Zeno's Paradoxes (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeZeno's ParadoxesFirst published Tue Apr 30, 2002; substantive revision Fri Mar 26, 2004Almost everything that we know about Zeno of Elea is to be found in theopening pages of Plato's Parmenides. There we learn that Zenowas nearly 40 years old when Socrates was a young man, say 20. SinceSocrates was born in 469 BC we can estimate a birth date for Zenoaround 490 BC. Beyond this, really all we know is that he was close toParmenides (Plato reports the gossip that they were lovers when Zenowas young), and that he wrote a book of paradoxes defending Parmenides'philosophy. Sadly this book has not survived, and what we know of hisarguments is second-hand, principally through Aristotle and hiscommentators (here I have drawn particularly on Simplicius, who, thoughwriting a thousand years after Zeno, apparently possessed at least someof his book). There were apparently 40 ‘paradoxes ofplurality’, attempting to show that ontological pluralism — abelief in the existence of many things rather than only one — leads toabsurd conclusions; of these paradoxes only two definitely survive,though a third argument can probably be attributed to Zeno. Aristotlespeaks of a further four arguments against motion (and by extensionchange generally), all of which he gives and attempts to refute. Inaddition Aristotle attributes two other paradoxes to Zeno. Sadly again,almost none of these paradoxes are quoted in Zeno's original words bytheir various commentators, but in paraphrase. 1. Background2. The Paradoxes of Plurality 2.1 The Argument from Denseness 2.2 The Argument from Finite Size 2.3 The Argument from Complete Divisibility 3. The Paradoxes of Motion 3.1 The Dichotomy 3.2 Achilles and the Tortoise 3.3 The Arrow 3.4 The Stadium 4. Two More Paradoxes 4.1 The Paradox of Place 4.2 The Grain of Millet 5. Zeno's Influence on PhilosophyFurther ReadingsBibliographyOther Internet ResourcesRelated Entries1. BackgroundBefore we look at the paradoxes themselves it will be useful to sketchsome of their historical and logical significance. First, Zeno soughtto defend Parmenides by attacking his critics. Parmenides rejectedpluralism and the reality of any kind of change: for him all was oneindivisible, unchanging reality, and any appearances to the contrarywere illusions, to be dispelled by reason and revelation. Notsurprisingly, this philosophy found many critics, who ridiculed thesuggestion; after all it flies in the face of some of our most basicbeliefs about the world. (Interestingly, general relativity —particularly quantum general relativity — arguably provides a novel —if novelty is possible — argument for the Parmenidean denialof change: Belot and Earman, 2001.) In response to this criticism Zenodid something that may sound obvious, but which had a profound impacton Greek philosophy that is felt to this day: he attempted to show thatequal absurdities followed logically from the denial of Parmenides'views. You think that there are many things? Then you must concludethat everything is both infinitely small and infinitely big! You thinkthat motion is infinitely divisible? Then it follows that nothingmoves! (This is what a ‘paradox’ is: a demonstration that acontradiction or absurd consequence follows from apparently reasonableassumptions.) ‘Dialectic’, the technique of arguing for or against aposition by careful logical reasoning — and in particular thetechnique of arguing against a view by showing that it entailsunacceptable consequences — was a crucial innovation, which hasgoverned philosophical method ever since. In the absence of such amethod one can only defend a position by mystical revelation say, or byrhetorical rather than rational appeal, or by force perhaps. Andaccording to Aristotle, Zeno was the inventor of the method (inphilosophy of least, for such an approach has been a part ofmathematics for even longer). Later philosophers however — especiallyPlato and Aristotle — were far finer exponents of the approach.As we read the arguments it is crucial to keep this method in mind.They are always directed towards a more-or-less specific target: theviews of some person or school. We must bear in mind that the argumentsare ad hominem, not in the ‘bad sense’ that theyattack a person rather than his views but in the ‘goodsense’ that they are formulated against a particularphilosopher's assertions. They work by temporarily supposing ‘forargument's sake’ that those assertions are true, and then arguingthat if they are then absurd consequences follow — that nothing movesfor example: they are ‘reductio ad absurdum’arguments. Then, if the argument is logically valid, and the conclusiongenuinely unacceptable, the assertions must be false after all. Thuswhen we look at Zeno's arguments we must ask two related questions:whom or what position is Zeno attacking, and what exactly is assumedfor argument's sake? If we find that Zeno makes hidden assumptionsbeyond what the position under attack commits one to, then the absurdconclusion can be avoided by denying one of the hidden assumptions,while maintaining the position. Indeed commentators at least sinceAristotle have responded to Zeno in this way.So whom do Zeno's arguments attack? There is a huge literaturedebating Zeno's exact historical target. As we shall discuss brieflybelow, some say that the target was a technical doctrine of thePythagoreans, but most today see Zeno as opposing common-sense notionsof plurality and motion. I will approach the paradoxes in this spirit,and refer the reader to the literature concerning the interpretivedebate.That said, it is also the majority opinion that — with certainqualifications — Zeno's paradoxes reveal some problems that cannot beresolved without the full resources of mathematics as worked out in theNineteenth century (and perhaps beyond). This is not (necessarily) tosay that modern mathematics is required to answer any of the problemsthat Zeno explicitly wanted to raise; arguably Aristotle and otherancients had replies that would — or should — have satisfied Zeno.(Nor yet should we conclude that Zeno's work had any direct influenceon the history of mathematics, though surely the kind of worries thathe raised did.) However, as mathematics developed, and more thought wasgiven to the paradoxes, new difficulties arose fromthem; these difficulties require modern mathematics for theirresolution. These new difficulties arise partly in response to theevolution in our understanding of what mathematical rigor demands:solutions that would satisfy Aristotle's standards of rigor would notsatisfy ours. Thus we shall push several of the paradoxes from theircommon sense formulations to their resolution in modern mathematics.(Another qualification: I will offer resolutions in terms of‘standard’ mathematics, but other modern formulations arealso capable of dealing with Zeno.)2. The Paradoxes of Plurality2.1 The Argument from DensenessIf there are many, they must be as many as they are andneither more nor less than that. But if they are as many as they are,they would be limited. If there are many, things that are areunlimited. For there are always others between the things that are, andagain others between those, and so the things that are are unlimited.