Classics Zeno of Elea
by
John Palmer
  • LAST MODIFIED: 10 March 2015
  • DOI: 10.1093/obo/9780195389661-0186

Introduction

Zeno of Elea (c. 490–post-450 BCE) is an early Greek philosopher famous for developing a set of ingenious paradoxes that challenge ordinary assumptions regarding plurality and motion. In Plato’s Parmenides, Zeno is made to endorse a description of his arguments as all aiming to show that there are not many things. Simplicius in his commentary on Aristotle’s Physics quotes one such argument and substantial portions of another. However, Zeno’s most famous paradoxes, reported and criticized by Aristotle in Physics 6, purport to show the incoherence of ordinary assumptions regarding the occurrence of motion rather than the existence of a plurality of things. In these paradoxes of motion, Zeno argued that it is impossible to traverse a stadium, that Achilles can never overtake the tortoise, that the moving arrow is actually at rest, and that a scenario involving bodies moving in opposite directions past one another shows that half the time is equal to its double. Because Aristotle’s reporting is often quite brief and comes amidst his own critical responses, reconstruction of Zeno’s original reasoning is challenging and controversial. A problem with some reconstructions is that, by trying to make Zeno’s arguments proof against Aristotle’s criticisms or by otherwise introducing notions he is unlikely to have employed, they produce arguments that are merely Zenonian rather than plausibly those of the historical Zeno. Zeno’s brilliance and stunning originality are nevertheless apparent. His paradoxes have had a lasting impact through the attempts, from Aristotle down to the present day, to respond to the problems they raise. They have spurred natural philosophers, mathematicians, and scientists to deploy ever more sophisticated tools and to develop more precise conceptions of space, time, motion, and material structure in order to respond to their challenges. We may never know just what led Zeno to develop his famous paradoxes. Because his paradoxes tend to problematize the application of quantitative conceptions to physical bodies and to extensions as ordinarily conceived, they may have originated in reflection upon Pythagorean efforts to apply mathematical notions to the natural world. While it is typically said that Zeno aimed to defend the paradoxical monism of his Eleatic mentor, Parmenides, the Platonic evidence on which this view has resided ultimately fails to support it. A more plausible view, going back to Aristotle, regards Zeno as an influential precursor of sophistic antilogic and eristic disputation whose paradoxical arguments were not driven by a specific doctrinal agenda.

General Overviews

The items in this section provide comprehensive overviews of what is known of Zeno and his paradoxes. The works may be consulted for an orientation to the evidence, the arguments, and the issues they raise before delving more deeply into particular problems and controversies. In his studies of the stadium and the arrow, Gregory Vlastos sets a new standard of analytic precision in the reconstruction of Zeno’s paradoxes that is reflected in Vlastos 1967, which remains worth consulting. Makin 1998 is similar in scope and style but has the advantage of having taken account of more-recent developments in scholarship. The fuller treatment in Barnes 1982 not only provides judicious reconstructions but also assesses responses to the major paradoxes from antiquity to the modern era. Kirk, et al. 1983, cited under Texts, Translations, and Commentaries, also provides a good overview along with its presentation of the Greek text of the source material. There are also several useful overviews online. Palmer 2012 presents the evidence for Zeno’s life and writings and for both the major and minor paradoxes followed by discussion of Zeno’s purposes. Less historically focused are the overviews in Huggett 2010 and Dowden 2013, both of which focus on modern mathematical resolutions of the paradoxes.

  • Barnes, Jonathan. 1982. The Presocratic philosophers. 2d ed. London: Routledge & Kegan Paul.

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    Chapters 12 and 13 on Zeno are among the best in this study of reason in early Greek philosophy. Barnes discusses all the extant arguments attributed to Zeno and concisely surveys lines of response to the major paradoxes from antiquity to the modern era.

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    • Dowden, Bradley. 2013. Zeno’s paradoxes. Internet encyclopedia of philosophy.

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      An account of the paradoxes followed by review of their treatment from Aristotle to the modern era. The focus here is on the “standard solution,” based on calculus, the rest of standard real analysis, and classical mechanics, and various problems it raises.

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      • Huggett, Nick. 2010. Zeno’s paradoxes. In The Stanford encyclopedia of philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford Univ.

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        Provides reconstructions of the paradoxes that take them from their commonsense formulations to their resolutions made possible by the resources of modern mathematics.

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        • Makin, Stephen. 1998. Zeno of Elea. In The Routledge encyclopedia of philosophy. Vol. 9. Edited by Edward Craig, 843–853. London and New York: Routledge.

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          Provides solid reconstructions of the major paradoxes and briefly explores strategies for response.

