Hence, if we think that objects are composed in the same way as the line, it follows that despite appearances, this version of the argument does not cut objects into parts whose total size we can properly discuss.
You might think that this problem could be fixed by taking the elements of the chains to be segments with no endpoint to the right. Then the first of the two chains we considered no longer has the half-way point in any of its segments, and so does not pick out that point.
What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesi the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain … the body will either consist of points and its constituents will be without magnitude or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance.
But if it consists of points, it will not possess any magnitude. Aristotle On Generation and Corruption , a Once again what matters is that the body is genuinely composed of such parts, not that anyone has the time and tools to make the division; and remembering from the previous section that one does not obtain such parts by repeatedly dividing all parts in half.
So suppose the body is divided into its dimensionless parts. And, the argument concludes, even if they are points, since these are unextended the body itself will be unextended: surely any sum—even an infinite one—of zeroes is zero. Could that final assumption be questioned? There is no way to label all the points in the line with the infinity of numbers 1, 2, 3, … , and so there are more points in a line segment than summands in a Cauchy sum. In short, the analysis employed for countably infinite division does not apply here.
So suppose that you are just given the number of points in a line and that their lengths are all zero; how would you determine the length? It turns out that that would not help, because Cauchy further showed that any segment, of any length whatsoever and indeed an entire infinite line have exactly the same number of points as our unit segment. Thus we answer Zeno as follows: the argument assumed that the size of the body was a sum of the sizes of point parts, but that is not the case; according to modern mathematics, a geometric line segment is an uncountable infinity of points plus a distance function.
Hence, if one stipulates that the length of a line is the sum of any complete collection of proper parts, then it follows that points are not properly speaking parts of a line unlike halves, quarters, and so on of a line.
Like the other paradoxes of motion we have it from Aristotle, 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 she must run half-way, as Aristotle says. Now she must also run half-way to the half-way point—i. There is no problem at any finite point in this series, but what if the halving is carried out infinitely many times? The resulting series contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. And now there is a problem, for this description of her run has her travelling an infinite number of finite distances, which, Zeno would have us conclude, must take an infinite time, which is to say it is never completed.
And since the argument does not depend on the distance or who or what the mover is, it follows that no finite distance can ever be traveled, which is to say that all motion is impossible.
A couple of common responses are not adequate. But this would not impress Zeno, who, as a paid up Parmenidean, held that many things are not as they appear: it may appear that Diogenes is walking or that Atalanta is running, but appearances can be deceptive and surely we have a logical proof that they are in fact not moving at all.
Thus each fractional distance has just the right fraction of the finite total time for Atalanta to complete it, and thus the distance can be completed in a finite time. Aristotle felt that this reply should satisfy Zeno, however he also realized Physics , a15 that it could not be the end of the matter.
However, Aristotle did not make such a move. Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Next, Aristotle takes the common-sense view that time is like a geometric line, and considers the time it takes to complete the run. But how could that be? He claims that the runner must do something at the end of each half-run to make it distinct from the next: she must stop, making the run itself discontinuous.
It is hard—from our modern perspective perhaps—to see how this answer could be completely satisfactory. In the first place it assumes that a clear distinction can be drawn between potential and actual infinities, something that was never fully achieved. Or perhaps Aristotle did not see infinite sums as the problem, but rather whether completing an infinity of finite actions is metaphysically and conceptually and physically possible.
Finally, the distinction between potential and actual infinities has played no role in mathematics since Cantor tamed the transfinite numbers—certainly the potential infinite has played no role in the modern mathematical solutions discussed here. This paradox turns on much the same considerations as the last. On the face of it Achilles should catch the tortoise after 1s, at a distance of 1m from where he starts and so 0. But in the time he takes to do this the tortoise crawls a little further forward.
So next Achilles must reach this new point. But in the time it takes Achilles to achieve this the tortoise crawls forward a tiny bit further.
And so on to infinity: every time that Achilles reaches the place where the tortoise was, the tortoise has had enough time to get a little bit further, and so Achilles has another run to make, and so Achilles has an infinite number of finite catch-ups to do before he can catch the tortoise, and so, Zeno concludes, he never catches the tortoise.
One aspect of the paradox is thus that Achilles must traverse the following infinite series of distances before he catches the tortoise: first 0. And so everything we said above applies here too. But what the paradox in this form brings out most vividly is the problem of completing a series of actions that has no final member—in this case the infinite series of catch-ups before Achilles reaches the tortoise.
