How does zeno support parmenides

Therefore, the magnitude of each of the many is limitless. Taken as a whole, then, this elaborate tour de force of an argument purports to have shown that, if there are many things, each of them must have simultaneously no magnitude and unlimited magnitude. Aristotle is most concerned with Zeno in Physics 6, the book devoted to the theory of the continuum.

In Physics 6. The ancient commentators on this chapter provide little additional information. He says no more about this argument here but alludes to his earlier discussion of it in Physics 6. Subsequently, in Physics 8. The argument Aristotle is alluding to in these passages gets its name from his mention in Topics 8. The following reconstruction attempts to remain true to this evidence and thus to capture something of how Zeno may originally have argued.

For anyone S to traverse the finite distance across a stadium from p 0 to p 1 within a limited amount of time, S must first reach the point half way between p 0 and p 1 , namely p 2. Before S reaches p 2 , S must first reach the point half way between p 0 and p 2 , namely p 3. Again, before S reaches p 3 , S must first reach the point half way between p 0 and p 3 , namely p 4. There is a half way point again to be reached between p 0 and p 4. In fact, there is always another half way point that must be reached before reaching any given half way point, so that the number of half way points that must be reached between any p n and any p n-1 is unlimited.

But it is impossible for S to reach an unlimited number of half way points within a limited amount of time. Therefore, it is impossible for S to traverse the stadium or, indeed, for S to move at all; in general, it is impossible to move from one place to another.

Simplicius adds the identification of the slowest runner as the tortoise in Ph. Aristotle remarks that this argument is merely a variation on the Dichotomy, with the difference that it does not depend on dividing in half the distance taken Ph. Whether this is actually the case is debatable. During the time it takes Achilles to reach the point from which the tortoise started t 0 , the tortoise will have progressed some distance d 1 beyond that point, namely to t 1 , as follows:.

Therefore, the slowest runner in the race, the tortoise, will never be overtaken by the fastest runner, Achilles. Epiphanius, Against the Heretics 3. Thus, according to Aristotle, the moving arrow A is actually standing still. The argument for this conclusion seems to be as follows: What moves is always, throughout the duration of its motion, in the now, that is to say, in one instant of time after another. So, throughout its flight, A is in one instant of time after another.

So A is resting at t. Thus A is resting at every instant of its flight, and this amounts to the moving arrow always being motionless or standing still.

This description suggests a final position as represented in Diagram 2. Apparently, Zeno somehow meant to infer from the fact that the leading B moves past two A s in the same time it moves past all four C s that half the time is equal to its double. The challenge is to develop from this less than startling fact anything more than a facile appearance of paradox. Since it is stressed that all the bodies are of the same size and that the moving bodies move at the same speed, Zeno would appear to have relied on some such postulate as that a body in motion proceeding at constant speed will move past bodies of the same size in the same amount of time.

He could have argued that in the time it takes all the C s to move past all the B s, the leading B moves past two A s or goes two lengths, and the leading B also moves past four C s or goes four lengths. According to the postulate, then, the time the leading B travels must be the same as half the time it travels.

Unfortunately, the evidence for this particular paradox does not enable us to determine just how Zeno may in fact have argued. Aristotle also gestures toward two additional ingenious arguments by Zeno, versions of which were also known to Simplicius.

The version of this argument known to Simplicius represents Zeno as engaged in a fictional argument with Protagoras, wherein he makes the point that if a large number of millet seeds makes a sound for example, when poured out in a heap , then one seed or even one ten-thousandth of a seed should also make its own sound for example, in that process Simp.

The evidence nonetheless suggests that Zeno anticipated reasoning related to that of the sorites paradox, apparently invented more than a century later.

Eudemus fr. Zeno would appear to have argued as follows. Everything that is is in something, namely a place. If a place is something, then it too must be in something, namely some further place. If this second place is something, it must be in yet another place; and the same reasoning applies to this and each successive place ad infinitum.

Thus, if there is such a thing as place, there must be limitless places everywhere, which is absurd. Therefore, there is no such thing as place. This argument could well have formed part of a more elaborate argument against the view that there are many things, such as that if there are many things, they must be somewhere, i.

This is, however, only speculation. After the portion of the exchange between Socrates and Zeno quoted above sect. Socrates virtually accuses Zeno of having plotted with Parmenides to conceal the fundamental identity of their conclusions. With so many readers of Plato accustomed to taking Socrates as his mouthpiece in the dialogues, it is not surprising that this passage has served as the foundation for the common view of Zeno as Parmenidean legatee and defender, by his own special means, of Eleatic orthodoxy.