(Simplicius(a) On Aristotle's Physics, 140.29)This first argument, given in Zeno's words according to Simplicius,attempts to show that there could not be many things, on pain ofcontradiction. Assume then that there are many things. First, he saysthat any collection must contain some definite number of things,neither more nor fewer. But if you have a definite number of things, hefurther concludes, you must have a finite — ‘limited’ —number of them; he implicitly assumes that to have infinitely manythings is not to have any particular number of them. Second, imagineany collection of things arranged in space — imagine them lined up inone dimension for definiteness. Between and two of them, he claims, isa third; and in between these three elements another two; and anotherfour between these five; and so on without end. Therefore the limitedcollection is also ‘unlimited’, which is a contradiction,and hence our original assumption must be false: there are not manythings after all. At least, so Zeno's reasoning runs.But why are there ‘always others between the things thatare’? (In modern terminology, why must objects always be‘densely’ ordered?) Suppose that I had imagined acollection of ten apples lined up; then there is indeed another applebetween the sixth and eighth, but there is none between the seventhand eighth! On the assumption that Zeno is not simply confused, whatdoes he have in mind? There are two possibilities: first, one mighthold that for any pair of physical objects (two apples say) toactually be two distinct objects and not just one (a‘double-apple’) there must be a third between them,physically separating them, even if it is just air. And one mightthink that for these three to be distinct, there must be two moreobjects separating them, and so on (this view presupposes that theirbeing made of different substances is not sufficient to render themdistinct). Second, one might hold that any body has parts that can bedensely ordered. Of course 1/2s, 1/4s, 1/8s and so on of apples arenot dense — some such parts are adjacent — but there maybe sufficiently small parts — call them‘point—parts’ — that are. Indeed, if betweenany two point-parts there lies a finite distance, and if point-partscan be arbitrarily close, then they are dense; a third lies at thehalf-way point of any two. In particular, familiar geometric pointsare like this, and hence are dense.And thus we should read the argument as follows: if you suppose thatthe world contains many things, then you are faced with acontradiction, for the collection must be both finite and infinite —finite because it contains a definite number of things, and infinitebecause they are dense. The assumption that any definite number isfinite seems intuitive, but we now know, thanks to the work of Cantorin the Nineteenth century, how to understand infinite numbers in a waythat makes them just as definite as finite numbers. One central elementof the this theory of the ‘transfinite numbers’ is aprecise definition of when two infinite collections are the same size,and when one is bigger than the other — with such a definition in handit is then possible to order the infinite numbers just as the finitenumbers are ordered. For example, both the fractions and geometricpoints in a line are dense, but there are different, definite infinitenumbers of them. (See Further Reading below for references tointroductions to these mathematical ideas.) Of course, settling themathematical question of whether infinite numbers can be definitedoesn't show that real physical objects actually have geometric pointparts, all it shows is that it is a logical possibility.2.2 The Argument from Finite Size… if it should be added to something else thatexists, it would not make it any bigger. For if it were of no size andwas added, it cannot increase in size. And so it follows immediatelythat what is added is nothing. But if when it is subtracted, the otherthing is no smaller, nor is it increased when it is added, clearly thething being added or subtracted is nothing. (Simplicius(a) OnAristotle's Physics,139.9) But if it exists, each thing must have some size and thickness, andpart of it must be apart from the rest. And the same reasoning holdsconcerning the part that is in front. For that too will have size andpart of it will be in front. Now it is the same thing to say this onceand to keep saying it forever. For no such part of it will be last, norwill there be one part not related to another. Therefore, if there aremany things, they must be both small and large; so small as not to havesize, but so large as to be unlimited. (Simplicius(a) OnAristotle's Physics, 141.2)Once again we have Zeno's own words. According to his conclusion,there are three parts to this argument, but only two survive. Thefirst — missing — argument purports to show that if manythings exist then they must have no size at all. Second, from thisZeno argues that it follows that they do not exist at all; since theresult of joining (or removing) a sizeless object to anything is nochange at all, he concludes that the thing added (or removed) isliterally nothing. The argument to this point is a self-containedrefutation of pluralism, but Zeno goes on to generate a furtherproblem for someone who continues to urge the existence of aplurality. This third part of the argument is rather badly put but itseems to run something like this: suppose there is a plurality, sosome spatially extended object exists (after all, he's just arguedthat inextended things do not exist). Since it is extended, it has twospatially distinct parts (one ‘in front’ of theother). And the parts exist, so they have extension, and so they alsoeach have two spatially distinct parts; and so on without end. Andhence, the final line of argument seems to conclude, the object, if itis extended at all, is infinite in extent.But what could justify this final step? It doesn't seem that becausean object has two parts it must be infinitely big! And neither does itfollow from any other of the divisions that Zeno describes here; four,eight, sixteen, or whatever finite parts make a finite whole. Again,surely Zeno is aware of these facts, and so must have something else inmind, presumably the following: he assumes that if the infinite seriesof divisions he describes were repeated infinitely many times then adefinite collection of parts would result. And notice that he doesn'thave to assume that anyone could actually carry out the divisions —there's not enough time and knives aren't sharp enough — just that anobject can be geometrically decomposed into such parts (neither does heassume that these parts are what we would naturally categorize asdistinct physical objects like apples, cells, molecules, electrons orso on, but only that they are geometric parts of these objects). Now,if — as a pluralist might well accept — such parts exist, it followsfrom the second part of his argument that they are extended, and, heapparently assumes, an infinite sum of finite parts is infinite.Here we should note that there are two ways he may be envisioningthe result of the infinite division.First, one could read him as first dividing the object into 1/2s,then one of the 1/2s — say the second — into two 1/4s, then one ofthe 1/4s — say the second again — into two 1/8s and so on. In thiscase the result of the infinite division results in an endless sequenceof pieces of size 1/2 the total length, 1/4 the length, 1/8 the length… . And then so the total length is (1/2 + 1/4 + 1/8 +…) of the length, which Zeno concludes is an infinite distance,so that the pluralist is committed to the absurdity that finite bodiesare ‘so large as to be unlimited’.What is often pointed out in response is that Zeno gives us no reasonto think that the sum is infinite rather than finite. He might havehad the intuition that any infinite sum of finite quantities, since itgrows endlessly with each new term must be infinite, but one mightalso take this kind of example as showing that some infinite sums areafter all finite. Thus, contrary to what he thought, Zeno has notproven that the absurd conclusion follows. However, what is not alwaysappreciated is that the pluralist is not off the hook so easily, forit is not enough just to say that the sum might be finite,she must also show that it is finite — otherwise weremain uncertain about the tenability of her position. As anillustration of the difficulty faced here consider the following: manycommentators speak as if it is simply obvious that the infinite sum ofthe fractions is 1, that there is nothing to infinite summation. Butwhat about the following sum: 1 − 1 + 1 − 1 + 1 −… . Obviously, it seems, the sum can be rewritten (1− 1) + (1 − 1) + … = 0 + 0 + … = 0. Surelythis answer seems as intuitive as the sum of fractions. But this sumcan also be rewritten 1 − (1 − 1 + 1 − 1 + …)= 1 − 0 — since we've just shown that the term inparentheses vanishes — = 1. Relying on intuitions about how toperform infinite sums leads to the conclusion that 1 = 0. Until onecan give a theory of infinite sums that can give a satisfactory answerto any problem, one cannot say that Zeno's infinite sum is obviouslyfinite. Such