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          • Palmer, John. 2012. Zeno of Elea. In the Stanford encyclopedia of philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford Univ.

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            Surveys the evidence for Zeno’s life and writings, presents the primary evidence for the major and minor paradoxes with reconstructions of their logic, and discusses Zeno’s likely purposes in propounding the paradoxes.

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            • Vlastos, Gregory. 1967. Zeno of Elea. In The encyclopedia of philosophy. Vol. 8. Edited by Paul Edwards, 369–379. New York and London: Macmillan.

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              Provides reconstructions and discussions of all the extant paradoxes that aim to isolate their more questionable premises, followed by a brief discussion of their influence. Reprinted in Gregory Vlastos, Studies in Greek Philosophy, Volume 1: The Presocratics, edited by Daniel W. Graham (Princeton, NJ: Princeton Univ. Press, 1993), pp. 241–263.

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              Texts, Translations, and Commentaries

              Most of our knowledge of Zeno’s paradoxes comes second hand via the testimony of Aristotle’s Physics. Only a few fragments of Zeno’s own writing have survived, primarily in Simplicius’s commentary on Aristotle’s work. Diels and Kranz 1951–1952 has long been the standard collection of the fragments of the Presocratics and sophists, together with testimonia pertaining to their lives and thought. The Greek texts of the fragments and testimonia relevant to Zeno’s arguments are presented along with English translation and commentary in Lee 1936, which remains useful for its comprehensiveness despite some outmoded interpretations. Kirk, et al. 1983 is uniquely useful in presenting Greek text and English translation interspersed with good discussion of most of the important issues bearing on a historical understanding of the paradoxes. Graham 2010 likewise presents the fragments and a select set of testimonia with facing English translation. Although his introduction and commentary are less helpful than the discussion in Kirk, et al. 1983, Graham usefully includes a number of ancient testimonia not included in that edition or in Diels and Kranz 1951–1952. Students without Greek requiring simply an English translation of the fragments and major testimonia are well served by Waterfield 2000.

              • Diels, Hermann, and Walter Kranz. 1951–1952. Die Fragmente der Vorsokratiker. 6th ed. 3 vols. Berlin: Weidmann.

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                Chapter 29 of this standard edition of the Presocratics and sophists presents a collection of untranslated ancient testimonia on Zeno’s life and thought followed by the text with German translation of the three genuine fragments in Simplicius (B1-3) and one probably spurious fragment in Diogenes Laertius (B4).

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                • Graham, Daniel. W. 2010. The texts of early Greek philosophy: The complete fragments and selected testimonies of the major Presocratics. Vol. 1. Cambridge, UK: Cambridge Univ. Press. 245–270.

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                  Chapter 7 on Zeno presents the Greek texts of the fragments and the ancient testimonia most relevant to the reconstruction and assessment of Zeno’s arguments and purposes with facing English translation and brief introduction and commentary. See pp. 245–270.

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                  • Kirk, Geoffrey S., John E. Raven, and Malcolm Schofield. 1983. The Presocratic philosophers: A critical history with a selection of texts. 2d ed. Cambridge, UK: Cambridge Univ. Press.

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                    Chapter 9 presents the Greek text of the fragments and major testimonia with English translation interspersed with judicious discussion of the major issues that makes this one of the best orientations available.

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                    • Lee, Henry D. P. 1936. Zeno of Elea: A text, with translation and notes. Cambridge, UK: Cambridge Univ. Press.

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                      A collection of virtually all the passages from Aristotle, Simplicius, Philoponus, and Themistius, and other ancient authors relevant to the reconstruction of Zeno’s arguments accompanied by English translation and commentary. More comprehensive than other presentations of the ancient textual evidence.

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                      • Waterfield, Robin. 2000. The first philosophers: The Presocratics and the sophists. Oxford and New York: Oxford Univ. Press.

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                        Presents in its chapter on Zeno (pp. 69–81) English translation without the Greek text of the fragments and major testimonia, accompanied by some pages of introduction and brief bibliography.

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                        Monographs and Collections

                        The articles collected in Salmon 1970 provide a good introduction to modern mathematical responses to the paradoxes. Particularly valuable is the set of articles by Black, Wisdom, Thomson, Benacerraf, and Grünbaum on the problem of “infinity machines.” The modest surviving evidence for Zeno’s life and thought means that he has been the subject of few monographs. Caveing 1982 and Ferber 1995 both undertake to provide truly comprehensive treatments, though with different aims. The Caveing study is valuable for its discussion of a fuller range of ancient testimonia than is typical in most reconstructions, but it is marred somewhat by its commitment to the discredited notion that Zeno was taking particular aim at Pythagorean mathematics. The analysis in Ferber 1995 is more philosophically and scientifically inclined, and it presents some ingenious reconstructions in support of its thesis that all the paradoxes rest on the same basic assumptions regarding the structure of space and time. The approach in Barnes, et al. 2011 to Zeno’s argument that if there are many things, they must be so large as to be infinite, is likewise primarily philosophical in its consideration of the adequacy of the responses that have attempted to isolate some basic error in Zeno’s reasoning.