Zeno this time replies that Socrates has not altogether grasped the truth about his book. First, he says, the book had nothing like the pretensions Socrates has ascribed to it Prm.

Zeno is made to explain his actual motivation as follows:. For not only does Parmenides end up examining the relation of his One to other things, which would have been impossible if his doctrine entailed their non-existence, but the relation other things have to the One actually proves responsible in a way for their existence.

Zeno cannot be supposing that his arguments against plurality entailed the doctrine of Parmenides when that doctrine is represented in this same dialogue by Parmenides himself as something altogether more involved than the simple thesis that only one thing exists. What Plato actually suggests is that Zeno aimed to show those whose superficial understanding of Parmenides had led them to charge him with flying in the face of common sense, that common sense views concerning unity and plurality are themselves riddled with latent contradictions.

Many men had mocked Parmenides: Zeno mocked the mockers. His logoi were designed to reveal the inanities and ineptitudes inherent in the ordinary belief in a plural world; he wanted to startle, to amaze, to disconcert.

However, whether the historical Zeno was actually involved in anything like the dialectical context Plato envisages for him must remain uncertain. The more mature Zeno seems a little embarrassed by the combative manner evident in the arguments of his younger days, as well he might since that spirit would have come to be seen as typical of the eristic controversialists who sprang up in the sophistic era.

In the Alcibiades , Socrates reports that Pythodorus and Callias each paid Zeno a hundred minae to become clever and skilled in argument Alc. Teaching for payment is of course one hallmark of the professional educators who styled themselves experts in wisdom. Precisely what Aristotle meant by this remains a matter of speculation, given that Aristotle also attributes the invention of dialectic to Socrates Arist. For Aristotle, then, Zeno was a controversialist and paradox-monger, whose arguments were nevertheless both sophisticated enough to qualify him as the inventor of dialectic and were important for forcing clarification of concepts fundamental to natural science.

Should we then think of Zeno as a sophist? The skill Plutarch attributes to Zeno, still evident in the fragmentary remains of his arguments, is just the kind of skill in argument manifested in a great deal of sophistic practice.

His apparent demonstrations of how the common-sense view is fraught with contradiction made him an influential precursor of sophistic antilogic and eristic disputation. It is not surprising that someone like Isocrates should have viewed Zeno as a sophist to be classed with Protagoras and Gorgias. While he perhaps does not fit exactly into any of these categories, still his development of sophisticated methods of argumentation to produce apparent proofs of the evidently false conclusions that motion is impossible and that there are not in fact many things made it quite natural for Plato, Aristotle, Isocrates, and others to refer to him under all these labels.

Several of the paradoxes involve no specifically mathematical notions at all. The Achilles is perhaps the best example since it employs only very ordinary notions, such as getting to where another has started from. The other extant arguments for the most part deploy similarly prosaic notions: being somewhere or being in a place, being in motion, moving past something else, getting halfway there, being of some size, having parts, being one, being like, being the same, and so on.

Where Zeno seems to have leapt ahead of earlier thinkers is in deploying specifically quantitative concepts, most notably quantitative concepts of limit peras and the lack of limit to apeiron. Earlier Greek thinkers had tended to speak of limitedness and unlimitedness in ways suggesting a qualitative rather than a quantitative notion.

They had an immediate impact on Greek physical theory. His arguments, perhaps more than anything else, forced the Greek natural philosophers to develop properly physical theories of composition as opposed to the essentially chemical theories of earlier thinkers such as Empedocles.

That mathematicians and physicists have worked ever since to develop responses to the more ingenious of his paradoxes is remarkable, though perhaps not surprising, for immunity to his paradoxes might be taken as a condition upon the adequacy of our most basic physical concepts.

He may even have offered his collection of paradoxes to provoke deeper consideration of the adequacy of theretofore unexamined notions.

References in this bibliography to items prior to are more selective than those to more recent items. For a nearly exhaustive and annotated listing of Zenonian scholarship down to , consult L. Paquet, M. Roussel, and Y. The long standard collection of the fragments of the Presocratics and sophists, together with testimonia pertaining to their lives and thought, has been:. For the English reader, the fragments and testimonia of the Presocratics and sophists are now most usefully presented in:.

Life and Writings 2. The Extant Paradoxes 2. Is this not what you say? For if there were many things, they would incur impossibilities. So is this what your arguments intend, nothing other than to maintain forcibly, contrary to everything normally said, that there are not many things?

And do you think that each of your arguments is a proof of this very point, so that you consider yourself to be furnishing just as many proofs that there are not many things as the arguments you have written? Is this what you say, or do I not understand correctly? Bibliography Further Reading References in this bibliography to items prior to are more selective than those to more recent items. Comprehensive accounts of Zeno and his arguments may be found in: Barnes, J.

