Why zeno was wrong

They are always directed towards a more-or-less specific target: the views of some person or school. Then, if the argument is logically valid, and the conclusion genuinely unacceptable, the assertions must be false after all. If we find that Zeno makes hidden assumptions beyond what the position under attack commits one to, then the absurd conclusion can be avoided by denying one of the hidden assumptions, while maintaining the position.

Indeed commentators at least since Aristotle have responded to Zeno in this way. As we shall discuss briefly below, some say that the target was a technical doctrine of the Pythagoreans, but most today see Zeno as opposing common-sense notions of plurality and motion. We shall approach the paradoxes in this spirit, and refer the reader to the literature concerning the interpretive debate.

This is not necessarily to say that modern mathematics is required to answer any of the problems that Zeno explicitly wanted to raise; arguably Aristotle and other ancients had replies that would—or should—have satisfied Zeno. However, as mathematics developed, and more thought was given to the paradoxes, new difficulties arose from them; these difficulties require modern mathematics for their resolution.

Thus we shall push several of the paradoxes from their common sense formulations to their resolution in modern mathematics. Between any two of them, he claims, is a third; and in between these three elements another two; and another four between these five; and so on without end.

So our original assumption of a plurality leads to a contradiction, and hence is false: there are not many things after all. Let us consider the two subarguments, in reverse order. Suppose that we had imagined a collection of ten apples lined up; then there is indeed another apple between the sixth and eighth, but there is none between the seventh and eighth!

On the assumption that Zeno is not simply confused, what does he have in mind? And one might think that for these three to be distinct, there must be two more objects separating them, and so on this view presupposes that their being made of different substances is not sufficient to render them distinct. So perhaps Zeno is arguing against plurality given a certain conception of physical distinctness.

But second, one might also hold that any body has parts that can be densely ordered. Indeed, if between any two point-parts there lies a finite distance, and if point-parts can be arbitrarily close, then they are dense; a third lies at the half-way point of any two. In particular, familiar geometric points are like this, and hence are dense. So perhaps Zeno is offering an argument regarding the divisibility of bodies.

Can this contradiction be escaped? The assumption that any definite number is finite seems intuitive, but we now know, thanks to the work of Cantor in the Nineteenth century, how to understand infinite numbers in a way that makes them just as definite as finite numbers. With such a definition in hand it is then possible to order the infinite numbers just as the finite numbers are ordered: for example, there are different, definite infinite numbers of fractions and geometric points in a line, even though both are dense.

See Further Reading below for references to introductions to these mathematical ideas, and their history. Though of course that only shows that infinite collections are mathematically consistent, not that any physically exist. But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever.

For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited.

According to his conclusion, there are three parts to this argument, but only two survive. The first—missing—argument purports to show that if many things exist then they must have no size at all.

Second, from this Zeno argues that it follows that they do not exist at all; since the result of joining or removing a sizeless object to anything is no change at all, he concludes that the thing added or removed is literally nothing.

The argument to this point is a self-contained refutation of pluralism, but Zeno goes on to generate a further problem for someone who continues to urge the existence of a plurality. And the parts exist, so they have extension, and so they also each have two spatially distinct parts; and so on without end.

And hence, the final line of argument seems to conclude, the object, if it is extended at all, is infinite in extent. But what could justify this final step? And neither does it follow 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 in mind, presumably the following: he assumes that if the infinite series of divisions he describes were repeated infinitely many times then a definite collection of parts would result.

Now, if—as a pluralist might well accept—such parts exist, it follows from the second part of his argument that they are extended, and, he apparently assumes, an infinite sum of finite parts is infinite. Here we should note that there are two ways he may be envisioning the result of the infinite division.

What is often pointed out in response is that Zeno gives us no reason to think that the sum is infinite rather than finite. He might have had the intuition that any infinite sum of finite quantities, since it grows endlessly with each new term must be infinite, but one might also take this kind of example as showing that some infinite sums are after all finite. Thus, contrary to what he thought, Zeno has not proven that the absurd conclusion follows.