But just what is the problem? Therefore, nowhere in his run does he reach the tortoise after all. Thinking in terms of the points that Achilles must reach in his run, 1m does not occur in the sequence 0. Thus the series of catch-ups does not after all completely decompose the run: the final point—at which Achilles does catch the tortoise—must be added to it. So is there any puzzle? Arguably yes. But does such a strange sequence—comprised of an infinity of members followed by one more—make sense mathematically?
If not then our mathematical description of the run cannot be correct, but then what is? Fortunately the theory of transfinites pioneered by Cantor assures us that such a series is perfectly respectable. It was realized that the order properties of infinite series are much more elaborate than those of finite series. Any way of arranging the numbers 1, 2 and 3 gives a series in the same pattern, for instance, but there are many distinct ways to order the natural numbers: 1, 2, 3, … for instance.
Or 2, 3, 4, … , 1, which is just the same kind of series as the positions Achilles must run through. Since the ordinals are standardly taken to be mathematically legitimate numbers, and since the series of points Achilles must pass has an ordinal number, we shall take it that the series is mathematically legitimate. Diogenes Laertius Lives of Famous Philosophers , ix.
Consider an arrow, apparently in motion, at any instant. But the entire period of its motion contains 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 the arrow is at rest during any instant. It follows immediately if one assumes that an instant lasts 0s: whatever speed the arrow has, it will get nowhere if it has no time at all.
But what if one held that the smallest parts of time are finite—if tiny—so that a moving arrow might actually move some distance during an instant?
One way of supporting the assumption—which requires reading quite a lot into the text—starts by assuming that instants are indivisible. Then suppose that an arrow actually moved during an instant. Note that this argument only establishes that nothing can move during an instant, not that instants cannot be finite.
So then, nothing moves during any instant, but time is entirely composed of instants, so nothing ever moves. A first response is to point out that determining the velocity of the arrow means dividing the distance traveled in some time by the length of that time.
The answer is correct, but it carries the counter-intuitive implication 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 traveled during any instant. So when does the arrow actually move? How does it get from one place to another at a later moment? The text is rather cryptic, but is usually interpreted along the following lines: picture three sets of touching cubes—all exactly the same—in relative motion. Then a contradiction threatens because the time between the states is unequivocal, not relative—the process takes some non-zero time and half that time.
The general verdict is that Zeno was hopelessly confused about relative velocities in this paradox. But could Zeno have been this confused? Sattler, , argues against this and other common readings of the stadium. Now, as a point moves continuously along a line with no gaps, there is a correspondence between the instants of time and the points on the line—to each instant a point, and to each point an instant. If we then, crucially, assume that half the instants means half the time, we conclude that half the time equals the whole time, a contradiction.
We saw above, in our discussion of complete divisibility, the problem with such reasoning applied to continuous lines: any line segment has the same number of points, so nothing can be inferred from the number of points in this way—certainly not that half the points here, instants means half the length or time.
The paradox fails as stated. This issue is subtle for infinite sets: to give a different example, 1, 2, 3, … is in correspondence with 2, 4, 6, …, and so there are the same number of each. So there is no contradiction in the number of points: the informal half equals the strict whole a different solution is required for an atomic theory, along the lines presented in the final paragraph of this section.
Imagine two wheels, one twice the radius and circumference of the other, fixed to a single axle. Let them run down a track, with one rail raised to keep the axle horizontal, for one turn of both wheels [they turn at the same rate because of the axle]: each point of each wheel makes contact with exactly one point of its rail, and every point of each rail with exactly one point of its wheel. Does the assembly travel a distance equal to the circumference of the big wheel?
Of the small? Something else? After one eighth of a minute I switch is back on and so on, each time halving the length of time I wait before I switch the lamp on or off as appropriate I have very quick reflexes.
So at this point, is the lamp on or off? And will it have made a difference if the lamp was initially on rather than off? As with Zeno's original version of Achilles, these arguments are based on the infinite divisibility of time, and the paradox that results can be seen to illustrating that time is not infinitely divisible in this way. Interestingly, as mentioned above, the Achilles paradox was only one of 40 arguments Zeno is thought to have produced, and in another of his arguments called the Arrow , Zeno also shows that the assumption that the universe consists of finite, indivisible elements is apparently incorrect.