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? Conversely, if one insisted that if they pass then there must be a moment when they are level, then it shows that cannot be a shortest finite interval—whatever it is, just run this argument against it.

However, why should one insist on this assumption? The problem is that one naturally imagines quantized space as being like a chess board, on which the chess pieces are frozen during each quantum of time. Then one wonders when the red queen, say, gets from one square to the next, or how she gets past the white queen without being level with her. But the analogy is misleading. It is better to think of quantized space as a giant matrix of lights that holds some pattern of illuminated lights for each quantum of time.

In this analogy a lit bulb represents the presence of an object: for instance a series of bulbs in a line lighting up in sequence represent a body moving in a straight line. We will discuss them briefly for completeness. When he sets up his theory of place—the crucial spatial notion in his theory of motion—Aristotle lists various theories and problems that his predecessors, including Zeno, have formulated on the subject.

The argument again raises issues of the infinite, since the second step of the argument argues for an infinite regress of places. However, Aristotle presents it as an argument against the very idea of place, rather than plurality thereby likely taking it out of context. It is hard to feel the force of the conclusion, for why should there not be an infinite series of places of places of places of …?

Presumably the worry would be greater for someone who like Aristotle believed that there could not be an actual infinity of things, for the argument seems to show that there are. But as we have discussed above, today we need have no such qualms; there seems nothing problematic with an actual infinity of places. The problem then is not that there are infinitely many places, but just that there are many. And Aristotle might have had this concern, for in his theory of motion, the natural motion of a body is determined by the relation of its place to the center of the universe: an account that requires place to be determinate, because natural motion is.

But supposing that one holds that place is absolute for whatever reason, then for example, where am I as I write? In context, Aristotle is explaining that a fraction of a force many not produce the same fraction of motion.

We describe this fact as the effect of friction. However, while refuting this premise Aristotle does not explain what role it played for Zeno, and we can only speculate. One speculation is that our senses reveal that it does not, since we cannot hear a single grain falling. In this final section we should consider briefly the impact that Zeno has had on various philosophers; a search of the literature will reveal that these debates continue. The Pythagoreans: For the first half of the Twentieth century, the majority reading—following Tannery —of Zeno held that his arguments were directed against a technical doctrine of the Pythagoreans.

According to this reading they held that all things were composed of elements that had the properties of a unit number, a geometric point and a physical atom: this kind of position would fit with their doctrine that reality is fundamentally mathematical.

We have implicitly assumed that these arguments are correct in our readings of the paradoxes. The Atomists: Aristotle On Generation and Corruption b34 claims that our third argument—the one concerning complete divisibility—was what convinced the atomists that there must be smallest, indivisible parts of matter.

What they realized was that a purely mathematical solution was not sufficient: the paradoxes not only question abstract mathematics, but also the nature of physical reality. So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space.

Salmon offers a nice example to help make the point: since alcohol dissolves in water, if you mix the two you end up with less than the sum of their volumes, showing that even ordinary addition is not applicable to every kind of system. Our belief that the mathematical theory of infinity describes space and time is justified to the extent that the laws of physics assume that it does, and to the extent that those laws are themselves confirmed by experience.

While it is true that almost all physical theories assume that space and time do indeed have the structure of the continuum, it is also the case that quantum theories of gravity likely imply that they do not. While no one really knows where this research will ultimately lead, it is quite possible that space and time will turn out, at the most fundamental level, to be quite unlike the mathematical continuum that we have assumed here.

Most starkly, our resolution to the Dichotomy and Achilles assumed that the complete run could be broken down into an infinite series of half runs, which could be summed. If not, and assuming that Atalanta and Achilles can complete their tasks, their complete runs cannot be correctly described as an infinite series of half-runs, although modern mathematics would so describe them.

What infinity machines are supposed to establish is that an infinite series of tasks cannot be completed—so any completable task cannot be broken down into an infinity of smaller tasks, whatever mathematics suggests. Infinitesimals: Finally, we have seen how to tackle the paradoxes using the resources of mathematics as developed in the Nineteenth century. For a long time it was considered one of the great virtues of this system that it finally showed that infinitesimal quantities, smaller than any finite number but larger than zero, are unnecessary.

Analogously, Bell explains how infinitesimal line segments can be introduced into geometry, and comments on their relation to Zeno. Reeder, , argues that non-standard analysis is unsatisfactory regarding the arrow, and offers an alternative account using a different conception of infinitesimals. The construction of non-standard analysis does however raise a further question about the applicability of analysis to physical space and time: it seems plausible that all physical theories can be formulated in either terms, and so as far as our experience extends both seem equally confirmed.

