DOCSLIB.ORG

Zeno of Elea

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Zeno of Elea Zeno of Elea Biography Zeno, commonly known as ‘Zeno of Elea’, was born in the 5th century B.C.E. in Elea in the Ancient Greek civilization (also known by its Roman name, Velia, and located in present day region of Campania in southern Italy). Biographical details of Zeno’s life are based primarily on Plato’s accounts in his book called Parmenides. Zeno’s birth year of 490 B.C.E. is approximated by Plato’s statement that Zeno was ​ ​ roughly 40 years old when Socrates was a young man [“Zeno’s Paradoxes” Stanford Encyclopedia of ​ ​ Philosophy]. And he died at the age of about 60 in roughly 430 B.C.E. Zeno was a disciple of Parmenides ​ of Elea, the founder of the Eleatic school of philosophy, and Zeno is considered to be Parmenides’ most important pupil albeit Parmenides also had significant impact on Plato [Parmenides by Plato]. ​ ​ In terms of Zeno’s life and work, very little is reliably known. And the handful of details that have emerged are from Plato’s accounts that were based on either gossip (for example, the claim that Zeno and Parmenides were lovers [“Zeno’s Paradoxes” Stanford Encyclopedia of Philosophy]) or what modern day ​ ​ ​ scholars claim to simply be fabrications of Plato for narrative purposes (for example, the story of Parmenides taking Zeno to Athens where he had encounters with Socrates [“Zeno’s Paradoxes” Internet ​ Encyclopedia of Philosophy]). There’s another story about Zeno’s life that’s reported by Diogenes ​ Laertius 700 years later in the 300 C.E. whereby Zeno had been arrested for supplying weapons to rebels fighting against a tyrant ruler of Elea. When caught, he wished to whisper something privately to the tyrant and is said to have bit his ear and didn’t let go until being stabbed to death [“Zeno’s Paradoxes” Internet Encyclopedia of Philosophy]. ​ Introduction to Paradoxes A paradox is defined as a “seemingly absurd or self-contradictory statement that when investigated may prove to be well founded or false” [Oxford Dictionary]. Even though a paradox is against the common ​ ​ ​ opinion, it is still a proposition that has been reached through some arguments. Therefore, in order to prove or disprove a paradoxical proposition one needs to carefully establish the validity of the arguments that lead to the absurd conclusion. Zeno can’t be regarded merely as a sophist (a philosopher who reasons with clever but fallacious arguments) because it does not appear that his aim was to simply confound by verbal trickery [Zeno’s ​ Paradoxes by Wesley C. Salmon (page 6)]. At his time, mathematics had been brought to a high level of sophistication but one that was founded upon a series of conclusions and many believe that Zeno fervently sought to expose these conclusions. But regardless of this historical debate as to whether or not his arguments were specifically targeted against the Pythagoreans or against some other philosophical school, his arguments did compel mathematicians to distinguish arithmetic from geometry [“Zeno and the Mathematicians” by G.E.L. Owen Proceedings of the Aristotelian Society. N.S. LVIII (1957-58), ​ ​ 199-222]. His paradoxes are believed to have been designed to portray that attempts to provide a mathematical description of motion are quite easily entangled in contradictions [Modern Science and the ​ Refutation of the Paradoxes of Zeno by Adolf Grunbaum]. ​ If we take a closer look at the history of ancient greek mathematics, we notice that greek mathematicians were studying the concepts of continuity and things infinitely large or infinitely small. In particular they reached the following two hypotheses: that a finite length can be divided infinitely many times or a finite length is a composition of many small indivisible parts. There were two schools that subscribed to these hypotheses. The platonic school of thought accepted the fact that lengths can be divided infinitely many times. There was another school that accepted the second hypothesis, namely the school of Democritus rejects the theory of continuity and believed in the theory of the “atom”. However, where does Zeno and the Eleatic school lie? The answer is nowhere. The Eleatic school rejects both hypotheses, because they think that this is a fake problem. They believe that the concept of a “thing” is one that cannot be altered. The impact of his arguments exposing problems central to our understanding and our theories of space, time, continuity and infinity are illustrated by Bertrand Russell: “Zeno’s arguments, in some form, have afforded grounds for almost all [theories on these topics that] have been constructed from his time to our own” [Zeno’s Paradoxes