So, here is where the real paradox of Zeno lies. In his arguments, he manages to show that the universe can neither be continuous infinitely divisible nor discrete discontinuous, that is made up of finite,indivisible parts. This seeming contradiction in the nature of reality is echoed by concepts from an area developed over years after Zeno lived, the Theory of Relativity.
For example, light is now thought of as having a dual nature, behaving sometimes as a particle or photon discrete , and at other times like a wave continuous. In fact even Zeno's belief in monism - in a static, unchanging reality - which was the basis for his producing the arguments in the first place, seems oddly similar to cosmologists ideas about ' worldlines ' the 'history' of a particle in spacetime where 'the entire history of each worldline already exists as a completed entity in the plenum of space time' read more.
So Zeno's paradoxes still challenge our understanding of space and time, and these ancient arguments have surprising resonance with some of the most modern concepts in science. Rachel Thomas is an assistant editor of Plus. The argument the author proposes doesn't really solve Zeno's paradox. Seeing outside consciousness is impossible. So, imho, Zeno lives! Seeing outside consciousness would lead to have no perception about anything in this world. Besides this, the argument that author provides is perfectly ok, but in my opinion you misunderstood it.
What matters here is not the division of space only, but the division of time too. You can repeatedly divide a distance as much as you wish, but time is not slowing down in any way - that's what the author says, so in fact there is no paradox here whatsoever.
I agree with the other comment that arguing against the infinite divisibility of space and time doesn't really get to grips with the paradox. My reason though is that it doesn't really matter whether or not you can go on dividing and subdividing the journey forever, you just don't want to anyway. Right from the start it seems pointless and irrelevant as a means of describing motion in this context. When this is no longer so, as with very fast objects over very small distances then quantum notions like superposition and probability clouds help out.
Nevertheless Zeno exploits the pedant in us which recognises an essential contradiction between motion and position, the fact or feeling that strictly speaking we can't really say where any moving object is, however practically useful it may be to do so.
That once specified, a position is really a stop even if the object continues moving, and stops can go on being specified forever. The antidote to pedantry is humourous exaggeration, so here goes:.
The cut can be made at a rational number or at an irrational number. Here are examples of each:. Otherwise, the cut defines an irrational number which, loosely speaking, fills the gap between A and B, as in the definition of the square root of 2 above.
By defining reals in terms of rationals this way, Dedekind gave a foundation to the reals, and legitimized them by showing they are as acceptable as actually-infinite sets of rationals. But what exactly is an actually-infinite or transfinite set, and does this idea lead to contradictions? This question needs an answer if there is to be a good theory of continuity and of real numbers.
In the s, Cantor clarified what an actually-infinite set is and made a convincing case that the concept does not lead to inconsistencies. That solution recommends using very different concepts and theories than those used by Zeno. In brief, the argument for the Standard Solution is that we have solid grounds for believing our best scientific theories, but the theories of mathematics such as calculus and Zermelo-Fraenkel set theory are indispensable to these theories, so we have solid grounds for believing in them, too.
Therefore, we should accept the Standard Solution. To be optimistic, the Standard Solution represents a counterexample to the claim that philosophical problems never get solved.
To be less optimistic, the Standard Solution has its drawbacks and its alternatives, and these have generated new and interesting philosophical controversies beginning in the last half of the 20th century, as will be seen in later sections. The primary alternatives contain different treatments of calculus from that developed at the end of the 19th century.
Did Zeno make mistakes? And was he superficial or profound? These questions are a matter of dispute in the philosophical literature. The majority position is as follows. If we give his paradoxes a sympathetic reconstruction, he correctly demonstrated that some important, classical Greek concepts are logically inconsistent, and he did not make a mistake in doing this, except in the Moving Rows Paradox, the Paradox of Alike and Unlike and the Grain of Millet Paradox, his weakest paradoxes.
Zeno did assume that the classical Greek concepts were the correct concepts to use in reasoning about his paradoxes, and now we prefer revised concepts, though it would be unfair to say he blundered for not foreseeing later developments in mathematics and physics. Zeno probably created forty paradoxes, of which only the following ten are known.
Only the first four have standard names, and the first two have received the most attention. The ten are of uneven quality. Zeno and his ancient interpreters usually stated his paradoxes badly, so it has taken some clever reconstruction over the years to reveal their full force. Below, the paradoxes are reconstructed sympathetically, and then the Standard Solution is applied to them.
