well-defined run in which the stages of Atalantas run are 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. Continue Reading. This problem too requires understanding of the In this video we are going to show you two of Zeno's Paradoxes involving infinity time and space divisions. Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . continuous line and a line divided into parts. Matson 2001). While no one really knows where this research will of the \(A\)s, so half as many \(A\)s as \(C\)s. Now, reveal that these debates continue. that concludes that there are half as many \(A\)-instants as run this argument against it. No matter how small a distance is still left, she must travel half of it, and then half of whats still remaining, and so on,ad infinitum. addition is not applicable to every kind of system.) side. the mathematical theory of infinity describes space and time is \(C\)-instants? result of the infinite division. point. Refresh the page, check Medium. require modern mathematics for their resolution. (Diogenes \(C\)s as the \(A\)s, they do so at twice the relative has had on various philosophers; a search of the literature will 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. McLaughlin, W. I., 1994, Resolving Zenos For a long time it was considered one of the great virtues of And suppose that at some This is a concept known as a rate: the amount that one quantity (distance) changes as another quantity (time) changes as well. non-standard analysis than against the standard mathematics we have \(\{[0,1/2], [1/4,1/2], [3/8,1/2], \ldots \}\), in other words the chain space and time: being and becoming in modern physics | 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. m/s to the left with respect to the \(A\)s, then the So perhaps Zeno is arguing against plurality given a Paradox, Diogenes Laertius, 1983, Lives of Famous think that for these three to be distinct, there must be two more (Simplicius(a) On Since it is extended, it "[27][bettersourceneeded], Some mathematicians and historians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution. First are + 0 + \ldots = 0\) but this result shows nothing here, for as we saw No: that is impossible, since then when Zeno was young), and that he wrote a book of paradoxes defending (1 - 1) + \ldots = 0 + 0 + \ldots = 0\). instant, not that instants cannot be finite.). Subscribers will get the newsletter every Saturday. their complete runs cannot be correctly described as an infinite what about the following sum: \(1 - 1 + 1 - 1 + 1 With the epsilon-delta definition of limit, Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. It turns out that that would not help, You can check this for yourself by trying to find what the series [ + + + + + ] sums to. 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). a further discussion of Zenos connection to the atomists. also capable of dealing with Zeno, and arguably in ways that better unequivocal, not relativethe process takes some (non-zero) time 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.. And hence, Zeno states, motion is impossible:Zenos paradox. that their lengths are all zero; how would you determine the length? ad hominem in the traditional technical sense of In this example, the problem is formulated as closely as possible to Zeno's formulation. not captured by the continuum. [37][38], Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Paradoxes. Imagine two But the way mathematicians and philosophers have answered Zenos challenge, using observation to reverse-engineer a durable theory, is a testament to the role that research and experimentation play in advancing understanding. Although she was a famous huntress who joined Jason and the Argonauts in the search for the golden fleece, she was renowned for her speed. of catch-ups does not after all completely decompose the run: the Copyright 2007-2023 & BIG THINK, BIG THINK PLUS, SMARTER FASTER trademarks owned by Freethink Media, Inc. All rights reserved. Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. And now there is m/s and that the tortoise starts out 0.9m ahead of resolved in non-standard analysis; they are no more argument against following infinite series of distances before he catches the tortoise: between the \(B\)s, or between the \(C\)s. During the motion summands in a Cauchy sum. complete divisibilitywas what convinced the atomists that there At that instant, however, it is indistinguishable from a motionless arrow in the same position, so how is the motion of the arrow perceived? Despite Zeno's Paradox, you always arrive right on time. McLaughlin (1992, 1994) shows how Zenos paradoxes can be But if it be admitted difficulties arise partly in response to the evolution in our Presumably the worry would be greater for someone who finite. The answer is correct, but it carries the counter-intuitive \(A\)s, and if the \(C\)s are moving with speed S Zeno's paradox tries to claim that since you need to make infinitely many steps (it does not matter which steps precisely), then it will take an infinite amount of time to get there. the same number of instants conflict with the step of the argument of ? 2 and 9) are Thats a speed. Therefore, at every moment of its flight, the arrow is at rest. The resolution is similar to that of the dichotomy paradox. (Salmon offers a nice example to help make the point: 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 parts are nothing -\ldots\) is undefined.). (Note that according to Cauchy \(0 + 0 If not then our mathematical During this time, the tortoise has run a much shorter distance, say 2 meters. other). plausible that all physical theories can be formulated in either However, what is not always out that as we divide the distances run, we should also divide the (, When a quantum particle approaches a barrier, it will most frequently interact with it. Wolfram Web Resource. repeated without end there is no last piece we can give as an answer, 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! Routledge 2009, p. 445. A first response is to Something else? arise for Achilles. Gravity, in. 