a theory was not fully worked out until the Nineteenthcentury by Cauchy. (In Cauchy's system 1/2 + 1/4 + … = 1 but 1− 1 + 1 − … is undefined.)Second, it could be that Zeno means that the object is divided inhalf, then both the 1/2s are both divided in half, then the 1/4s areall divided in half and so on. In this case the pieces at anyparticular stage are all the same finite size, and so one could concludethat the result of carrying on the procedure infinitely would be piecesthe same size, which if they exist — according to Zeno — is greaterthan zero; but an infinity of equal extended parts is indeed infinitelybig.But this line of thought can be resisted. First, suppose that theprocedure just described completely divides the object intonon-overlapping parts. (There is a problem with this supposition thatwe will see just below.) It involves doubling the number of piecesafter every division and so after N divisions there are2N pieces. But it turns out that for any naturalor infinite number, N, 2N >N, and so the number of (supposed) parts obtained by theinfinity of divisions described is an even larger infinity. This is noimmediate difficulty since, as we mentioned above, infinities come indifferent sizes. The number of times everything is divided in two issaid to be ‘countably infinite’: there is a countableinfinity of things in a collection if they can be labeled by thenumbers 1, 2, 3, … without remainder on either side. But thenumber of pieces the infinite division produces is ‘uncountablyinfinite’, which means that there is no way to label them 1, 2,3, … without missing some of them — in fact infinitely many ofthem. However, Cauchy's definition of an infinite sum only applies tocountably infinite series of numbers, and so does not apply to thepieces we are considering. However, we could consider just countablymany of them, whose lengths according to Zeno — since he claims theyare all equal and non-zero — will sum to an infinite length; clearlythe length of all of the pieces could not be less thanthis.At this point the pluralist who believes that Zeno's divisioncompletely divides objects into non-overlapping parts (see the nextparagraph) could respond that the parts in fact have no extension,even though they exist. That would block the conclusion that finiteobjects are infinite, but it seems to push her back to the other hornof Zeno's argument, for how can all these zero length pieces make up anon-zero sized whole? (Note that according to Cauchy 0 + 0 + 0 +… = 0 but this result shows nothing here, for as we saw thereare uncountably many pieces to add up — more than are added in thissum.) We shall postpone this question for the discussion of the nextparadox, where it comes up explicitly.The second problem with interpreting the infinite division as arepeated division of all parts is that it does not divide an objectinto distinct parts, if objects are composed in the natural way. Tosee this, let's ask the question of what parts are obtained by thisdivision into 1/2s, 1/4s, 1/8s, .... Since the division is repeatedwithout end there is no last piece we can give as an answer, and so weneed to think about the question in a different way. If we supposethat an object can be represented by a line segment of unit length,then the division produces collections of segments, where the first iseither the first or second half of the whole segment, the second isthe first or second quarter, or third or fourth quarter, and ingeneral the segment produced by N divisions is either thefirst or second half of the previous segment. For instance, writingthe segment with endpoints a and b as[a,b], some of these collections (technically knownas 'chains' since the elements of the collection are ordered by size)would start {[0,1], [0,1/2], [1/4,1/2], [1/4,3/8], ...}. (When Iargued before that Zeno's division produced uncountably many pieces ofthe object, what I should have said more carefully is that it producesuncountably many chains like this.)The question of which parts the division picks out is then thequestion of which part any given chain picks out; it's natural to saythat a chain picks out the part of the line which is contained inevery one of its elements. Consider for instance the chain{[0,1/2], [1/4,1/2], [3/8,1/2], ...}, in other words the chain thatstarts with the left half of the line and for which every otherelement is the right half of the previous one. The half-way point isin every one of the segments in this chain; it's the right-handendpoint of each one. But no other point is in all its elements:clearly no point beyond half-way is; and pick any point pbefore half-way, if you take right halves of [0,1/2] enough times, theleft-hand end of the segment will be to the right of p. Thusthe only part of the line that is in all the elements of this chain isthe half-way point, and so that is the part of the line picked out bythe chain. The problem is that by parallel reasoning, the half-waypoint is also picked out by the distinct chain {[1/2,1], [1/2,3/4],[1/2,5/8], ...}, where each segment after the first is the left halfof the preceding one. And so both chains pick out the same piece ofthe line: the half-way point. (And many other pairs of chains have thesame problem.) Thus Zeno's Dichotomy, interpreted as a repeateddivision of all parts into half, doesn't divide the line intodistinct parts. Hence, if we think that objects are composed in thesame way as the line, it follows that despite appearances, thisversion of the Dichotomy does not cut objects into parts whose totalsize we can properly discuss.(You might think that this problem could be fixed by taking theelements of the chains to be segments with no endpoint to theright. Then the first of the two chains we considered no longer hasthe half-way point in any of its segments, and so does not pick outthat point. The problem now is that it fails to pick out any part ofthe line: the previous reasoning showed that it doesn't pick out anypoint greater than or less than the half-way point, and now it doesn'tpick out that point either!)2.3 The Argument from Complete Divisibility… whenever a body is by nature divisible through andthrough, whether by bisection, or generally by any method whatever,nothing impossible will have resulted if it has actually been divided… though perhaps nobody in fact could so divide it. What then will remain? A magnitude? No: that is impossible, sincethen there will be something not divided, whereas ex hypothesithe body was divisible through and through. But if it beadmitted that neither a body nor a magnitude will remain … thebody will either consist of points (and its constituents willbe without magnitude) or it will be absolutely nothing. If thelatter, then it might both come-to-be out of nothing and exist as acomposite of nothing; and thus presumably the whole body will benothing but an appearance. But if it consists of points, it will notpossess any magnitude. (Aristotle On Generation andCorruption, 316a19)These words are Aristotle's not Zeno's, and indeed the argument is noteven attributed to Zeno by Aristotle. However we have Simplicius'opinion ((a) On Aristotle's Physics, 139.24) that itoriginates with Zeno, which is why it is included here. Aristotlebegins by hypothesizing that some body is completely divisible,‘through and through’; the second step of the argumentmakes clear that he means by this that it is divisible into parts thatthemselves have no size — parts with any magnitude remainincompletely divided. (Once again what matters is that the body isgenuinely composed of such parts, not that anyone has the time andtools to make the division; and remembering from the previous sectionthat one does not obtain such parts by repeatedly dividing all partsin half.) So suppose the body is divided into its dimensionlessparts. These parts could either be nothing at all — as Zeno arguedabove — or ‘point-parts’. If the parts are nothing thenso is the body: it's just an illusion. And, the argument concludes,even if they are points, since these are unextended the body itselfwill be unextended: surely any sum — even an infinite one — ofzeroes is zero.One could of course point out that it is only assumed that aninfinity of zeroes is