These reconstructions use just one of several reasonable schemes for presenting the paradoxes, but the present article does not explore the historical research about the variety of interpretive schemes and their relative plausibility. Achilles, whom we can assume is the fastest runner of antiquity, is racing to catch the tortoise that is slowly crawling away from him. Both are moving along a linear path at constant speeds.
In order to catch the tortoise, Achilles will have to reach the place where the tortoise presently is. However, by the time Achilles gets there, the tortoise will have crawled to a new location.
Achilles will then have to reach this new location. By the time Achilles reaches that location, the tortoise will have moved on to yet another location, and so on forever. Zeno claims Achilles will never catch the tortoise. This argument shows, he believes, that anyone who believes Achilles will succeed in catching the tortoise and who believes more generally that motion is physically possible is the victim of illusion.
There is no evidence that Zeno used a tortoise rather than a slow human. Zeno is assuming that space and time are infinitely divisible; they are not discrete or atomistic. It implies that Zeno is assuming Achilles cannot achieve his goal because. Maybe he is just guessing that the sum of an infinite number of terms could somehow be well-defined and be infinite.
Here is a graph using the methods of the Standard Solution showing the activity of Achilles as he chases the tortoise and overtakes it. For ease of understanding, Zeno and the tortoise are assumed to be point masses or infinitesimal particles, each moving at a constant velocity that is, a constant speed in one direction. Achilles travels a distance d 1 in reaching the point x 1 where the tortoise starts, but by the time Achilles reaches x 1 , the tortoise has moved on to a new point x 2.
When Achilles reaches x 2 , having gone an additional distance d 2 , the tortoise has moved on to point x 3 , requiring Achilles to cover an additional distance d 3 , and so forth.
This sequence of non-overlapping distances or intervals or sub-paths is an actual infinity, but happily the geometric series converges. Similar reasoning would apply if Zeno were to have made assumptions 2 or 3 above about there not being enough time for Achilles or there being too many places for him to run.
More will be said about assumption 5 in Section 5c when we discuss supertasks. As Aristotle realized, the Dichotomy Paradox is just the Achilles Paradox in which Achilles stands still ahead of the tortoise. In his Progressive Dichotomy Paradox, Zeno argued that a runner will never reach the stationary goal line on a straight racetrack.
The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on.
The runner cannot reach the final goal, says Zeno. Why not? The Standard Solution argues instead that the sum of this infinite geometric series is one, not infinity. The problem of the runner getting to the goal can be viewed from a different perspective.
According to the Regressive version of the Dichotomy Paradox, the runner cannot even take a first step. Here is why. Any step may be divided conceptually into a first half and a second half. Like the Achilles Paradox, this paradox also concludes that any motion is impossible. This is key to solving the Dichotomy Paradox according to the Standard Solution.
It is basically the same treatment as that given to the Achilles. Aristotle, in Physics Z9, said of the Dichotomy that it is possible for a runner to come in contact with a potentially infinite number of things in a finite time provided the time intervals becomes shorter and shorter. Aristotle said Zeno assumed this is impossible, and that is one of his errors in the Dichotomy.
However, Aristotle merely asserted this and could give no detailed theory that enables the computation of the finite amount of time. Today the calculus is used to provide the Standard Solution with that detailed theory.
There is another detail of the Dichotomy that needs resolution. Think of how you would distinguish an arrow that is stationary in space from one that is flying through space, given that you look only at a snapshot an instantaneous photo of them.
Would there be any difference? That is, during any indivisible moment or instant it is at the place where it is. But places do not move. So, if in each moment, the arrow is occupying a space equal to itself, then the arrow is not moving in that moment. The reason it is not moving is that it has no time in which to move; it is simply there at the place.
It cannot move during the moment because that motion would require an even smaller unit of time, but the moment is indivisible. So, the arrow is never moving. By a similar argument, Zeno can establish that nothing else moves.
The Standard Solution to the Arrow Paradox requires the reasoning to use our contemporary theory of speed from calculus. This theory defines instantaneous motion, that is, motion at an instant, without defining motion during an instant.
The modern difference between rest and motion, as opposed to the difference in antiquity, has to do with what is happening at nearby moments and—contra Zeno—has nothing to do with what is happening during a moment. Some researchers have speculated that the Arrow Paradox was designed by Zeno to attack discrete time and space rather than continuous time and space. This is not clear, and the Standard Solution works for both. Yet regardless of how long the instant lasts, there still can be instantaneous motion, namely motion at that instant provided the object is in a different place at some other instant.