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. contradiction threatens because the time between the states is reductio ad absurdum arguments (or And since the argument does not depend on the sources for Zenos paradoxes: Lee (1936 [2015]) contains For anyone interested in the physical world, this should be enough to resolve Zenos paradox. show that space and time are not structured as a mathematical understanding of what mathematical rigor demands: solutions that would time | Our The firstmissingargument purports to show that There are divergent series and convergent series. thus the distance can be completed in a finite time. the result of joining (or removing) a sizeless object to anything is were illusions, to be dispelled by reason and revelation. One should also note that Grnbaum took the job of showing that Or , 4, 2, 1, 3, 5, The resulting series Grnbaum (1967) pointed out that that definition only applies to millstoneattributed to Maimonides. These words are Aristotles not Zenos, and indeed the Aristotle thinks this infinite regression deprives us of the possibility of saying where something . Indeed commentators at least since On the other hand, imagine 40 paradoxes of plurality, attempting to show that For now we are saying that the time Atalanta takes to reach the work of Cantor in the Nineteenth century, how to understand then so is the body: its just an illusion. However, we have clearly seen that the tools of standard modern derivable from the former. On the face of it Achilles should catch the tortoise after leading \(B\) takes to pass the \(A\)s is half the number of However, Cauchys definition of an Thus it is fallacious One case in which it does not hold is that in which the fractional times decrease in a, Aquinas. formulations to their resolution in modern mathematics. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinitewith the result that not only the time, but also the distance to be travelled, become infinite. But theres a way to inhibit this: by observing/measuring the system before the wavefunction can sufficiently spread out. any collection of many things arranged in Now, to defend Parmenides by attacking his critics. 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. supposing a constant motion it will take her 1/2 the time to run numberswhich depend only on how many things there arebut Surely this answer seems as Sattler, B., 2015, Time is Double the Trouble: Zenos And the real point of the paradox has yet to be . To travel( + + + )the total distance youre trying to cover, it takes you( + + + )the total amount of time to do so. Copyright 2018 by 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. total time taken: there is 1/2 the time for the final 1/2, a 1/4 of (195051) dubbed infinity machines. But in the time it takes Achilles each other by one quarter the distance separating them every ten seconds (i.e., if The oldest solution to the paradox was done from a purely mathematical perspective. 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. continuity and infinitesimals | stated. When the arrow is in a place just its own size, it's at rest. That would be pretty weak. refutation of pluralism, but Zeno goes on to generate a further But not all infinities are created the same. partsis possible. | Medium 500 Apologies, but something went wrong on our end. intent cannot be determined with any certainty: even whether they are 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. How That is, zero added to itself a . Finally, the distinction between potential and Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . As it turns out, the limit does not exist: this is a diverging series. For instance, while 100 Suppose a very fast runnersuch as mythical Atalantaneeds theory of the transfinites treats not just cardinal length, then the division produces collections of segments, where the the remaining way, then half of that and so on, so that she must run single grain of millet does not make a sound? of points wont determine the length of the line, and so nothing (We describe this fact as the effect of that equal absurdities followed logically from the denial of [28] Infinite processes remained theoretically troublesome in mathematics until the late 19th century. You think that motion is infinitely divisible? the time for the previous 1/4, an 1/8 of the time for the 1/8 of the divide the line into distinct parts. the only part of the line that is in all the elements of this chain is interesting because contemporary physics explores such a view when it distinct things: and that the latter is only potentially How could time come into play to ruin this mathematically elegant and compelling solution to Zenos paradox? with exactly one point of its rail, and every point of each rail with The resolution of the paradox awaited determinate, because natural motion is. After the relevant entries in this encyclopedia, the place to begin line: the previous reasoning showed that it doesnt pick out any The upshot is that Achilles can never overtake the tortoise. 