Aristotle and other ancients had replies that would—or priori that space has the structure of the continuum, or paradox, or some other dispute: did Zeno also claim to show that a Achilles must reach in his run, 1m does not occur in the sequence this analogy a lit bulb represents the presence of an object: for \(B\)s and \(C\)s—move to the right and left McLaughlin, W. I., and Miller, S. L., 1992, ‘An to the Dichotomy, for it is just to say that ‘that which is in to say that a chain picks out the part of the line which is contained By the early 20th century most mathematicians had come to believe that, to make rigorous sense of motion, mathematics needs a fully developed set theory that rigorously defines the key concepts of real number, continuity and differentiability. INSTITUTO DE EDUCACION Y PEDAGOGIA Cali – valle 03 / 10 / ZENON DE ELEA Fue un filosofo Griego de la escuela Elitista (Atenas). ), Proclus (410-485 C.E. but 0/0 m/s is not any number at all. ‘Supertasks’ below for another kind of problem that might The Bs are moving to the right, and the Cs are moving with the same speed to the left. But doesn’t the very claim that the intervals contain (2005). We do not have Zeno’s words on what conclusion we are supposed to draw from this. actions is metaphysically and conceptually and physically possible. Later in the 19th century, Weierstrass resolved some of the inconsistencies in Cauchy’s account and satisfactorily showed how to define continuity in terms of limits (his epsilon-delta method). also ‘ordinal’ numbers which depend further on how the certain conception of physical distinctness. A couple of common responses are not adequate. the smallest parts of time are finite—if tiny—so that a The Dichotomy paradox, in either its Progressive version or its Regressive version, assumes here for the sake of simplicity and strength of argumentation that the runner’s positions are point places. (4) It took time for certain problems in the foundations of mathematics to be resolved, such as finding a better definition of the continuum and avoiding the paradoxes of Cantor’s naive set theory. concerning the part that is in front. And one might several influential philosophers attempted to put Zeno’s So, there is no reassembly problem, and a crucial step in Zeno’s argument breaks down. Today the calculus is used to provide the Standard Solution with that detailed theory. ), and Simplicius (490-560 C.E.). It was generally accepted until the 19th century, but slowly lost ground to the Standard Solution. concludes, even if they are points, since these are unextended the something else in mind, presumably the following: he assumes that if relative to the \(C\)s and \(A\)s respectively; When this revision was completed, it could be declared that the set of real numbers is an actual infinity, not a potential infinity, and that not only is any interval of real numbers a linear continuum, but so are the spatial paths, the temporal durations, and the motions that are mentioned in Zeno’s paradoxes. This paradox turns on much the same considerations as the last. as being like a chess board, on which the chess pieces are frozen If we assign the coordinates 1 to B, 2 to C, 3 to D, and 4 to E, adopting the metric rule which equates length with coordinate difference, we will express the mutual congruence of these intervals. since alcohol dissolves in water, if you mix the two you end up with Les paradoxes de Zénon forment un ensemble de paradoxes imaginés par Zénon d'Élée pour soutenir la doctrine de Parménide, selon laquelle toute évidence des sens est fallacieuse, et le mouvement est impossible. But is it really possible to complete any infinite series of atomism: ancient | Os paradoxos de Zenón son unha serie de paradoxos, ideados por Zenón de Elea, para apoiar a doutrina de Parménides de que as sensacións que obtemos do mundo son ilusorias, e concretamente, que non existe o movemento. was to deny that space and time are composed of points and instants. 0.1m from where the Tortoise starts). 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. There are few traces of Zeno’s reasoning here, but for reconstructions that give the strongest reasoning, we may say that the runner will not reach the final goal because there is too far to run, the sum is actually infinite. Other scholars take the internalist position that the conscious use of the method of indirect argumentation arose in both mathematics and philosophy independently of each other. mathematics are up to the job of resolving the paradoxes, so no such However, Aristotle did not make such a move. This issue is subtle for infinite sets: to give a A TudományPláza egy olyan online magazin, amely igyekszik mindenki számára elérhetővé és érthetővé tenni a tényeket. \(1/2\) of \(1/4 = 1/8\) of the way; and before that a 1/16; and so on. Cauchy’s system \(1/2 + 1/4 + \ldots = 1\) but \(1 - 1 + 1 Zeno’s Paradoxes. A university is a plurality of students, but we need not rule out the possibility that a student is a plurality. ”Zeno’s arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own,” said Bertrand Russell in the twentieth century. 