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. The derivative is defined in terms of the ratio of infinitesimals, in the style of Leibniz, rather than in terms of a limit as in standard real analysis in the style of Weierstrass. What makes them nonstandard is especially that they contain infinitely large hyper integers.
For nonstandard calculus one needs nonstandard models of real analysis rather than just of arithmetic. An important feature demonstrating the usefulness of nonstandard analysis is that it achieves essentially the same theorems as those in classical calculus.
McLaughlin believes this approach to the paradoxes is the only successful one, but commentators generally do not agree with that conclusion, and consider it merely to be an alternative solution.
See Dainton pp. Abraham Robinson in the s resurrected the infinitesimal as an infinitesimal number, but F. Lawvere in the s resurrected the infinitesimal as an infinitesimal magnitude. Smooth infinitesimal analysis retains the intuition that a continuum should be smoother than the continuum of the Standard Solution.
Unlike both standard analysis and nonstandard analysis whose real number systems are set-theoretical entities and are based on classical logic, the real number system of smooth infinitesimal analysis is not a set-theoretic entity but rather an object in a topos of category theory, and its logic is intuitionist Harrison, , p.
For more discussion see note 11 in Dainton pp. What influence has Zeno had? He had none in the East, but in the West there has been continued influence and interest up to today. Before Zeno, philosophers expressed their philosophy in poetry, and he was the first philosopher to use prose arguments.
This new method of presentation was destined to shape almost all later philosophy, mathematics, and science. An oft-repeated story tells of his bravery under torture and the painful death which he endured. It is possible that Zeno wrote more than one work, but he is best known for a single volume of epicheiremata attacks on the postulates of Parmenides's opponents.
Only fragments of this work have survived, but a fairly clear idea of his methods may be found in the summaries given by Aristotle and the 6th-century A. Neoplatonist Simplicius. Zeno seems to have had no constructive theories of his own to set forth, and some of his destructive arguments seem to apply equally well to the conclusions drawn by Parmenides. Zeno's original contribution to thought was the method of deduction which he developed to work out two sets of contradictory conclusions from a given postulate.
A text with Translation and Commentary Cambridge, F Cajori, The history of Zeno's arguments on motion, Amer. H Frankel, Zeno of Elea's attacks on plurality, Amer. Philology 63 , 1 - 25 ; - Bucuresti Ser. Acta Logica 10 , 5 - Mathesis 3 1 , 3 - J Lear, A note on Zeno's arrow, Phronesis 26 , 91 - Aristotelian Soc.
Mathesis 6 3 , - P Urbani, Zeno's paradoxes and mathematics : a bibliographic contribution Italian , Arch. G Vlastos, A note on Zeno's arrow, Phronesis 11 , 3 - G Vlastos, Zeno's race course, J. M Zangari, Zeno, zero and indeterminate forms: Instants in the logic of motion, Australasian Journal of Philosophy 72 , - The remaining argument, the antinomy of large and small see 2.
Socrates might easily have been taking it for granted that, for Zeno, such motion goes along automatically with plurality. More formal reconstructions are possible and available. But such efforts can come at the cost of historical accuracy, which is the primary goal of this article. But if they are just as many as they are, they will be limited.
If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. This is the only Zenonian antinomy that has the appearance of being preserved in its entirety. The argument here may be reconstructed as follows. Its overall structure is: If there are many things, then there must be finitely many things; and if there are many things, then there must be infinitely many things.
The assumption that there are many things is thus supposed to have been shown to lead to contradiction, namely, that things are both finitely and infinitely many. The particular argument for the first arm of the antinomy seems to be simply: If there are many things, then they must be just so many as they are.
If the many things are just so many as they are, they must be finitely many. Therefore, if there are many things, then there must be finitely many things. In fact, the argument depends on a postulate specifying a necessary condition upon two things being distinct, rather than on division per se , and it may be reconstructed as follows: If there are many things, they must be distinct, that is, separate from one another.
Postulate: Any two things will be distinct or separate from one another only if there is some other thing between them. Two representative things, x 1 and x 2 , will be distinct only if there is some other thing, x 3 , between them. In turn, x 1 and x 3 will be distinct only if there is some other thing, x 4 , between them.
Since the postulate can be repeatedly applied in this manner unlimited times, between any two distinct things there will be limitlessly many other things. Therefore, if there are many things, then there must be limitlessly many things. Indeed, in this argument he shows that what has neither magnitude nor thickness nor bulk would not even exist. And so what was added would just be nothing. And the same account applies to the part out ahead. For that part too will have magnitude and will have part of it out ahead.
Indeed, it is the same to say this once as always to keep saying it; for no such part of it will be last, nor will one part not be related to another. Each of the many is the same as itself and one.
Whatever has magnitude can be divided into distinguishable parts; whatever has distinguishable parts is not everywhere the same as itself; thus, whatever has magnitude is not everywhere, and so is not genuinely, the same as itself.
Whatever is not the same as itself is not genuinely one. Thus, whatever has magnitude is not genuinely one. Therefore, each of the many has no magnitude. Each of the many has some magnitude and thickness from the lemma. Whatever has some magnitude and thickness will have distinguishable parts, so that each of the many will have parts.
If x is one of the many, then x will have parts. Since each of these parts of x has some magnitude and thickness, each of these parts will have its own parts, and these parts will in turn have parts of their own, and so on, and so on, without limit. Thus each of the many will have a limitless number of parts.
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.
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