itself zero, and deny that assumption. However ithas a strong intuitive pull, and once again one should show how anydimensionless points actually do make an extended whole. FortunatelyGrünbaum (1967) showed how this is possible according to themodern mathematical treatment of a line. Consider a line segment ofunit length. At its most basic level the segment is just a set ofpoints — if you take any spatial part of it, all you have is a pointor set of points. Now Cantor gave a beautiful, astounding and extremelyinfluential ‘diagonal’ proof that the number of points inthe segment is uncountably infinite: there is no way to labelall the points in the line with the infinity of numbers 1, 2,3, … . As we noted above, it follows that we cannot applythe Cauchy definition of infinite sums to the points of the line, andso happily we cannot immediately conclude that because they all havezero length so does the whole line. But that still leaves open thequestion of how the line gets extension from its inextended points.So suppose that you are just given the number of points in a lineand that their lengths are all zero; how would you determine thelength? Do we need a new definition, one that extends Cauchy's touncountably infinite sums? It turns out that that would not help,because Cauchy further showed that any segment, of any lengthwhatsoever (and indeed an entire infinite line) have exactly thesame number of points as our unit segment. So knowing the numberof points won't determine the length of the line, and so nothing likefamiliar addition — in which the whole is determined by the parts —is possible. Instead we must think of the distance properties of a lineas logically posterior to its point composition: first we havea set of points (ordered in a certain way, so that there is some fact,for example, about which of any two is before the other) thenwe define a function of two points which specifies how far apart theyare (and which satisfies such conditions as that the distance betweenA and B plus the distance between B andC equals the distance between A and C —assuming that C is not between A and B). Byanalogy, the maiden names of a married couple do not determine theirsurname: they could take either maiden name or hyphenate, or take awholly new name if they choose. Thus we answer Zeno as follows: theargument assumed that the size of the body was a sum of the sizes ofthe point parts, but that is not the case; according to modernmathematics, a line is an uncountable infinity of points plus adistance function. (Note that Grünbaum used the fact that thepoint composition fails to determine a length to support his‘conventionalist’ view that a line has no determinatelength at all, independent of a standard of measurement.)3. The Paradoxes of Motion3.1 The DichotomyThe first asserts the non-existence of motion on the groundthat that which is in locomotion must arrive at the half-way stagebefore it arrives at the goal. (Aristotle Physics,239b11)This paradox is known as the ‘dichotomy’ because itinvolves repeated division into two (like the second paradox ofplurality). Like the other paradoxes of motion we have it fromAristotle, who sought to refute it.Suppose a very fast runner — such as mythical Atalanta —needs to run for the bus. Clearly before she reaches the bus stop shemust run half-way, as Aristotle says. There's no problem there;supposing a constant motion it will take her 1/2 the time to runhalf-way there and 1/2 the time to run the rest of the way. Now shemust also run half-way to the half-way point — i.e., a 1/4 ofthe total distance — before she reaches the half-way point, butagain she is left with a finite number of finite lengths to run, andplenty of time to do it. And before she reaches 1/4 of the way shemust reach 1/2 of 1/4 = 1/8 of the way; and before that a 1/16; and soon. There is no problem at any finite point in this series, but whatif the halving is carried out infinitely many times? The resultingseries contains no first distance to run, for any possible firstdistance could be divided in half, and hence would not be first afterall. However it does contain a final distance, namely 1/2 of the way;and a penultimate distance, 1/4 of the way; and a third to lastdistance, 1/8 of the way; and so on. Thus the series of distances thatAtalanta is required to run is: …, then 1/16 of the way, then1/8 of the way, then 1/4 of the way, and finally 1/2 of the way (ofcourse we are not suggesting that she stops at the end ofeach segment and then starts running at the beginning of the next— we are thinking of her continuous run being composed of suchparts). And now there is a problem, for this description of her runhas her travelling an infinite number of finitedistances, which, Zeno would have us conclude, must take an infinitetime, which is to say it is never completed. And since the argumentdoes not depend on the distance or who or what the mover is, itfollows that no finite distance can ever be traveled, which is to saythat all motion is impossible. (Note that the paradox could easily begenerated in the other direction so that Atalanta must first run halfway, then half the remaining way, then half of that and so on, so thatshe must run the following endless sequence of fractions of the totaldistance: 1/2, 1/4, 1/8 ….)A couple of common responses are not adequate. One might — asSimplicius ((a) On Aristotle's Physics, 1012.22) tells usDiogenes the Cynic did by silently standing and walking — point outthat it is a matter of the most common experience that things in factdo move, and that we know very well that Atalanta would have notrouble reaching her bus stop. But this would not impress Zeno, who asa paid up Parmenidean held that many things are not as they appear: itmay appear that Diogenes is walking or that Atalanta is running, butappearances can be deceptive and surely we have a logical proof thatthey are in fact not moving at all. And if one doesn't accept thatZeno has given a proof that motion is illusory — as we hopefully donot — one then owes an account of what is wrong with his argument: hehas given reasons why motion is impossible, and so an adequateresponse must show why those reasons are not sufficient. And it won'tdo simply to point out that there are some ways of cutting upAtalanta's run — into just two halves, say — in which there is noproblem. For if you accept all of the steps in Zeno's argument thenyou must accept his conclusion (assuming that he has reasoned in alogically deductive way): it's not enough to show an unproblematicdivision, you must also show why the given division isunproblematic.Another response — given by Aristotle himself — is to point out thatas we divide the distances run we should also divide the total timetaken: there is 1/2 the time for the final 1/2, a 1/4 of the time forthe previous 1/4, an 1/8 of the time for the 1/8 of the run and soon. Thus each fractional distance has just the right fraction of thefinite total time for Atalanta to complete it, and thus the distancecan be completed in a finite time. Aristotle felt that this replyshould satisfy Zeno, however he also realized (Physics,263a15) that this could not be the end of the matter (and surely Zenowould have made the same point if presented with Aristotle'sresponse). For now we are saying that the time Atalantatakes to reach the bus stop is composed of an infinite number offinite pieces — …, 1/8, 1/4, and 1/2 (of the total time) —and isn't that an infinite time?Of course, one could again claim that some infinite sums in fact havefinite totals, and in particular that the sum of these pieces is 1× the total time, which is of course finite (and again acomplete solution would demand a rigorous account of infinitesummation, like Cauchy's). However, Aristotle did not make such amove. What he said is worth noting because it had a considerableinfluence on later thinking about Zeno. In his response Aristotle drewa sharp distinction between what he termed a ‘continuous’line and a line divided into parts. Consider a simple division of aline into two: on the one hand there is the undivided line, and on theother the line with a mid-point selected as the boundary of the twohalves. Aristotle claims that these are two