However, what is not always appreciated is that the pluralist is not off the hook so easily, for it is not enough just to say that the sum might be finite, she must also show that it is finite—otherwise we remain uncertain about the tenability of her position. As an illustration of the difficulty faced here consider the following: many commentators speak as if it is simply obvious that the infinite sum of the fractions is 1, that there is nothing to infinite summation.

Surely this answer seems as intuitive as the sum of fractions. Such a theory was not fully worked out until the Nineteenth century by Cauchy. In this case the pieces at any particular stage are all the same finite size, and so one could conclude that the result of carrying on the procedure infinitely would be pieces the same size, which if they exist—according to Zeno—is greater than zero; but an infinity of equal extended parts is indeed infinitely big.

But this line of thought can be resisted. First, suppose that the procedure just described completely divides the object into non-overlapping parts.

There is a problem with this supposition that we will see just below. This result poses no immediate difficulty since, as we mentioned above, infinities come in different sizes. However, we could consider just countably many of them, whose lengths according to Zeno—since he claims they are all equal and non-zero—will sum to an infinite length; the length of all of the pieces could not be less than this.

We shall postpone this question for the discussion of the next paradox, where it comes up explicitly. The second problem with interpreting the infinite division as a repeated division of all parts is that it does not divide an object into distinct parts, if objects are composed in the natural way.

Since the division is repeated without end there is no last piece we can give as an answer, and so we need to think about the question in a different way. Thus the only part of the line that is in all the elements of this chain is the half-way point, and so that is the part of the line picked out by the chain.

In fact, it follows from a postulate of number theory that there is exactly one point that all the members of any such a chain have in common. And so both chains pick out the same piece of the line: the half-way point. And so on for many other pairs of chains. 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. 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. The Standard Solution to this interpretation of the paradox accuses Zeno of mistakenly assuming that there is no lower bound on the size of something that can make a sound.

There is no problem, we now say, with parts having very different properties from the wholes that they constitute. The iterative rule is initially plausible but ultimately not trustworthy, and Zeno is committing both the fallacy of division and the fallacy of composition.

When a bushel of millet grains crashes to the floor, it makes a sound. The bushel is composed of individual grains, so they, too, make an audible sound. But if you drop an individual millet grain or a small part of one or an even smaller part, then eventually your hearing detects no sound, even though there is one. Therefore, you cannot trust your sense of hearing.

This reasoning about our not detecting low amplitude sounds is similar to making the mistake of arguing that you cannot trust your thermometer because there are some ranges of temperature that it is not sensitive to.

So, on this second interpretation, the paradox is also easy to solve. Regarding the Dichotomy Paradox, Aristotle is to be applauded for his insight that Achilles has time to reach his goal because during the run ever shorter paths take correspondingly ever shorter times. Aristotle had several criticisms of Zeno. His second complaint was that Zeno should not suppose that lines contain indivisible points. Here is how Aristotle expressed the point:. For motion…, although what is continuous contains an infinite number of halves, they are not actual but potential halves.

Physics a If the units are actual, it is not possible: if they are potential, it is possible. Physics b Aristotle denied the existence of the actual infinite both in the physical world and in mathematics, but he accepted potential infinities there.

By calling them potential infinities he did not mean they have the potential to become actually infinite; potential infinity is a technical term that suggests a process that has not been completed. The term actual infinite does not imply being actual or real. It implies being complete, with no dependency on some process in time. A potential infinity is an unlimited iteration of some operation—unlimited in time. Aristotle claimed correctly that if Zeno were not to have used the concept of actual infinity and of indivisible point, then the paradoxes of motion such as the Achilles Paradox and the Dichotomy Paradox could not be created.

Here is why doing so is a way out of these paradoxes. Zeno said that to go from the start to the finish line, the runner Achilles must reach the place that is halfway-there, then after arriving at this place he still must reach the place that is half of that remaining distance, and after arriving there he must again reach the new place that is now halfway to the goal, and so on.

These are too many places to reach. Zeno made the mistake, according to Aristotle, of supposing that this infinite process needs completing when it really does not need completing and cannot be completed; the finitely long path from start to finish exists undivided for the runner, and it is the mathematician who is demanding the completion of such a process.