by Wesley C. Salmon (page 5)]. ​ ​ Zeno’s Method Zeno’s method of argumentation and proof is also believed to be amongst the first cases of reductio ad ​ absurdum (proof by contradiction) whereby he divided cases into separate arguments with each argument depending on some hypothesis. And his proof was based on reducing the hypothesis to absurdity [“Zeno and the Mathematicians” by G.E.L. Owen Proceedings of the Aristotelian Society. N.S. LVIII (1957-58), ​ ​ 199-222]. Another important characteristic of Zeno’s arguments was their ad hominem nature. His arguments ​ ​ assume the premises granted by his opponents and by the virtue of granting these premises, he imposes a need for his opponents to deny other (often absurd) consequences that can be deduced from the premises [The Problem of Infinity Considered Historically by Bertrand Russell]. ​ ​ The combination of these methods led to Zeno’s invention (or at the very least, a systematic and widespread establishment) of the dialectic method . Aristotle attributing credit to Zeno for this invention is regarded as an important reason for why Zeno has maintained an historically important position in the development of logic [Zeno’s Paradoxes by Wesley C. Salmon (page 7)]. The dialectic method ​ necessitates a formal dialogue between two speakers and relies on an extensive use of rebuttals by reductio ad absurdum. This process was revolutionary because it embodied a shift away from poetry to a ​ prose that required explicit premises and conclusions [“Zeno’s Paradoxes” Internet Encyclopedia of ​ Philosophy]. ​ Paradox 1: The Dichotomy The Dichotomy paradox is one the most interesting paradoxes proposed by Zeno. Aristotle mentions this paradox extensively in his book Physics, and he is the one that gave the name Dichotomy to the paradox. ​ ​ The Dichotomy paradox goes as follows: Suppose we have an object or human being named A. Also suppose we have a starting point S and an endpoint F that are different from each other, and suppose that A wants to move from S to F. Then the following happens: (1) There is an infinite sequence sequence Qp of points between S and F (namely, the point half-way from S to F, the point half-way from that point to F, the point half-way from that point to F and so on.) (2) If A moves from S to F, when it reaches F it has touched one by one in a finite time all the points in Qp . So we have the following: (a) if A moves from S to F, when it reaches F it has touched one by one all the points in a sequence of infinite points. (b) It is impossible to have touched one by one in a finite time all the points in an infinite sequence. Therefore, (3) A does not move from S to F. This is the progressive version of the paradox, there is another version that is called the regressive version, and we shall see and analyze it later. Now we shall analyze the assumptions made by Zeno and establish their validity. As we know if we want to disprove the conclusion of this argument, we need to show that at least one of the assumptions does not hold. Lets try and show that the argument “it is impossible to have touched one by one in a finite time all the points in an infinite sequence”. First, we need to define what an enumerable set is. A set (finite or infinite) is called enumerable when every element in the set can be included in a single list that has a first entry, second entry, etc. For example the set of natural numbers ℕ (an infinite set) is enumerable, with the function f(x) = x , identity function being an enumeration. We want to show that the sequence Qp is enumerable. J.A Faris in The Paradoxes of Zeno argues that this sequence is enumerable for the following reasons. ​ Faris quotes the book Computability and Logic by Boolos and Jeffrey and assumes that there is a being ​ ​ that he names Zeus that has all the human and superhuman powers and it can enumerate all the elements of such a sequence in a finite time. Since Zeus has all this powers it is logical that he can “touch” all the points in the sequence Qp in a finite time. Boolos and Jeffrey provide the following argument to support the previous claim: If a set is enumerable, Zeus can enumerate it in one second by writing out an infinite list faster and faster. He spends ½ second writing the first entry in the list, ¼ second writing the second entry, ⅛ writing the third, etc. He writes each entry in half time he spent on its predecessor. At no point during the one second interval has he written out the whole list, however when one second has passed, the list is complete. [Computability and Logic, Boolos and Jeffrey]. This argument is irrefutable, simply ​ ​ because the being named Zeus has no reason of not being able to complete such a task.