distinct things: and thatthe later is only ‘potentially’ derivable from theformer. Next, Aristotle takes the common-sense view that time is likea geometric line, and considers the time it takes to complete therun. We can again distinguish the two cases: on the one hand there isthe continuous run from start to finish, and on the other there is therun divided into Zeno's infinity of half-runs. The former is‘potentially infinite’ in the sense that it could bedivided into latter ‘actual infinity’. Here's the crucialstep: Aristotle thinks that since these times aregeometrically distinct they must be physicallydistinct. But how could that be? He claims that the runner must dosomething at the end of each half-run to make it distinct from thenext: she must stop. (Why stop rather than cough or something? Becauseif the time is discontinuous then so is the motion.) And soAristotle's full answer to the paradox is that Zeno's question —whether the infinite series of runs is possible or not — isambiguous. One the one hand, the answer is ‘yes’ if onemeans the potentially infinite series that form the continuous run. Onthe other the answer is ‘no’ if one means the actualinfinity of pieces that form the discontinuous run.It is hard — from our modern perspective perhaps — to see how thisanswer could be completely satisfactory. In the first place it assumesthat a clear distinction can be drawn between potential and actualinfinities, something that was never fully achieved. Second, supposethat Zeno's problem turns on the claim that infinite sums of finitequantities are invariably infinite. Then Aristotle's distinction willonly help if he can explain why potentially infinite sums are in factfinite (and couldn't I potentially add 1 + 1 + 1 + …, which doesnot have a finite total); or if he can give a reason why potentiallyinfinite sums just don't exist. Or perhaps Aristotle did not seeinfinite sums as the problem, but rather whether completing an infinityof finite actions is metaphysically and conceptually and physicallypossible, an idea discussed at length in recent years: see‘Supertasks’ below. In this case we need an account ofactions that makes precise the sense in which the continuous run isindeed a single action (using rest to individuate motions seemsproblematic, for humans are probably never completely still, and yet weperform distinct motions — breathing, eating, skipping and so on).Finally, the distinction between potential and actual infinities hasplayed no role in mathematics since Cantor tamed the transfinitenumbers — certainly the potential infinite has played no role in themodern mathematical solutions discussed here.One last point: Zeno's argument seeks most obviously to establishthe impossibility of motion, but he also intended it (and the followingarguments) as further refutations of plurality — certainly, Platointerprets Zeno's intentions in this way. How might the argument seekto establish this conclusion? Presumably Zeno has in mind the view thatspatial (and perhaps temporal) distances have a plurality of parts;parts which are infinitely divisible into two. Given that assumption,supposedly finite distances (or times) can be decomposed into aninfinity of finite parts with no first (or alternatively, last) one.And how can such distances be finite after all? And if the pluralistalso believes in motion, how can such a distance be traversed? It seemsit cannot be.3.2 Achilles and the TortoiseThe [second] argument was called "Achilles," accordingly,from the fact that Achilles was taken [as a character] in it, and theargument says that it is impossible for him to overtake the tortoisewhen pursuing it. For in fact it is necessary that what is to overtake[something], before overtaking [it], first reach the limit from whichwhat is fleeing set forth. In [the time in] which what is pursuingarrives at this, what is fleeing will advance a certain interval, evenif it is less than that which what is pursuing advanced … .And in the time again in which what is pursuing will traverse this[interval] which what is fleeing advanced, in this time again what isfleeing will traverse some amount … . And thus in everytime in which what is pursuing will traverse the [interval] which whatis fleeing, being slower, has already advanced, what is fleeing willalso advance some amount. (Simplicius(b) On Aristotle'sPhysics, 1014.10)This paradox turns on much the same considerations as the last.Imagine Achilles chasing a tortoise, and suppose that Achilles isrunning at 1 m/s, that the tortoise is crawling at 0.1m/s and that the tortoise starts out 0.9 m ahead ofAchilles. On the face of it Achilles should catch the tortoise after1s, at a distance of 1m from where he starts (and so0.1m from where the Tortoise starts). We could break Achilles'motion up as we did Atalanta's, into halves, or we could do it asfollows: before Achilles can catch the tortoise he must reach the pointwhere the tortoise started. But in the time he takes to do this thetortoise crawls a little further forward. So next Achilles must reachthis new point. But in the time it takes Achilles to achieve this thetortoise crawls forward a tiny bit further. And so on to infinity:every time that Achilles reaches the place where the tortoise was thetortoise has had enough time to get a little bit further, and soAchilles has another run to make, and so Achilles has in infinitenumber of finite catch-ups to do before he can catch the tortoise, andso, Zeno concludes, he never catches the tortoise.One aspect of the paradox is thus that Achilles must traverse thefollowing infinite series of distances before he catches the tortoise:first 0.9m, then an additional 0.09m, then0.009m, … . These are the series of distancesahead that the tortoise reaches at the start of each of Achilles'catch-ups. Looked at this way the puzzle is identical to the Dichotomy,for it is just to say that ‘that which is in locomotion mustarrive [nine tenths of the way] before it arrives at the goal’.And so everything we said above applies here too.But what the paradox in this form brings out most vividly is theproblem of completing a series of actions that has no final member —in this case the infinite series of catch-ups before Achilles reachesthe tortoise. But just what is the problem? Perhaps the following:Achilles' run to the point at which he should reach the tortoise can,it seems, be completely decomposed into the series of catch-ups, noneof which take him to the tortoise. Therefore, nowhere in his run doeshe reach the tortoise after all. But if this is what Zeno had in mindit won't do. Of course Achilles doesn't reach the tortoise at any pointof the sequence, for every run in the sequence occurs beforewe expect Achilles to reach it! Thinking in terms of the points thatAchilles must reach in his run, 1m does not occur in the sequence0.9m, 0.99m, 0.999m, … , so ofcourse he never catches the tortoise during that sequence of runs! Theseries of catch-ups does not after all completely decompose the run:the final point — at which Achilles does catch the tortoise — must beadded to it. So is there any puzzle? Arguably yes.Achilles run passes through the sequence of points 0.9m,0.99m, 0.999m, … , 1m. But doessuch a strange sequence — comprised of an infinity of members followedby one more — make sense mathematically? If not then our mathematicaldescription of the run cannot be correct, but then what is? Fortunatelythe theory of transfinites pioneered by Cantor assures us that such aseries is perfectly respectable. It was realized that the orderproperties of infinite series are much more elaborate than those offinite series. Any way of arranging the numbers 1, 2 and 3 gives aseries in the same pattern, for instance, but there are many distinctways to order the natural numbers: 1, 2, 3, … for instance. Or… , 3, 2, 1. Or … , 4, 2, 1, 3, 5,… . Or 2, 3, 4, … , 1, which is just the samekind of series as the positions Achilles must run through. Thus thetheory of the transfinites treats not just ‘cardinal’numbers — which depend only on how many things there are — but also‘ordinal’ numbers which depend further on how the thingsare arranged. Since the ordinals are standardly taken to bemathematically legitimate numbers, and