Without using that concept of a completed infinity there is no paradox. Aristotle is correct about this being a treatment that avoids paradox. From what Aristotle says, one can infer between the lines that he believes there is another reason to reject actual infinities: doing so is the only way out of these paradoxes of motion.

Today we know better. Leibniz accepted actual infinitesimals, but other mathematicians and physicists in European universities during these centuries were careful to distinguish between actual and potential infinities and to avoid using actual infinities. Given 1, years of opposition to actual infinities, the burden of proof was on anyone advocating them.

Bernard Bolzano and Georg Cantor accepted this burden in the 19th century. The key idea is to see a potentially infinite set as a variable quantity that is dependent on being abstracted from a pre-exisiting actually infinite set. Bolzano argued that the natural numbers should be conceived of as a set, a determinate set, not one with a variable number of elements.

Cantor argued that any potential infinity must be interpreted as varying over a predefined fixed set of possible values, a set that is actually infinite. He put it this way:. However, this domain cannot itself be something variable…. Thus each potential infinite…presupposes an actual infinite. Cantor The same can be said for sets of real numbers. Aristotle had said mathematicians need only the concept of a finite straight line that may be produced as far as they wish, or divided as finely as they wish, but Cantor would say that this way of thinking presupposes a completed infinite continuum from which that finite line is abstracted at any particular time.

Cantor provided the missing ingredient—that the mathematical line can fruitfully be treated as a dense linear ordering of uncountably many points, and he went on to develop set theory and to give the continuum a set-theoretic basis which convinced mathematicians that the concept was rigorously defined. These ideas now form the basis of modern real analysis.

Zeno said Achilles cannot achieve his goal in a finite time, but there is no record of the details of how he defended this conclusion. He might have said the reason is i that there is no last goal in the sequence of sub-goals, or, perhaps ii that it would take too long to achieve all the sub-goals, or perhaps iii that covering all the sub-paths is too great a distance to run.

Zeno might have offered all these defenses. In attacking justification ii , Aristotle objects that, if Zeno were to confine his notion of infinity to a potential infinity and were to reject the idea of zero-length sub-paths, then Achilles achieves his goal in a finite time, so this is a way out of the paradox. However, an advocate of the Standard Solution says Achilles achieves his goal by covering an actual infinity of paths in a finite time, and this is the way out of the paradox.

The discussion of whether Achilles can properly be described as completing an actual infinity of tasks rather than goals will be considered in Section 5c. Physics , a In modern real analysis, a continuum is composed of points, but Aristotle, ever the advocate of common sense reasoning, claimed that a continuum cannot be composed of points.

Aristotle believed a line can be composed only of smaller, indefinitely divisible lines and not of points without magnitude. Similarly a distance cannot be composed of point places and a duration cannot be composed of instants. In addition to complaining about points, Aristotelians object to the idea of an actual infinite number of them.

Aristotle recommends not allowing Zeno to appeal to instantaneous moments and restricting Zeno to saying motion be divided only into a potential infinity of intervals. So, at any time, there is a finite interval during which the arrow can exhibit motion by changing location. So the arrow flies, after all. However, the Standard Solution agrees with Zeno that time can be composed of indivisible moments or instants, and it implies that Aristotle has mis-diagnosed where the error lies in the Arrow Paradox.

Regarding the Paradox of the Grain of Millet, Aristotle said that parts need not have all the properties of the whole, and so grains need not make sounds just because bushels of grains do. Physics , a, 22 And if the parts make no sounds, we should not conclude that the whole can make no sound.

It would have been helpful for Aristotle to have said more about what are today called the Fallacies of Division and Composition that Zeno is committing. The Standard Solution uses contemporary concepts that have proved to be more valuable for solving and resolving so many other problems in mathematics and physics.

Nevertheless, there is a significant minority in the philosophical community who do not agree, as we shall see in the sections that follow. The following—once presumably safe—intuitions or assumptions must be rejected:. Item 8 was undermined when it was discovered that the continuum implies the existence of fractal curves. However, the loss of intuition 1 has caused the greatest stir because so many philosophers object to a continuum being constructed from points.