Load more

Recommended publications
  • Zeno of Elea: Where Space, Time, Physics, and Philosophy Converge
    Western Kentucky University TopSCHOLAR® Honors College Capstone Experience/Thesis Honors College at WKU Projects Fall 2007 Zeno of Elea: Where Space, Time, Physics, and Philosophy Converge An Everyman’s Introduction to an Unsung Hero of Philosophy William Turner Western Kentucky University Follow this and additional works at: http://digitalcommons.wku.edu/stu_hon_theses Part of the Other Philosophy Commons, Other Physics Commons, and the Philosophy of Science Commons Recommended Citation Turner, William, "Zeno of Elea: Where Space, Time, Physics, and Philosophy Converge An Everyman’s Introduction to an Unsung Hero of Philosophy" (2007). Honors College Capstone Experience/Thesis Projects. Paper 111. http://digitalcommons.wku.edu/stu_hon_theses/111 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Honors College Capstone Experience/ Thesis Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. P │ S─Z─T │ P Zeno of Elea: Where Space, Time, Physics, and Philosophy Converge An Everyman’s Introduction to an Unsung Hero of Philosophy Will Turner Western Kentucky University Abstract Zeno of Elea, despite being among the most important of the Pre-Socratic philosophers, is frequently overlooked by philosophers and scientists alike in modern times. Zeno of Elea’s arguments on have not only been an impetus for the most important scientific and mathematical theories in human history, his arguments still serve as a basis for modern problems and theoretical speculations. This is a study of his arguments on motion, the purpose they have served in the history of science, and modern applications of Zeno of Elea’s arguments on motion.
    [Show full text]
  • Diogenes Laertius, Vitae Philosophorum, Book Five
    Binghamton University The Open Repository @ Binghamton (The ORB) The Society for Ancient Greek Philosophy Newsletter 12-1986 The Lives of the Peripatetics: Diogenes Laertius, Vitae Philosophorum, Book Five Michael Sollenberger Mount St. Mary's University, [email protected] Follow this and additional works at: https://orb.binghamton.edu/sagp Part of the Ancient History, Greek and Roman through Late Antiquity Commons, Ancient Philosophy Commons, and the History of Philosophy Commons Recommended Citation Sollenberger, Michael, "The Lives of the Peripatetics: Diogenes Laertius, Vitae Philosophorum, Book Five" (1986). The Society for Ancient Greek Philosophy Newsletter. 129. https://orb.binghamton.edu/sagp/129 This Article is brought to you for free and open access by The Open Repository @ Binghamton (The ORB). It has been accepted for inclusion in The Society for Ancient Greek Philosophy Newsletter by an authorized administrator of The Open Repository @ Binghamton (The ORB). For more information, please contact [email protected]. f\îc|*zx,e| lîâ& The Lives of the Peripatetics: Diogenes Laertius, Vitae Philosoohorum Book Five The biographies of six early Peripatetic philosophers are con­ tained in the fifth book of Diogenes Laertius* Vitae philosoohorum: the lives of the first four heads of the sect - Aristotle, Theophras­ tus, Strato, and Lyco - and those of two outstanding members of the school - Demetrius of Phalerum and Heraclides of Pontus, For the history of two rival schools, the Academy and the Stoa, we are for­ tunate in having not only Diogenes' versions in 3ooks Four and Seven, but also the Index Academicorum and the Index Stoicorum preserved among the papyri from Herculaneum, But for the Peripatos there-is no such second source.