since the series of pointsAchilles must pass has an ordinal number, we shall take it that theseries is mathematically legitimate. (Again, see‘Supertasks’ below for another kind of problem that mightarise for Achilles’.)3.3 The ArrowThe third is … that the flying arrow is at rest,which result follows from the assumption that time is composed ofmoments … . he says that if everything when it occupies anequal space is at rest, and if that which is in locomotion is always ina now, the flying arrow is therefore motionless. (AristotlePhysics, 239b.30) Zeno abolishes motion, saying "What is in motion moves neither inthe place it is nor in one in which it is not". (Diogenes LaertiusLives of Famous Philosophers, ix.72)This argument against motion explicitly turns on a particular kindof assumption of plurality: that time is composed of moments (or‘nows’) and nothing else. Consider an arrow,apparently in motion, at any instant. First, Zeno assumes that ittravels no distance during that moment — ‘it occupies an equalspace’ for the whole instant. But the entire period of its motioncontains only instants, all of which contain an arrow at rest, and so,Zeno concludes, the arrow cannot be moving.An immediate concern is why Zeno is justified in assuming that thearrow is at rest during any instant. It follows immediately if oneassumes that an instant lasts 0s: whatever speed the arrow has, it willget nowhere if it has no time at all. But what if one held that thesmallest parts of time are finite — if tiny — so that a moving arrowmight actually move some distance during an instant? One way ofsupporting the assumption — which requires reading quite a lot intothe text we have — is to assume that instants are indivisible. Thensuppose that an arrow actually moved during an instant. It would be atdifferent locations at the start and end of the instant, which impliesthat the instant has a ‘start’ and an ‘end’,which in turn implies that it has at least two parts, and so isdivisible, and so is not an indivisible moment at all. (Note that thisargument only establishes that nothing can move during an instant, notthat instants cannot be finite.)So then, nothing moves during any instant, but time is entirelycomposed of instants, so nothing ever moves. A first response is topoint out that determining the velocity of the arrow means dividing thedistance traveled in some time by the length of that time. But —assuming from now on that instants have zero duration — this formulamakes no sense in the case of an instant: the arrow travels 0min the 0s the instant lasts, but 0/0 m/s is not anynumber at all. Thus it is fallacious to conclude from the fact that thearrow doesn't travel any distance in an instant that it is at rest;whether it is in motion at an instant or not depends on whether ittravels any distance in a finite interval that includes theinstant in question.The answer is correct, but it carries the counter-intuitiveimplication that motion is not something that happens at any instant,but rather only over finite periods of time. Think about it this way:time, as we said, is composed only of instants. No distance is traveledduring any instant. So when does the arrow actually move? How does itget from one place to another at a later moment? There's only oneanswer: the arrow gets from point X at time 1 to pointY at time 2 simply in virtue of being at successiveintermediate points at successive intermediate times — the arrow neverchanges its position during an instant but only over intervals composedof instants, by the occupation of different positions at differenttimes. In Bergson's memorable words — which he thought expressed anabsurdity — ‘movement is composed of immobilities’ (1911,308): getting from X to Y is a matter of occupyingexactly one place in between at each instant (in the right order ofcourse). For a recent discussion of this issue see Arntzenius (2000).3.4 The StadiumThe fourth argument is that concerning equal bodies[AA] which move alongside equal bodies in the stadium fromopposite directions — the ones from the end of the stadium[CC], the others from the middle [BB] — at equalspeeds, in which he thinks it follows that half the time is equal toits double…. And it follows that the C has passed allthe As and the B half; so that the time is half… . And at the same time it follows that the firstB has passed all the Cs. (Aristotle Physics,239b33)The final paradox of motion runs as follows: picture three sets ofthree touching cubes — all nine exactly the same, with side Lm — in relative motion. One set — the As— are at rest, and the others — the Bs andCs — move from the left and right respectively, at aconstant equal speed, S m/s. And suppose that at some momentthe rightmost B is perfectly aligned with the middleA, and the leftmost C with the rightmost A:thus the edges of the rightmost B and leftmost C areexactly lined up. That is they are arranged as shown. AAA BBB CCCNow consider the later time at which the rightmost A andB are aligned; since the speeds of the Bs andCs are equal, at this moment the middle A will bealigned with the leftmost C. That is consider the moment whenthe blocks are configured thus. AAA BBB CCC This motion requires the rightmost B to move one block — adistance L m — to the right, at a speed of S m/s, soit takes L/S s. And the same motion also requires the leftmostC to move from just to the right of the rightmost Binto alignment with the middle B, a distance a distance of2L m. So far so good, but now Zeno concludes thatsince the Cs are moving at S m/s, the motion mustalso take 2L/S s. And hence ‘half the time[L/S] is equal to its double [2L/S]’, since oneand the same motion seems to take both times.The unanimous verdict on Zeno is that he was hopelessly confusedabout relative velocity in this paradox. If the Bs are movingwith speed S m/s to the right with respect to the As and ifthe Cs are moving with speed S m/s to the left withrespect to the As then the Cs are moving with speedS+S = 2S m/s to the left with respect to theBs. And so, as expected it takes the Cs2L/2S = L/S s to complete the motion afterall.This resolution notwithstanding, recent philosophers have attemptedto put a new spin on Zeno's argument (it's arguable whether Zenohimself had anything like what follows in mind). This argument opposedthe view that space and time are ‘quantized’, composed ofsmallest finite parts. Suppose they are and that L m is the‘quantum’ of length and that the two moments considered areseparated by a single quantum of time. Now something strange hashappened, for the rightmost B and the leftmost C haveclearly passed each other during the motion, and yet there is no momentat which they are level: since the two moments are separated by thesmallest possible time, there can be no moment between them — it wouldbe a time smaller than the smallest time from the two moments weconsidered. Conversely, if one insisted that if they pass then theremust be a moment when they are level, then it shows that cannot be ashortest finite interval — whatever it is, just run this argumentagainst it. However, why should one insist on this assumption? Theproblem is that one naturally imagines quantized space as being like achess board, on which the chess pieces are frozen during each quantumof time. Then one wonders when the red queen, say, gets from one squareto the next, or how she gets past the white queen without being levelwith her. But the analogy is misleading. It is better to think ofquantized space as a giant matrix of lights that holds some pattern ofilluminated lights for each quantum of time. In this analogy a lit bulbrepresents the presence of an object: for instance a series of bulbs ina line lighting up in sequence represent a body moving in a straightline. In this case there is no temptation to ask when the light‘gets’ from one bulb to the next — or in analogy how thebody moves from one location to the next.4. Two More Paradoxes4.1 The Paradox of PlaceZeno's difficulty demands an explanation; for if everythingthat exists has a place, place too will have a place, and so on adinfinitum. (Aristotle Physics, 209a23)When he sets up his theory of place — the crucial spatial notion inhis theory of motion — Aristotle