To re-emphasize this crucial point, note that both Zeno and 21st century mathematical physicists agree that the arrow cannot be in motion within or during an instant an instantaneous time , but the physicists will point out that the arrow can be in motion at an instant in the sense of having a positive speed at that instant its so-called instantaneous speed , provided the arrow occupies different positions at times before or after that instant so that the instant is part of a period in which the arrow is continuously in motion.
If we do not pay attention to what happens at nearby instants, it is impossible to distinguish instantaneous motion from instantaneous rest, but distinguishing the two is the way out of the Arrow Paradox. Zeno would have balked at the idea of motion at an instant, and Aristotle explicitly denied it.
The Arrow Paradox is refuted by the Standard Solution with its new at-at theory of motion, but the paradox seems especially strong to someone who would prefer instead to say that motion is an intrinsic property of an instant, being some propensity or disposition to be elsewhere. But the speed at an instant is well defined. If we require the use of these modern concepts, then Zeno cannot successfully produce a contradiction as he tries to do by his assuming that in each moment the speed of the arrow is zero—because it is not zero.
According to Aristotle Physics, Book VI, chapter 9, ba18 , Zeno try to create a paradox by considering bodies that is, physical objects of equal length aligned along three parallel rows within a stadium. One track contains A bodies three A bodies are shown below ; another contains B bodies; and a third contains C bodies. Each body is the same distance from its neighbors along its track. The A bodies are stationary. The Bs are moving to the right, and the Cs are moving with the same speed to the left.
Here are two snapshots of the situation, before and after. They are taken one instant apart. Zeno points out that, in the time between the before-snapshot and the after-snapshot, the leftmost C passes two Bs but only one A, contradicting his very controversial assumption that the C should take longer to pass two Bs than one A.
The usual way out of this paradox is to reject that controversial assumption. Aristotle argues that how long it takes to pass a body depends on the speed of the body; for example, if the body is coming towards you, then you can pass it in less time than if it is stationary. Some analysts, for example Tannery , believe Zeno may have had in mind that the paradox was supposed to have assumed that both space and time are discrete quantized, atomized as opposed to continuous, and Zeno intended his argument to challenge the coherence of the idea of discrete space and time.
Well, the paradox could be interpreted this way. If so, assume the three objects A, B, and C are adjacent to each other in their tracks, and each A, B and C body are occupying a space that is one atom long.
Then, if all motion is occurring at the rate of one atom of space in one atom of time, the leftmost C would pass two atoms of B-space in the time it passed one atom of A-space, which is a contradiction to our assumption about rates. There is another paradoxical consequence. Look at the space occupied by left C object. During the instant of movement, it passes the middle B object, yet there is no time at which they are adjacent, which is odd.
However, most commentators suspect Zeno himself did not interpret his paradox this way. Zeno offered more direct attacks on all kinds of plurality. The first is his Paradox of Alike and Unlike. According to Plato in Parmenides , Zeno argued that the assumption of plurality—the assumption that there are many things—leads to a contradiction.
Consider a plurality of things, such as some people and some mountains. These things have in common the property of being heavy. But if they all have this property in common, then they really are all the same kind of thing, and so are not a plurality.
They are a one. By this reasoning, Zeno believes it has been shown that the plurality is one or the many is not many , which is a contradiction. Therefore, by reductio ad absurdum, there is no plurality, as Parmenides has always claimed. Plato immediately accuses Zeno of equivocating. A thing can be alike some other thing in one respect while being not alike it in a different respect.
Your having a property in common with some other thing does not make you identical with that other thing. Consider again our plurality of people and mountains.
People and mountains are all alike in being heavy, but are unlike in intelligence. And they are unlike in being mountains; the mountains are mountains, but the people are not. This paradox is also called the Paradox of Denseness. Suppose there exist many things rather than, as Parmenides would say, just one thing. So, there are three things. But between these, …. Therefore, there are no pluralities; there exists only one thing, not many things. Two objects can be distinct at a time simply by one having a property the other does not have.