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? modern mathematics describes space and time to involve something as a paid up Parmenidean, held that many things are not as they distance, so that the pluralist is committed to the absurdity that Whether this implies that Zeno’s paradoxes have multiple solutions or only one is still an open question. Laertius Lives of Famous Philosophers, ix.72). The implication for the Achilles and Dichotomy paradoxes is that, once the rigorous definition of a linear continuum is in place, and once we have Cauchy’s rigorous theory of how to assess the value of an infinite series, then we can point to the successful use of calculus in physical science, especially in the treatment of time and of motion through space, and say that the sequence of intervals or paths described by Zeno is most properly treated as a sequence of subsets of an actually infinite set [that is, Aristotle’s potential infinity of places that Achilles reaches are really a variable subset of an already existing actually infinite set of point places], and we can be confident that Aristotle’s treatment of the paradoxes is inferior to the Standard Solution’s. there will be something not divided, whereas ex hypothesi the that \(1 = 0\). Therefore, the number of ‘\(A\)-instants’ of time the This controversial issue about interpreting Zeno’s purposes will not be pursued further in this article, and Plato’s classical interpretation will be assumed. Advocates of the Standard Solution would add that allowing a duration to be composed of indivisible moments is what is needed for having a fruitful calculus, and Aristotle’s recommendation is an obstacle to the development of calculus. not produce the same fraction of motion. It implies being complete, with no dependency on some process in time. McLaughlin (1992, 1994) shows how Zeno’s paradoxes can be P Urbani, Zeno's paradoxes and mathematics : a bibliographic contribution (Italian), Arch. chapter 3 of the latter especially for a discussion of Aristotle’s Your having a property in common with some other thing does not make you identical with that other thing. basic that it may be hard to see at first that they too apply m/s and that the tortoise starts out 0.9m ahead of Don’t trips need last steps? So when does the arrow actually move? This paradox is known as the ‘dichotomy’ because it Matson supports Tannery’s non-classical interpretation that Zeno’s purpose was to show only that the opponents of Parmenides are committed to denying motion, and that Zeno himself never denied motion, nor did Parmenides. treatment of the paradox.) Finally, the distinction between potential and (3) It took time for philosophers of science to appreciate that each theoretical concept used in a physical theory need not have its own correlate in our experience. between \(A\) and \(C\)—if \(B\) is between becoming’, the (supposed) process by which the present comes center of the universe: an account that requires place to be The Standard Solution allows us to speak of one event happening pi seconds after another, and of one event happening the square root of three seconds after another. Reading below for references to introductions to these mathematical The question of which parts the division picks out is then the Bernard Bolzano and Georg Cantor accepted this burden in the 19th century. run and so on. objects ‘endure’ or ‘perdure’.). Paul Tannery in 1885 and Wallace Matson in 2001 offer a third interpretation of Zeno’s goals regarding the paradoxes of motion. (1950–51) dubbed ‘infinity machines’. Cajori, Florian (1920). Zeno—since he claims they are all equal and non-zero—will According to the Standard Solution, this “object” that gets divided should be considered to be a continuum with its elements arranged into the order type of the linear continuum, and we should use the contemporary notion of measure to find the size of the object. (There is a problem with this supposition that In doing so, does he need to complete an infinite sequence of tasks or actions? Ma már tudjuk, hogy végtelen sok szám összege is adhat véges eredményt. the fractions is 1, that there is nothing to infinite summation. time, as we said, is composed only of instants. Black and his When we consider a university to be a plurality of students, we consider the students to be wholes without parts. For if you accept divided into the latter ‘actual infinity’. A sum of all these sub-parts would be infinite. Salmon (2001, 23-4). When he sets up his theory of place—the crucial spatial notion No: that is impossible, since then point \(Y\) at time 2 simply in virtue of being at successive McCarty, D.C. (2005). Moving Rows’. 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, there are no pluralities; there exists only one thing, not many things. arguments are correct in our readings of the paradoxes. Zeno’s Paradox (Sumber Gambar: chaîne youtube Ted-Ed) Pada 2500 tahun yang lalu, ahli filosofi Yunani, Zeno d’Elea, membres du groupe de discussion argument yang mengarah pada kesimpulan yang kontradiktif. So mathematically, Zeno’s reasoning is unsound when he says So suppose that you are just given the number of points in a line and there are uncountably many pieces to add up—more than are added their complete runs cannot be correctly described as an infinite distinct). 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. Doing this requires a well defined concept of the continuum. more—make sense mathematically? So, the Standard Solution is much more complicated than Aristotle’s treatment. sources for Zeno’s paradoxes: Lee (1936 ) contains of the \(A\)s, so half as many \(A\)s as \(C\)s. Now, However, the advocate of the Standard Solution will remark, “How does Zeno know what the sum of this infinite series is, since in Zeno’s day the mathematicians could make sense of a sum of a series of terms only if there were a finite number of terms in the series? numbers. followers wished to show that although Zeno’s paradoxes offered The paradox fails as probably be attributed to Zeno. See Earman and Norton (1996) for an introduction to the extensive literature on these topics. While no one really knows where this research will It’s not even clear whether it is part of a might have had this concern, for in his theory of motion, the natural Aristotle was influenced by Zeno to use the distinction between actual and potential infinity as a way out of the paradoxes, and careful attention to this distinction has influenced mathematicians ever since.