lists various theories and problemsthat his predecessors, including Zeno, have formulated on the subject.One can again see here a problem for pluralism, for the second step ofthe argument concludes that there are many places. It is perhaps alittle hard to feel the full force of the conclusion, for why shouldthere not be an infinite series of places of places of places of…? Presumably the worry would be greater for someone who (likeAristotle) believed that there could not be an actual infinity ofthings, for the argument seems to show that there are. But certainlytoday we need have no such qualms; there seems nothing problematic withan actual infinity of places; indeed, it seems very natural to thinkthat every point of space is a distinct place, even if there are aninfinity of points.The only other way one might find the regress troubling is if onehad reason to suppose that objects must have ‘absolute’places, in the sense that there is always a unique answer to thequestion ‘where is it’? For example, where am I as I write?If the paradox is right then I'm in my place, and I'm also in myplace's place, and my place's place's place, and my … .Since I'm in all these places any might seem an appropriate answer tothe question. But why think that there must be a unique answer to thequestion? Why shouldn't I have many locations? At my desk, in myapartment, in Chicago, Illinois, USA, North America, the Earth, SolarSystem … . (In fact there is a reason that Aristotle mighthave had this concern about Zeno's argument, for in his theory ofmotion, the natural motion of a body is determined by the relation ofits place to the center of the universe: an account that only makessense if bodies can be attributed a unique place. Interestingly,Newton, in the Scholium to the principal Definitions in Book I of hisPrincipia, gives an argument along similar lines: he assumesthat every body has a unique, absolute velocity, and argues that only afixed matter-independent, ‘absolute’ space will providesuch uniqueness. That said, there is no evidence either that Zeno hadthis kind of argument in mind, or that Newton was influenced byAristotle in this regard.)4.2 The Grain of Millet… Zeno's reasoning is false when he argues thatthere is no part of the millet that does not make a sound; for there isno reason why any part should not in any length of time fail to movethe air that the whole bushel moves in falling. (AristotlePhysics, 250a19)This argument is a Parmenidean argument against the reliability ofthe senses. It goes like this: if you drop a sack of millet on thefloor then you hear a loud thud; but this noise is the result of thenoise made by every grain of millet in the sack; and the result of thenoise made by every part of every grain; therefore every part of everygrain makes a noise as it hits the ground. But now consider dropping atiny part of a grain; we know that we won't hear it. Therefore oursense of hearing is deceptive — there are noises it cannot hear — andso we should not trust it. Aristotle's response seems to be that thepart would not move as much air as the sack, but the paradox is notthat the part should make as much noise as the sack, but that it shouldmake some noise. A better reply is surely that not every disturbance inthe air is audible by us: that a measuring instrument is unreliableover some range is no argument that it is unreliable over everyrange.5. Zeno's Influence on PhilosophyIn this final section we should consider briefly the impact that Zenohas had on various philosophers; a search of the literature will revealthat these debates continue. The Pythagoreans: For the first half of the Twentieth century themajority reading — following Tannery (1885) — of Zeno held that hisarguments were directed against a technical doctrine of thePythagoreans. According to this reading they held that all things werecomposed of elements that had the properties of a unit number, ageometric point and a physical atom: this kind of position would fitwith their doctrine that reality is fundamentally mathematical.However, in the middle of the century a series of commentators(Vlastos, 1967, summarizes the argument and contains references)forcefully argued that Zeno's target was instead a common senseunderstanding of plurality and motion — one grounded in familiargeometrical notions — and indeed that the doctrine was not a majorpart of Pythagorean thought. We have implicitly assumed that thesearguments are correct in our readings of the paradoxes. That said,Tannery's interpretation still has its defenders (see e.g., Matson2001).The Atomists: Aristotle (On Generation and Corruption316b34) claims that our third argument — the one concerning completedivisibility — was what convinced the atomists that there must besmallest, indivisible parts of matter. See Abraham (1972) for a furtherdiscussion of Zeno's connection to the atomists.Temporal Becoming: In the early part of the Twentieth centuryseveral influential philosophers attempted to put Zeno's arguments towork in the service of a metaphysics of ‘temporalbecoming’, the (supposed) process by which the present comes intobeing. Such thinkers as Bergson (1911), James (1911, Ch 10 —11) andWhitehead (1929) argued that Zeno's paradoxes show that space and timeare not structured as a mathematical continuum: they argued that theway to preserve the reality of motion was to deny that space and timeare composed of points and instants. However, we have clearly seen thatthe tools of standard modern mathematics are up to the job of resolvingthe paradoxes, so no such conclusion seems warranted: if the presentindeed ‘becomes’, there is no reason to think that theprocess is not captured by the continuum.Applying the Mathematical Continuum to Physical Space and Time:Following a lead given by Russell (1929, 182-198), a number ofphilosophers — most notably Grünbaum (1967) — took up the taskof showing how modern mathematics could solve all of Zeno's paradoxes;their work has thoroughly influenced our discussion of the arguments.What they realized was that a purely mathematical solution was notsufficient: the paradoxes not only question abstract mathematics, butalso the nature of physical reality. So what they sought was anargument not only that Zeno posed no threat to the mathematics ofinfinity but also that that mathematics correctly describes objects,time and space. The idea that a mathematical law — say Newton's law ofuniversal gravity — may or may not correctly describe things isfamiliar, but some aspects of the mathematics of infinity — the natureof the continuum, definition of infinite sums and so on — seem sobasic that it may be hard to see at first that they too applycontingently. But surely they do: nothing guarantees apriori that space has the structure of the continuum, oreven that parts of space add up according to Cauchy's definition.(Salmon offers a nice example to help make the point: since alcoholdissolves in water, if you mix the two you end up with less than thesum of their volumes, showing that even ordinary addition is notapplicable to every kind of system.) Our belief that the mathematicaltheory of infinity describes space and time is justified to the extentthat the laws of physics assume that it does, and to the extent thatthose laws are themselves confirmed by experience. While it is truethat almost all physical theories assume that space and time do indeedhave the structure of the continuum, it is also the case that quantumtheories of gravity likely imply that they do not. While no one reallyknows where this research will ultimately lead, it is quite possiblethat space and time will turn out, at the most fundamental level, to bequite unlike the mathematical continuum that we have assumed here.One should also note that Grünbaum took the job ofshowing that modern mathematics describes space and time to involvesomething rather different from arguing that it is confirmed byexperience. The dominant view at the time (though not at present) wasthat scientific terms had meaning insofar as they referred directly toobjects of experience — such as ‘1 m ruler’ — or, if theyreferred to ‘theoretical’ rather than‘observable’ entities — such as ‘a point ofspace’ or ‘1/2 of 1/2 of … 1/2 a racetrack’ —then they obtained meaning by their logical relations — viadefinitions and theoretical laws — to such observation terms. ThusGrünbaum undertook an impressive program to give meaning to allterms involved in the modern theory of infinity, interpreted as anaccount of space and time.Supertasks: A further strand of thought concerns what Black(1950-51) dubbed ‘infinity machines’. Black and hisfollowers wished to show that although Zeno's paradoxes offered noproblem to mathematics, they showed that after all mathematics was notapplicable to space, time and motion. Most starkly, our resolution tothe Dichotomy and Achilles assumed that the complete run could bebroken down into an infinite series of half runs, which could besummed. But is it really possible to complete any infinite series ofactions: to complete was is known as a ‘supertask’? If not,and assuming that Atalanta and Achilles can complete their tasks, theircomplete runs cannot be correctly described as an infinite series ofhalf-runs, although modern mathematics would so describe them. Whatinfinity machines are supposed to establish is that an infinite seriesof tasks cannot be completed — so any completable task cannot bebroken down into an infinity of smaller tasks, whatever mathematicssuggests.Non-standard analysis: Finally, we have seen how to tackle theparadoxes using the resources of mathematics as developed in theNineteenth century. For a long time it was considered one the greatvirtues of this system that it finally showed how to do withoutinfinitesimal quantities, smaller than any finite number but largerthan zero. (Newton's calculus for instance effectively made use of suchnumbers, treating them sometimes as zero and sometimes as finite; theproblem with such an approach is that how to treat the numbers is amatter of intuition not rigor.) However, in the Twentieth centuryRobinson showed how to introduce infinitesimal numbers intomathematics: this is the system of ‘non-standard analysis’(the familiar system of real numbers, given a rigorous foundation byDedekind, is by contrast just ‘analysis’). And it has beenshown by McLaughlin (1992, 1994) that Zeno's paradoxes can also beresolved in non-standard analysis; they are no more argument againstnon-standard analysis than the standard mathematics we have assumedhere. It should be emphasized however that — contrary to McLaughlin'ssuggestions — there is no need for non-standard analysis to solve theparadoxes: either system is equally successful. (The construction ofnon-standard analysis does however raise a further question about theapplicability of analysis to physical space and time: it seemsplausible that all physical theories can be formulated in either terms,and so as far as our experience extends both seem equally confirmed.But they cannot both be true of space and time: either space hasinfinitesimal parts or it doesn't.)Further ReadingsAfter the relevant entries in this encyclopedia, the place to begin anyfurther investigation is Salmon (2001), which contains some of the mostimportant articles on Zeno up to 1970, and an impressivelycomprehensive bibliography of works in English in the Twentieth Century. One might also take a look at Huggett (1999, Ch. 3) for furthersource passages and discussion. Forintroductions to the mathematical ideas behind the modern resolutions,the Appendix to Salmon (2001) is a good start; Russell (1919) andCourant et al. (1996, Chs. 2 and 9) are also both wonderfulsources. Finally, three collections of original sources for Zeno'sparadoxes: Lee (1936) contains everything known, Kirk et al(1983, Ch. 9) contains a great deal of material (in English and Greek)with useful commentaries, and Cohen et al. (1995) also has themain passages.BibliographyAbraham, W. E., 1972, ‘The Nature of Zeno's Argument AgainstPlurality in DK 29 B I’, Phronesis 17: 40-52Aristotle, 1984, ‘ On Generation and Corruption’, A. A.Joachim (trans), in The Complete Works of Aristotle, J. Barnes(ed.), Princeton: Princeton University PressAristotle, 1984, ‘Physics’, W. D. Ross(trans), inThe Complete Works of Aristotle, J. Barnes (ed.), Princeton:Princeton University PressArntzenius, F., 2000, ‘Are There Really InstantaneousVelocities?’, The Monist, 83: 187-208Belot, G. and Earman, J., 2001, ‘Pre-Socratic QuantumGravity’, in Physics Meets Philosophy at the Planck Scale:Contemporary Theories in Quantum Gravity, C. Callender and N.Huggett (eds), Cambridge: Cambridge University PressBergson, H., 1911, Creative Evolution, A. Mitchell(trans.), New York: Holt, Reinhart and WinstonBlack, M., 1950, ‘Achilles and the Tortoise’,Analysis, 11: 91-101Cohen, S. M., Curd, P. and Reeve, C. D. C. (eds), 1995,Readings in Ancient Greek Philosophy From Thales to Aristotle,Indianapolis/Cambridge: Hackett Publishing Co. Inc.Courant, R., Robbins, H., and Stewart, I., 1996, What isMathematics? An Elementary Approach to Ideas and Methods, 2ndEdition, New York/Oxford: Oxford University PressDiogenes Laertius, 1983, ‘Lives of FamousPhilosophers’, p.273 of The Presocratic Philosophers: ACritical History with a Selection of Texts, 2nd Edition, G. S.Kirk, J. E. Raven and M. Schofield (eds), Cambridge, UK: CambridgeUniversity PressGrünbaum, A., 1967, Modern Science and Zeno'sParadoxes, Middletown: Connecticut Wesleyan University PressHuggett, N. (ed.), 1999, Space from Zeno to Einstein: ClassicReadings with a Contemporary Commentary, Cambridge, MA: MITPressJames, W., 1911, Some Problems of Philosophy, New York:Longmans, Green & Co.Lee, H. D. P. (ed.), 1967, Zeno of Elea, Amsterdam: AdofHakkertKirk, G. S., Raven J. E. and Schofield M. (eds), 1983, ThePresocratic Philosophers: A critical History with a Selection ofTexts, 2nd Edition, Cambridge, UK: Cambridge University PressMcLaughlin, W. I., and Miller, S. L., 1992, ‘AnEpistemological Use of Nonstandard Analysis to Answer Zeno's Objectionsagainst Motion’, Synthese, 92: 371-384McLaughlin, W. I., 1994, ‘Resolving Zeno's Paradoxes’,Scientific American, November: 84-89Matson, W. I., 2001, ‘Zeno Moves!’, in Essays inAncient Greek Philosophy VI: Before Plato, A. Preus (ed.), Albany:SUNY PressNewton, I., 1999, The Principia: Mathematical Principles ofNatural Philosophy, I. B. Cohen and A. M. Whitman (trans.),Berkeley: University of California PressPlato, 1997, ‘Parmenides’, M. L. Gill and P. Ryan(trans), in Plato: Complete Works, J. M. Cooper (ed.),Indianapolis/Cambridge: Hackett Publishing Co. Inc.Russell, B., 1929, Our Knowledge of the External World,New York: W. W. Norton & Co. Inc.Russell, B., 1919, Introduction to MathematicalPhilosophy, London: George Allen and Unwin LtdSalmon, W. C., 2001, Zeno's Paradoxes, 2nd Edition,Indianapolis: Hackett Publishing Co. Inc.Simplicius(a), 1995, ‘On Aristotle's Physics’, inReadings in Ancient Greek Philosophy From Thales to Aristotle,S. M. Chohen, P. Curd and C. D. C. Reeve (eds), Indianapolis/Cambridge:Hackett Publishing Co. Inc. 58-59Simplicius(b), 1989, On Aristotle's Physics 6, D. Konstan(trans.), London: Gerald Duckworth & Co. LtdTannery, P., 1885, ‘Le Concept Scientifique du continu: Zenond'Elee et Georg Cantor’, Revue Philosophique de la France etde l'Etranger, 20: 385Vlastos, G., 1967, ‘Zeno of Elea’, in TheEncyclopedia of Philosophy, P. Edwards (ed.), New York: TheMacmillan Co. and The Free PressWhitehead, A. N., 1929, Process and Reality, New York: TheMacmillan Co.Other Internet Resources[Please contact the author with suggestions.] Related Entries Aristotle | atomism: ancient | Cantor, Georg | Dedekind, Richard | infinity | Parmenides | Plato | Pythagoras | quantum mechanics | Simplicius | space and time: supertasks | timeAcknowledgmentsThis entry is dedicated to the late Wesley Salmon, who did so much toeducate philosophers about the significance of Zeno's paradoxes. Thosefamilar with his work will see that this discussion owes a great dealto him; I hope that he would find it satisfactory. This material isbased upon work supported by National Science Foundation GrantSES-0004375. I would also like to thank Eliezer Dorr for bringing tomy attention some problems with my original formulation of theDichotomy. Copyright © 2004 byNick Huggett<huggett@uic.edu> |
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