Suppose there exist many things rather than, as Parmenides says, just one thing. Then every part of any plurality is both so small as to have no size but also so large as to be infinite, says Zeno. If there is a plurality, then it must be composed of parts which are not themselves pluralities. Now, why are the parts of pluralities so large as to be infinite? Well, the parts cannot be so small as to have no size since adding such things together would never contribute anything to the whole so far as size is concerned.
So, the parts have some non-zero size. If so, then each of these parts will have two spatially distinct sub-parts, one in front of the other. Each of these sub-parts also will have a size. The front part, being a thing, will have its own two spatially distinct sub-parts, one in front of the other; and these two sub-parts will have sizes. Ditto for the back part. And so on without end. A sum of all these sub-parts would be infinite.
Therefore, each part of a plurality will be so large as to be infinite. A university is a plurality of students, but we need not rule out the possibility that a student is a plurality.
When we consider a university to be a plurality of students, we consider the students to be wholes without parts. But for another purpose we might want to say that a student is a plurality of biological cells.
Zeno is confused about this notion of relativity, and about part-whole reasoning; and as commentators began to appreciate this they lost interest in Zeno as a player in the great metaphysical debate between pluralism and monism.
A second error occurs in arguing that the each part of a plurality must have a non-zero size. The contemporary notion of measure developed in the 20th century by Brouwer, Lebesgue, and others showed how to properly define the measure function so that a line segment has nonzero measure even though the singleton set of any point has a zero measure. The measure of the line segment [a, b] is b — a; the measure of a cube with side a is a 3. This theory of measure is now properly used by our civilization for length, volume, duration, mass, voltage, brightness, and other continuous magnitudes.
Interest was rekindled in this topic in the 18th century. Boscovich in as being collections of point masses. Each point mass is a movable point carrying a fixed mass. This idealization of continuous bodies as if they were compositions of point particles was very fruitful; it could be used to easily solve otherwise very difficult problems in physics.
This is the most challenging of all the paradoxes of plurality. Consider the difficulties that arise if we assume that an object theoretically can be divided into a plurality of parts. According to Zeno, there is a reassembly problem.
Imagine cutting the object into two non-overlapping parts, then similarly cutting these parts into parts, and so on until the process of repeated division is complete. There are three possibilities. In that case the original objects will be a composite of nothing, and so the whole object will be a mere appearance, which is absurd. So, the original object is composed of elements of zero size.
Adding an infinity of zeros yields a zero sum, so the original object had no size, which is absurd. If so, these can be further divided, and the process of division was not complete after all, which contradicts our assumption that the process was already complete.
In summary, there were three possibilities, but all three possibilities lead to absurdity. So, objects are not divisible into a plurality of parts. Simplicius says this argument is due to Zeno even though it is in Aristotle On Generation and Corruption , a, b34 and a and is not attributed there to Zeno, which is odd.
Aristotle says the argument convinced the atomists to reject infinite divisibility. The argument has been called the Paradox of Parts and Wholes, but it has no traditional name. The Standard Solution says we first should ask Zeno to be clearer about what he is dividing. Is it concrete or abstract? When dividing a concrete, material stick into its components, we reach ultimate constituents of matter such as quarks and electrons that cannot be further divided.
These have a size, a zero size according to quantum electrodynamics , but it is incorrect to conclude that the whole stick has no size if its constituents have zero size. On the other hand, is Zeno dividing an abstract path or trajectory? If so, then choice 2 above is the one to think about. The size length, measure of a point-element is zero, but Zeno is mistaken in saying the total size length, measure of all the zero-size elements is zero.
The size of the object is determined instead by the difference in coordinate numbers assigned to the end points of the object. An object extending along a straight line that has one of its end points at one meter from the origin and other end point at three meters from the origin has a size of two meters and not zero meters.
There are two common interpretations of this paradox. According to the first, which is the standard interpretation, when a bushel of millet or wheat grains falls out of its container and crashes to the floor, it makes a sound. Since the bushel is composed of individual grains, each individual grain also makes a sound, as should each thousandth part of the grain, and so on to its ultimate parts.
But this result contradicts the fact that we actually hear no sound for portions like a thousandth part of a grain, and so we surely would hear no sound for an ultimate part of a grain. Yet, how can the bushel make a sound if none of its ultimate parts make a sound? There seems to be appeal to the iterative rule that if a millet or millet part makes a sound, then so should a next smaller part. Perhaps he would conclude it is a mistake to suppose that whole bushels of millet have millet parts.
This is an attack on plurality.
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