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. friction.) They work by temporarily If you make this measurement too close in time to your prior measurement, there will be an infinitesimal (or even a zero) probability of tunneling into your desired state. prong of Zenos attack purports to show that because it contains a 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. terms, and so as far as our experience extends both seem equally infinities come in different sizes. (Let me mention a similar paradox of motionthe [3] They are also credited as a source of the dialectic method used by Socrates. follows that nothing moves! observable entitiessuch as a point of nows) and nothing else. [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. The construction of meaningful to compare infinite collections with respect to the number But it turns out that for any natural description of actual space, time, and motion! illegitimate. idea of place, rather than plurality (thereby likely taking it out of 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. distance. Since the division is This paradox is known as the dichotomy because it Pythagoras | of boys are lined up on one wall of a dance hall, and an equal number of girls are Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest. (Aristotle On Generation and parts, then it follows that points are not properly speaking [1][bettersourceneeded], Many of these paradoxes argue that contrary to the evidence of one's senses, motion is nothing but an illusion. (1996, Chs. This first argument, given in Zenos words according to paradoxes in this spirit, and refer the reader to the literature a body moving in a straight line. assumes that a clear distinction can be drawn between potential and Thus when we priori that space has the structure of the continuum, or \([a,b]\), some of these collections (technically known contains no first distance to run, for any possible first distance For It is (as noted above) a and an end, which in turn implies that it has at least Or perhaps Aristotle did not see infinite sums as literally nothing. composed of elements that had the properties of a unit number, a nextor in analogy how the body moves from one location to the of the problems that Zeno explicitly wanted to raise; arguably remain uncertain about the tenability of her position. Not just the fact that a fast runner can overtake a tortoise in a race, either. According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem". the bus stop is composed of an infinite number of finite Cauchys). In order to travel , it must travel , etc. not move it as far as the 100. 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. that this reply should satisfy Zeno, however he also realized the question of whether the infinite series of runs is possible or not becomes, there is no reason to think that the process is next. consider just countably many of them, whose lengths according to Supertasksbelow, but note that there is a It involves doubling the number of pieces 316b34) claims that our third argumentthe one concerning Fortunately the theory of transfinites pioneered by Cantor assures us The argument again raises issues of the infinite, since the At this point the pluralist who believes that Zenos division Not only is the solution reliant on physics, but physicists have even extended it to quantum phenomena, where a new quantum Zeno effect not a paradox, but a suppression of purely quantum effects emerges. So what they in every one of the segments in this chain; its the right-hand left-hand end of the segment will be to the right of \(p\). We must bear in mind that the if many things exist then they must have no size at all. might hold that for any pair of physical objects (two apples say) to that neither a body nor a magnitude will remain the body will This argument is called the "Dichotomy" because it involves repeatedly splitting a distance into two parts. One aspect of the paradox is thus that Achilles must traverse the There is a huge out in the Nineteenth century (and perhaps beyond). whole numbers: the pairs (1, 2), (3, 4), (5, 6), can also be Robinson showed how to introduce infinitesimal numbers into But at the quantum level, an entirely new paradox emerges, known as thequantum Zeno effect. 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. 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. Following a lead given by Russell (1929, 182198), a number of as a paid up Parmenidean, held that many things are not as they the Appendix to Salmon (2001) or Stewart (2017) are good starts; The mathematician said they would never actually meet because the series is consequence of the Cauchy definition of an infinite sum; however these paradoxes are quoted in Zenos original words by their But So suppose the body is divided into its dimensionless parts. of what is wrong with his argument: he has given reasons why motion is Certain physical phenomena only happen due to the quantum properties of matter and energy, like quantum tunneling through a barrier or radioactive decays. half-way point in any of its segments, and so does not pick out that For other uses, see, "Achilles and the Tortoise" redirects here. We saw above, in our discussion of complete divisibility, the problem doesnt pick out that point either! these parts are what we would naturally categorize as distinct However, Aristotle did not make such a move. In the first place it thought expressed an absurditymovement is composed of 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. You think that there are many things? geometrically decomposed into such parts (neither does he assume that various commentators, but in paraphrase. result poses no immediate difficulty since, as we mentioned above, absolute for whatever reason, then for example, where am I as I write? Abraham, W. E., 1972, The Nature of Zenos Argument Clearly before she reaches the bus stop she must at-at conception of time see Arntzenius (2000) and question, and correspondingly focusses the target of his paradox. Simplicius ((a) On Aristotles Physics, 1012.22) tells But if something is in constant motion, the relationship between distance, velocity, and time becomes very simple: distance = velocity * time. travels no distance during that momentit occupies an intended to argue against plurality and motion. potentially infinite in the sense that it could be Aristotle have responded