    [Show full text]
  • THE PHILOSOPHY BOOK George Santayana (1863-1952)
    Georg Hegel (1770-1831) ................................ 30 Arthur Schopenhauer (1788-1860) ................. 32 Ludwig Andreas Feuerbach (1804-1872) ...... 32 John Stuart Mill (1806-1873) .......................... 33 Soren Kierkegaard (1813-1855) ..................... 33 Karl Marx (1818-1883).................................... 34 Henry David Thoreau (1817-1862) ................ 35 Charles Sanders Peirce (1839-1914).............. 35 William James (1842-1910) ............................ 36 The Modern World 1900-1950 ............................. 36 Friedrich Nietzsche (1844-1900) .................... 37 Ahad Ha'am (1856-1927) ............................... 38 Ferdinand de Saussure (1857-1913) ............. 38 Edmund Husserl (1859–1938) ....................... 39 Henri Bergson (1859-1941) ............................ 39 Contents John Dewey (1859–1952) ............................... 39 Introduction....................................................... 1 THE PHILOSOPHY BOOK George Santayana (1863-1952) ..................... 40 The Ancient World 700 BCE-250 CE..................... 3 Miguel de Unamuno (1864-1936) ................... 40 Introduction Thales of Miletus (c.624-546 BCE)................... 3 William Du Bois (1868-1963) .......................... 41 Laozi (c.6th century BCE) ................................. 4 Philosophy is not just the preserve of brilliant Bertrand Russell (1872-1970) ........................ 41 Pythagoras (c.570-495 BCE) ............................ 4 but eccentric thinkers that it is popularly Max Scheler
    [Show full text]
  • Seminar in Greek Philosophy Pyrrhonian Scepticism 1
    PHILOSOPHY 210: SEMINAR IN GREEK PHILOSOPHY 1 PYRRHONIAN SCEPTICISM Monte Johnson [email protected] UCSD Fall 2014 Course Description Pyrrhonian scepticism, as represented in the works of Sextus Empiricus, presents both a culmination and critique of the whole achievement of Greek philosophy, and was a major influence on the renaissance and the scientific revolutions of the seventeenth century, and continues to influence contemporary epistemology. In this seminar, we will get a general overview of Pyrrhonian scepticism beginning with the doxographies in Book IX of Diogenes Laertius, Lives of the Famous Philosophers (including Heraclitus, Xenophanes, Parmenides, Zeno of Elea, Leucippus, Democritus, Protagoras, Diogenes of Apollonia, Anaxarchus, Pyrrho, and Timon). The core part of the course will consist of a close reading and discussion of the three books of the Outlines of Pyrhhonian Scepticism by Sextus Empiricus, along with a more detailed examination of the treatment of logic, physics, and ethics in his Against the Dogmatists VII-XI. The last three weeks of the seminar will be devoted to student presentations relating Pyrrhonian scepticism to their own interests in philosophy (whether topical or historical). Goals • Learn techniques of interpreting and criticizing works of ancient philosophy in translation, including fragmentary works. • Obtain an overview of ancient scepticism, especially Pyrrhonian scepticism, its textual basis, predecessors, and influence upon later philosophy and science. • Conduct original research relating ancient skepticism to your own philosophical interests; compile an annotated bibliography and craft a substantial research paper. • Develop skills in discussing and presenting philosophical ideas and research, including producing handouts, leading discussions, and fielding questions. Evaluation 10% Participation and discussion.