Continuity is something given in perception, said Brentano, and not in a mathematical construction; therefore, mathematics misrepresents. But the Standard Solution needs to be thought of as a package to be evaluated in terms of all of its costs and benefits.

As a consequence, advocates of the Standard Solution say we must live with rejecting the eight intuitions listed above, and accept the counterintuitive implications such as there being divisible continua, infinite sets of different sizes, and space-filling curves.

They agree with the philosopher W. Peirce, James Thomson, Alfred North Whitehead , and Hermann Weyl argued in different ways that the standard mathematical account of continuity does not apply to physical processes, or is improper for describing those processes.

A minority of philosophers are actively involved in attempting to retain one or more of the eight intuitions listed in the previous section. An important philosophical issue is whether the paradoxes should be solved by the Standard Solution or instead by assuming that a line is not composed of points but of intervals, and whether use of infinitesimals is essential to a proper understanding of the paradoxes.

In doing so, does he need to complete an infinite sequence of tasks or actions? In other words, assuming Achilles does complete the task of reaching the tortoise, does he thereby complete a supertask , a transfinite number of tasks in a finite time?

At the end of the minute, an infinite number of tasks would have been performed. In fact, Achilles does this in catching the tortoise, Russell said. In the mid-twentieth century, Hermann Weyl, Max Black, James Thomson, and others objected, and thus began an ongoing controversy about the number of tasks that can be completed in a finite time.

That controversy has sparked a related discussion about whether there could be a machine that can perform an infinite number of tasks in a finite time. A machine that can is called an infinity machine. Let the machine switch the lamp on for a half-minute; then switch it off for a quarter-minute; then on for an eighth-minute; off for a sixteenth-minute; and so on. Would the lamp be lit or dark at the end of minute? Thomson argued that it must be one or the other, but it cannot be either because every period in which it is off is followed by a period in which it is on, and vice versa, so there can be no such lamp, and the specific mistake in the reasoning was to suppose that it is logically possible to perform a supertask.

The lamp could be either on or off at the limit. The limit of the infinite converging sequence is not in the sequence. Could some other argument establish this impossibility? The Thomson Lamp Argument has generated a great literature in philosophy. Here are some of the issues. Is the lamp logically impossible or physically impossible? Is the lamp metaphysically impossible? Was it proper of Thomson to suppose that the question of whether the lamp is lit or dark at the end of the minute must have a determinate answer?

Should we conclude that it makes no sense to divide a finite task into an infinite number of ever shorter sub-tasks? Is there an important difference between completing a countable infinity of tasks and completing an uncountable infinity of tasks?

See Earman and Norton for an introduction to the extensive literature on these topics. Constructivism is not a precisely defined position, but it implies that acceptable mathematical objects and procedures have to be founded on constructions and not, say, on assuming the object does not exist, then deducing a contradiction from that assumption.

Most constructivists believe acceptable constructions must be performable ideally by humans independently of practical limitations of time or money.

Although everyone agrees that any legitimate mathematical proof must use only a finite number of steps and be constructive in that sense, the majority of mathematicians in the first half of the twentieth century claimed that constructive mathematics could not produce an adequate theory of the continuum because essential theorems would no longer be theorems, and constructivist principles and procedures are too awkward to use successfully.

But thanks in large part to the later development of constructive mathematics by Errett Bishop and Douglas Bridges in the second half of the 20th century, most contemporary philosophers of mathematics believe the question of whether constructivism could be successful in the sense of producing an adequate theory of the continuum is still open [see Wolf p. Frank Arntzenius , Michael Dummett , and Solomon Feferman have done important philosophical work to promote the constructivist tradition.

Although Zeno and Aristotle had the concept of small, they did not have the concept of infinitesimally small, which is the informal concept that was used by Leibniz and Newton in the development of calculus.

In the 19th century, infinitesimals were eliminated from the standard development of calculus due to the work of Cauchy and Weierstrass on defining a derivative in terms of limits using the epsilon-delta method.

But in , C. Peirce advocated restoring infinitesimals because of their intuitive appeal. Unfortunately, he was unable to work out the details, as were all mathematicians—until when Abraham Robinson produced his nonstandard analysis.