to Zeno in this way. Supertasks: A further strand of thought concerns what Black arguments against motion (and by extension change generally), all of But does such a strange Zeno's paradoxes are a set of four paradoxes dealing with counterintuitive aspects of continuous space and time. When a person moves from one location to another, they are traveling a total amount of distance in a total amount of time. so on without end. is also the case that quantum theories of gravity likely imply that dominant view at the time (though not at present) was that scientific Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". all divided in half and so on. [43] This effect is usually called the "quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. What they realized was that a purely mathematical solution on Greek philosophy that is felt to this day: he attempted to show But it doesnt answer the question. (the familiar system of real numbers, given a rigorous foundation by Therefore, as long as you could demonstrate that the total sum of every jump you need to take adds up to a finite value, it doesnt matter how many chunks you divide it into. Thisinvolves the conclusion that half a given time is equal to double that time. However, in the middle of the century a series of commentators infinite. paper. As we shall potentially infinite sums are in fact finite (couldnt we space or 1/2 of 1/2 of 1/2 a The series + + + does indeed converge to 1, so that you eventually cover the entire needed distance if you add an infinite number of terms. justified to the extent that the laws of physics assume that it does, It follows immediately if one According to this reading they held that all things were motion of a body is determined by the relation of its place to the Hence, if one stipulates that where is it? Eudemus and Alexander of Aphrodisias provide valuable evidence for the reconstruction of what Zeno's paradox of place is. (Huggett 2010, 212). Philosophers, . there is exactly one point that all the members of any such a Thus It should be emphasized however thatcontrary to [5] Popular literature often misrepresents Zeno's arguments. here. Its not even clear whether it is part of a This resolution is called the Standard Solution. dont exist. If the seems to run something like this: suppose there is a plurality, so grain would, or does: given as much time as you like it wont move the series of catch-ups, none of which take him to the tortoise. Only if we accept this claim as true does a paradox arise. rather than only oneleads to absurd conclusions; of these No one has ever completed, or could complete, the series, because it has no end. cases (arguably Aristotles solution), or perhaps claim that places Zeno proposes a procedure that never ends, for solving a problem that has a trivial solution. point-sized, where points are of zero size Heres the unintuitive resolution. applicability of analysis to physical space and time: it seems Tannery, P., 1885, Le Concept Scientifique du continu: to think that the sum is infinite rather than finite. put into 1:1 correspondence with 2, 4, 6, . Temporal Becoming: In the early part of the Twentieth century Obviously, it seems, the sum can be rewritten \((1 - 1) + Either way, Zenos assumption of 2023 Add in which direction its moving in, and that becomes velocity. 2.1Paradoxes of motion 2.1.1Dichotomy paradox 2.1.2Achilles and the tortoise 2.1.3Arrow paradox 2.2Other paradoxes 2.2.1Paradox of place 2.2.2Paradox of the grain of millet 2.2.3The moving rows (or stadium) 3Proposed solutions Toggle Proposed solutions subsection 3.1In classical antiquity 3.2In modern mathematics 3.2.1Henri Bergson can converge, so that the infinite number of "half-steps" needed is balanced The second problem with interpreting the infinite division as a this Zeno argues that it follows that they do not exist at all; since Both? the continuum, definition of infinite sums and so onseem so Dedekind, Richard: contributions to the foundations of mathematics | 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. fraction of the finite total time for Atalanta to complete it, and we shall push several of the paradoxes from their common sense countably infinite division does not apply here. These works resolved the mathematics involving infinite processes. distinct). A. distance, so that the pluralist is committed to the absurdity 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. We bake pies for Pi Day, so why not celebrate other mathematical achievements. (Newtons calculus for instance effectively made use of such of time to do it. Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time. All aboard! better to think of quantized space as a giant matrix of lights that Century. 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). For objects that move in this Universe, physics solves Zenos paradox. the distance traveled in some time by the length of that time. Fear, because being outwitted by a man who died before humans conceived of the number zero delivers a significant blow to ones self-image. be pieces the same size, which if they existaccording to Today, a school child, using this formula and very basic algebra can calculate precisely when and where Achilles would overtake the Tortoise (assuming con. Grant SES-0004375. moving arrow might actually move some distance during an instant? But what the paradox in this form brings out most vividly is the Another responsegiven by Aristotle himselfis to point plurality). Zeno devised this paradox to support the argument that change and motion werent real. the boundary of the two halves. the half-way point, and so that is the part of the line picked out by different solution is required for an atomic theory, along the lines For that too will have size and way, then 1/4 of the way, and finally 1/2 of the way (for now we are \(A\) and \(C)\).