    [Show full text]
  • Zeno of Elea Born Around 489 BC This Zeno, As Opposed to Zeno Of
    Zeno of Elea Born around 489 BC This Zeno, as opposed to Zeno of Citium, is perhaps most famous for his paradoxes. These strange scenarios are designed to make the think about mathematics and physics in new and different ways. Zeno was opposed to many of the ideas of Pythagoras, supported many of the ideas of Parmenides. His paradoxes have been reconstructed in slightly different ways by modern scientists from his cryptic and fragmentary writings. • Let us suppose that reality is made up of units. These units are either with magnitude or without magnitude. If the former, then a line for example, as made up of units possessed of magnitude, will be infinitely divisible, since, however far you divide, the units will still have magnitude and so be divisible. But in this case, the line will be made up of an infinite number of units, each of which is possessed of magnitude. The line, then, must be infinitely great, as composed of an infinite number of bodies. Everything in the world, then, must be infinitely great, and a forteriori the world itself must be infinitely great. Suppose, on the other hand, that the units are without magnitude. In this case, the whole universe will also be without magnitude, since, however many units you add together, if none of them has any magnitude, then the whole collection of them will also be without magnitude. But if the universe is without any magnitude, it must be infinitely small. Indeed, everything in the universe must be infinitely small. Either everything in the universe is infinitely great, or everything in the universe is infinitely small.
    [Show full text]
  • Humor As Philosophical Subversion, Especially in the Skeptics
    Humor as Philosophical Subversion, Especially in the Skeptics Richard Bett 1. Introduction Aristotle is not exactly a comedian. He wrote about comedy in the lost second book of the Poetics, and, as discussed in another paper in this volume, he wrote about wittiness (εὐτραπελία) in his ethical works. But he does not exhibit much of either. What humor there is in Aristotle seems to fall into two main varieties. First, there is word-play that engages the reader’s attention, which can perhaps be seen as an instance of a technique he describes in Rhetoric 3.10, that of saying “smart things and things that create a good impression” (τὰ ἀστεῖα καὶ τὰ εὐδοκιµοῦντα, 1410b6).1 Early in the Nicomachean Ethics, he says that in endeavoring to determine the principles (ἀρχαί) of ethics, we should begin (ἀρκτέον) with things known to us (1095b2-4). A little later, introducing the idea of the function (ἔργον) of a human being, he asks whether we can seriously consider that a human being as such (as opposed to people in various occupations) is ἀργόν (1097b28- 30) – which is intentionally ambiguous between “without function” and “lazy.” In De Caelo, introducing the topic of minimal magnitudes, he says that positing such a minimal magnitude (τοὐλάχιστον) will make the biggest difference (τὰ µέγιστα) in mathematics (271b10-11). And in De Interpretatione, discussing names, he says that “non-human 1 Unless otherwise noted, translations are my own. In the case of Timon, I sometimes draw on translations in Bett 2000 and Bett 2015. In the case of Sextus I generally draw on Bett 1997, Bett 2005, and Bett 2012.
    [Show full text]
  • Early Greek Philosophy
    P1: GNK/ABS P2: CSS/SCM P3: CSS/SCM QC: ANG/ADS T1: ADS CB162/Long CB162-FM1 January 29, 1999 13:56 The Cambridge Companion to EARLY GREEK PHILOSOPHY Edited by A. A. Long University of California, Berkeley iii P1: GNK/ABS P2: CSS/SCM P3: CSS/SCM QC: ANG/ADS T1: ADS CB162/Long CB162-FM1 January 29, 1999 13:56 published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge cb2 2ru, uk http: //www.cup.cam.ac.uk 40 West 20th Street, New York, ny 10011-4211, usa http: //www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia c Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United States of America Typeset in Trump Medieval 10/13 pt. in LATEX[tb] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data The Cambridge companion to early Greek philosophy/edited by A. A. Long. p. cm. Includes bibliographical references (p. ) and indexes. isbn 0-521-44122-6 (hbk.) isbn 0-521-44667-8 (pbk.) 1. Philosophy, Ancient. 1. Long, A. A. B188.C35 1999 182 –dc21 98-38077 CIP isbn 0 521 44122 6 hardback isbn 0 521 44667 8 paperback iv P1: GNK/ABS P2: CSS/SCM P3: CSS/SCM QC: ANG/ADS T1: ADS CB162/Long CB162-FM1 January 29, 1999 13:56 contents Contributors page vii Preface xi Source abbreviations xv Lives and writings of the early Greek philosophers xvii Chronology xxix Map xxxi 1 The scope of early Greek philosophy a.