After the relevant entries in this encyclopedia, the place to begin any further investigation is Salmon , which contains some of the most important articles on Zeno up to , and an impressively comprehensive bibliography of works in English in the Twentieth Century.

One might also take a look at Huggett , Ch. For introductions to the mathematical ideas behind the modern resolutions, the Appendix to Salmon or Stewart are good starts; Russell and Courant et al. Aristotle atomism: ancient continuity and infinitesimals Dedekind, Richard: contributions to the foundations of mathematics Parmenides Plato Pythagoras quantum theory: quantum gravity set theory set theory: early development space and time: being and becoming in modern physics space and time: supertasks temporal parts time Zeno of Elea.

Those familiar with his work will see that this discussion owes a great deal to him; I hope that he would find it satisfactory. I would also like to thank Eliezer Dorr for bringing to my attention some problems with my original formulation of the argument from finite size, an anonymous referee for some thoughtful comments, and Georgette Sinkler for catching errors in earlier versions.

Background 2. The Paradoxes of Plurality 2. The Paradoxes of Motion 3. Two More Paradoxes 4. Background Before we look at the paradoxes themselves it will be useful to sketch some of their historical and logical significance. But if they are as many as they are, they would be limited.

If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited. For if it were of no size and was added, it cannot increase in size.

And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing.

For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if 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 is fleeing will traverse some amount …. And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount. Aristotle Physics , a23 When he sets up his theory of place—the crucial spatial notion in his theory of motion—Aristotle lists various theories and problems that his predecessors, including Zeno, have formulated on the subject.

Aristotle Physics , a19 In context, Aristotle is explaining that a fraction of a force many not produce the same fraction of motion. Further Readings After the relevant entries in this encyclopedia, the place to begin any further investigation is Salmon , which contains some of the most important articles on Zeno up to , and an impressively comprehensive bibliography of works in English in the Twentieth Century.

Bibliography Abraham, W. Barnes ed. Arntzenius, F. Bell, J. Belot, G. Callender and N. Huggett eds , Cambridge: Cambridge University Press. Bergson, H. Mitchell trans. Black, M. Cohen, S. In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of meters, for example. If we suppose that each racer starts running at some constant speed one very fast and one very slow , then after some finite time, Achilles will have run meters, bringing him to the tortoise's starting point.

During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

That which is in locomotion must arrive at the half-way stage before it arrives at the goal. Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on. This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible finite first distance could be divided in half, and hence would not be first after all.

Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.

An alternative conclusion, proposed by Henri Bergson, is that motion time and distance is not actually divisible. This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness.

It is also known as the Race Course paradox. Some, like Aristotle, regard the Dichotomy as really just another version of Achilles and the Tortoise. There are two versions of the dichotomy paradox. In the other version, before Homer could reach the end of the path, he must reach half of the distance to it. Before reaching the last half, he must complete the next quarter of the distance.

Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be accomplished before he could reach the end of the path.

Expressed this way, the dichotomy paradox is very much analogous to that of Achilles and the tortoise. If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.

In the arrow paradox also known as the fletcher's paradox , Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one duration-less instant of time, the arrow is neither moving to where it is, nor to where it is not.

It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points. Description of the paradox from the Routledge Dictionary of Philosophy :.

The argument is that a single grain of millet makes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, an absurd conclusion. Zeno is wrong in saying that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling.

In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially. This is a Parmenidean argument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound. For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius' commentary On Aristotle's Physics.

According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions.

Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes. Aristotle also distinguished "things infinite in respect of divisibility" such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same from things or distances that are infinite in extension "with respect to their extremities".

Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles. Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved.

Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time. Before BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller.

Modern calculus achieves the same result, using more rigorous methods see convergent series, where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent. These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.

Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times.

In this view motion is a function of position with respect to time. Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest.

Peter Lynds has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist. Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position for if it did, it could not be in motion , and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes.

For more about the inability to know both speed and location, see Heisenberg uncertainty principle. Another proposed solution is to question one of the assumptions Zeno used in his paradoxes particularly the Dichotomy , which is that between any two different points in space or time , there is always another point.

Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space, if not on time and space themselves.

According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem". According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.

Hans Reichenbach has proposed that the paradox may arise from considering space and time as separate entities. In a theory like general relativity, which presumes a single space-time continuum, the paradox may be blocked. Infinite processes remained theoretically troublesome in mathematics until the late 19th century.

The epsilon-delta version of Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.