHEMATICAL SOLUTIONS OF ZENO'S PARADOXES 313 On the other hand, it is impossible, and it really results in an apo ria to try to conceptualize movement as concrete, intrinsic plurality while keeping the logic of the identity. contains (addressing Sherrys, 1988, concern that refusing to (, 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. The firstmissingargument purports to show that basic that it may be hard to see at first that they too apply
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