    [Show full text]
  • The Fragments of Zeno and Cleanthes, but Having an Important
    ,1(70 THE FRAGMENTS OF ZENO AND CLEANTHES. ftonton: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. ambriDse: DEIGHTON, BELL, AND CO. ltip>ifl: F. A. BROCKHAUS. #tto Hork: MACMILLAX AND CO. THE FRAGMENTS OF ZENO AND CLEANTHES WITH INTRODUCTION AND EXPLANATORY NOTES. AX ESSAY WHICH OBTAINED THE HARE PRIZE IX THE YEAR 1889. BY A. C. PEARSON, M.A. LATE SCHOLAR OF CHRIST S COLLEGE, CAMBRIDGE. LONDON: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE. 1891 [All Rights reserved.] Cambridge : PBIXTKIi BY C. J. CLAY, M.A. AND SONS, AT THK UNIVERSITY PRKSS. PREFACE. S dissertation is published in accordance with thr conditions attached to the Hare Prize, and appears nearly in its original form. For many reasons, however, I should have desired to subject the work to a more under the searching revision than has been practicable circumstances. Indeed, error is especially difficult t<> avoid in dealing with a large body of scattered authorities, a the majority of which can only be consulted in public- library. to be for The obligations, which require acknowledged of Zeno and the present collection of the fragments former are Cleanthes, are both special and general. The Philo- soon disposed of. In the Neue Jahrbticher fur Wellmann an lofjie for 1878, p. 435 foil., published article on Zeno of Citium, which was the first serious of Zeno from that attempt to discriminate the teaching of Wellmann were of the Stoa in general. The omissions of the supplied and the first complete collection fragments of Cleanthes was made by Wachsmuth in two Gottingen I programs published in 187-i LS75 (Commentationes s et II de Zenone Citiensi et Cleaitt/ie Assio).
    [Show full text]
  • Ancient Greek Philosophy. Part 1. Pre-Socratic Greek Philosophers
    Ancient Greek Philosophy. Part 1. Pre-Socratic Greek philosophers. The pre-Socratic philosophers rejected traditional mythological explanations for the phenomena they saw around them in favor of more rational explanations. Many of them asked: From where does everything come? From what is everything created? How do we explain the plurality of things found in nature? How might we describe nature mathematically? The Milesian school was a school of thought founded in the 6th Century BC. The ideas associated with it are exemplified by three philosophers from the Ionian town of Miletus, on the Aegean coast of Anatolia: Thales, Anaximander, and Anaximenes. They introduced new opinions contrary to the prevailing viewpoint on how the world was organized. Philosophy of nature These philosophers defined all things by their quintessential substance (which Aristotle calls the arche) of which the world was formed and which was the source of everything. Thales thought it to be water. But as it was impossible to explain some things (such as fire) as being composed of this element, Anaximander chose an unobservable, undefined element, which he called apeiron. He reasoned that if each of the four traditional elements (water, air, fire, and earth) are opposed to the other three, and if they cancel each other out on contact, none of them could constitute a stable, truly elementary form of matter. Consequently, there must be another entity from which the others originate, and which must truly be the most basic element of all. The unspecified nature of the apeiron upset critics, which caused Anaximenes to define it as being air, a more concrete, yet still subtle, element.