At this point in time it was no longer reasonable to say that banishing infinitesimals from analysis was an intellectual advance. Robinson went on to create a nonstandard model of analysis using hyperreal numbers. The reciprocal of an infinitesimal is an infinite hyperreal number. These hyperreals obey the usual rules of real numbers except for the Archimedean axiom. Infinitesimal distances between distinct points are allowed, unlike with standard real analysis. For objects that move in this Universe, physics solves Zeno's paradox.

But at the quantum level, an entirely new paradox emerges, known as the quantum Zeno effect. Certain physical phenomena only happen due to the quantum properties of matter and energy, like quantum tunneling through a barrier or radioactive decays.

In order to go from one quantum state to another, your quantum system needs to act like a wave: its wavefunction spreads out over time. Eventually, there will be a non-zero probability of winding up in a lower-energy quantum state. This is how you can tunnel into a more energetically favorable state even when there isn't a classical path that allows you to get there.

Although the step of tunneling itself may be instantaneous, the traveling particles are still limited by the speed of light. Most physicists refer to this type of interaction as "collapsing the wavefunction," as you're basically causing whatever quantum system you're measuring to act "particle-like" instead of "wave-like. If you make this measurement too close in time to your prior measurement, there will only be an infinitesimal or even a zero probability of tunneling into your desired state.

If you keep your quantum system interacting with the environment, you can suppress the inherently quantum effects, leaving you with only the classical outcomes as possibilities. When a quantum particle approaches a barrier, it will most frequently interact with it.

But there is If you were to measure the position of the particle continuously, however, including upon its interaction with the barrier, this tunneling effect could be entirely suppressed via the quantum Zeno effect. The takeaway is this: motion from one place to another is possible, and it's because of the explicit physical relationship between distance, velocity and time that we can learn exactly how motion occurs in a quantitative sense.

Yes, in order to cover the full distance from one location to another, you have to first cover half that distance, then half the remaining distance, then half of what's left, etc. But the time it takes to do so also halves, and so motion over a finite distance always takes only a finite amount of time for any object in motion. Although this is still an interesting exercise for mathematicians and philosophers, not only is the solution reliant on physics, but physicists have even extended it to quantum phenomena, where a new quantum Zeno effect — not a paradox, but a suppression of purely quantum effects — emerges.

As in all scientific fields, the Universe itself is the final arbiter of how reality behaves. Thanks to physics, we at last understand how. This is a BETA experience. You may opt-out by clicking here. More From Forbes. Nov 11, , am EST. Achilles gives the Tortoise a head start of, say 10 m, since he runs at 10 ms -1 and the Tortoise moves at only 1 ms When Achilles reaches T 1 , the labouring Tortoise will have moved on 0. When Achilles reaches T 2 , the Tortoise will still be ahead by 0.

Each time Achilles reaches the point where the Tortoise was, the cunning reptile will always have moved a little way ahead. This seems very peculiar. We know that Achilles should pass the Tortoise after 1. But why in Zeno's argument does it seem that Achilles will never catch the tortoise? If you think of the distances Achilles has to travel, first 10 m to T 0 , then 1 m to T 1 , then 0.

Now it is a little clearer. As the distance that Achilles travels to catch the tortoise is the sum of a geometric series where the multiplier is less than one read more , we know that the distance is finite and equal to And as he only has to travel a finite distance, Achilles will obviously cover that distance in a finite time if he is traveling at a constant speed.

So how did Zeno manage to confuse us? Zeno's argument is based on the assumption that you can infinitely divide space the race track and time how long it takes to run. By dividing the race track into an infinite number of pieces, Zeno's argument turned the race into an infinite number of steps that seemed as if they would never end. However, each step is decreasing, and so dividing space and therefore time into smaller and smaller pieces implies that the passage of time is 'slowing down' and can never reach the moment where Achilles passes the Tortoise.

We know that time doesn't slow down in this way. The assumption that space and time is infinitely divisible is wrong more on the physical implications of the limiting process.

There are ways to rephrase the Achilles argument that can take our brains in a slightly different direction. In one example, known as Thomson's Lamp, we suspend our disbelief once again and consider a lamp with a switch that we press to turn on, and press again to turn off.