    [Show full text]
  • Kant and Zeno of Elea: Historical Precedents of the “Sceptical Method”1
    Kant and Zeno of Elea Artigos / Articles KANT AND ZENO OF ELEA: HISTORICAL PRECEDENTS OF THE “SCEPTICAL METHOD”1 Giuseppe Micheli2 ABSTRACT: For Kant’s interpretation of Zeno in KrV A502-507/B530-535, scholars have usually referred to Plato’s Phaedrus (261d); in reality the sources Kant uses are, on one hand, Brucker (who depends in turn on the pseudo-Aristotelian De Melisso, Xenophane, et Gorgia, 977 b 2-21), and, on the other, Plato’s Parmenides (135e6-136b1) and Proclus’ commentary on it, as quoted by Gassendi in a popular textbook he wrote on the history of logic. KEYWORDS: Contradictory Propositions. Contrary Propositions. Dialectic. Antinomy. Sceptical Method. Dogmatism. Scepticism. Apagogical Proof. In his “[…] critical solution [Entscheidung] of the cosmological conflict of reason with itself”, Kant explicitly recalls, as his predecessor and model, Zeno of Elea, who apparently had maintained “[…] that God (presumably for him this was nothing but the world) is neither finite nor infinite, is neither in motion nor at rest, and is neither similar nor dissimilar to any other thing. To those who judged him, it appeared that he wanted entirely to deny two mutually contradictory propositions, which is absurd”, and for this reason he “[…] was severely reprimanded by Plato as a wanton sophist”;3 unjustly, according to Kant (who considers more in detail the first of his propositions), since the propositions in question are not contradictory: if they were, by the principle of excluded middle, they could not both be false, and it would be sophistic to maintain their consistency. One is not a mere contradictory of the other, for it “[…] does not merely contradict the other, but says something 1 The quotations from Kant’s works are taken directly from the Academy edition (Kant’s Gesammelte Schriften, Berlin, 1902- ); besides the page and volume number, the line reference is also given where necessary.
    [Show full text]
  • Zeno's Paradoxes
    http://www.iep.utm.edu/zeno-par/ Go DEC JUN JUL ⍰ ❎ 297 captures 10 f 11 Jun 2010 - 10 Jun 2020 2019 2020 2021 ▾ About this capture Zeno’s Paradoxes In the fifth century B.C.E., Zeno of Elea offered arguments that led to conclusions contradicting what we all know from our physical experience—that runners run, that arrows fly, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because many of the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno was the first person to show that the concept of infinity is problematical. In the Achilles Paradox, Achilles races to catch a slower runner—for example, a tortoise that is crawling in a line away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least as far as the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run at least to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that any kind of motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion. Although practically no scholars today would agree with Zeno’s conclusion, we cannot escape the paradox by jumping up from our seat and chasing down a tortoise, nor by saying Zeno should have constructed a new argument in which Achilles takes better aim and runs to some other target place ahead of where the tortoise is.
    [Show full text]
  • Leibniz's Inversion of Zeno
    Leibniz’s Inversion of Zeno: Continuity of Motion, Substantial Action and Plurality1 R. T. W. Arthur Dept. of Philosophy Middlebury College Vermont, USA [email protected] 1 This paper was read at the workshop on Corporeal Substances and the Labyrinth of the Continuum, Università di Firenze, November 30, 2000. I am indebted to several of the participants for their helpful comments, and to Sam Levey, both in general for a long and invaluable correspondence on matters Leibnizian, and in particular for insightful comments on the final draft. Leibniz’s Inversion of Zeno 2 Itaque actio in corpore non nisi per aversionem quandam intelligi potest. Si vero ad vivum reseces, seu si momentum unumquodque inspicias, nulla est. Hinc sequitur Actiones proprias et momentaneas, earum esse rerum quae agendo non mutantur. (A VI.iii 566)2 In what follows I shall attempt to shed light on both the main themes of this volume by considering the relationship of Leibniz’s views to Zeno’s Paradoxes. I do not mean to suggest that Leibniz formulated his views as explicit responses to Zeno of Elea’s famous arguments against motion and plurality (it is probably more accurate to see him as responding to the whole tradition of thought prompted by them, from Plato and Aristotle through Sextus Empiricus and the Scholastics to Galileo Galilei and the moderns). Nevertheless I believe a direct comparison of the two is informative. For I think that in his treatment of the problems of continuous motion and plurality Leibniz employs a characteristic style of arguing—an argument schema—which can usefully be regarded as an inversion of Zeno’s typical way of reasoning.
    [Show full text]