                        • Barnes, Jonathan, et al. 2011. Eleatica 2008: Zenone e l’infinito. Edited by Livio Rossetti and Massimo Pulpito. Sankt Augustin, Germany: Academia Verlag.

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                          Presents the set of lectures delivered by Barnes at a conference organized by the Fondazione Alario per Elea Velia, followed by a set of subsequently commissioned comments and Barnes’s replies. The lectures focus on Fragment 1 and assesses the success of the physical, mathematical, and logical responses.

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                          • Caveing, Maurice. 1982. Zénon d’Elée: Prolégomènes aux doctrines du continu—Étude historique et critique des fragments et témoignages. Paris: Vrin.

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                            Approaches the arguments against plurality and motion by considering the full range of discussion in the ancient commentators, all with a view to reviving the idea that Zeno was targeting Pythagorean mathematicians.

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                            • Ferber, Rafael. 1995. Zenons Paradoxien der Bewegung und die Struktur von Raum und Zeit. 2d ed. Stuttgart: Franz Steiner.

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                              Undertakes to demonstrate that the paradoxes all assume that spatiotemporal points are indivisible and without extension and that any stretch of space or time is a continuum. Draws upon modern mathematics and physics to propose that the paradoxes may be defused by accepting the existence of spatiotemporal atoms. (First edition, Munich: C. H. Beck, 1981.)

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                              • Salmon, Wesley C., ed. 1970. Zeno’s paradoxes. Indianapolis, IN, and New York: Bobbs-Merrill.

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                                An influential collection reprinting studies by Bertrand Russell, Henri Bergson, Max Black, John Wisdom, James Thomson, Paul Benacerraf, G. E. L. Owen, and Adolf Grünbaum. The focus here is on efforts to resolve the paradoxes with the tools of modern mathematics.

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                                Bibliography

                                References in this article to items before 1980 are more selective than those published after that date. The relevant pages of Paquet, et al. 1989 provide a nearly exhaustive listing of scholarship on Zeno for the years prior to 1980. Navia 1993 is less systematic but still useful, particularly for the years 1980–1990. The annotated listings in L’Année Philologique and The Philosopher’s index may be consulted for bibliography for the years since.

                                • L’Année philologique. Paris: Société d’Édition “Les Belles Lettres”.

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                                  The bibliography of record for the field of classical studies. Also available online by subscription.

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                                  • Navia, Luis. E. 1993. The Presocratic philosophers: An annotated bibliography. New York and London: Garland.

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                                    The section on Zeno (pp. 649–704) contains over 200 annotated bibliographical entries.

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                                    • Paquet, Léonce, Michel Roussel, and Yvon Lafrance. 1989. Les Présocratiques: Bibliographie analytique, 1879–1980. 2 vols. Montreal: Bellarmin.

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                                      The second volume contains an annotated listing, chronologically arranged, of scholarship on Zeno through 1980 plus a set of well-designed indices that facilitate finding relevant items in other sections of the bibliography.

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                                      • The Philosopher’s index. Bowling Green, OH: Philosopher’s Information Center.

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                                        The most comprehensive bibliography of scholarly research in philosophy. Especially useful for articles in philosophy journals not indexed in L’Année Philologique. Print edition through March 2014, after which online only.

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                                        Zeno’s Writings and Cultural Context

                                        The most important evidence for Zeno’s writings is the fictionalized account of his book and its publication given toward the beginning of Plato’s Parmenides, where it is suggested that Zeno composed a single treatise comprising numerous arguments cast in the form of antinomies. The Suda, however, attributes four works to Zeno: Eristic Arguments, An Exegesis of Empedocles’ Works, Against the Philosophers, and On Nature. Ebert 2001 argues for the authenticity of the third of these titles. Proclus’s commentary on Plato’s Parmenides suggests that he was familiar with a work transmitted under Zeno’s name containing forty arguments. Dillon 1986 brings this fact to the attention of scholars. Tarrant 1990 extends the work of Dillon 1986. Both Dillon and Tarrant are ultimately cautious about the amount of genuinely Zenonian material the Forty Logoi may have contained. The fact is that we have no better evidence regarding the form in which Zeno originally presented his arguments than the story Plato has him tell in the opening of the Parmenides. Historians of philosophy have primarily considered the place of Zeno’s paradoxes in the development of Presocratic philosophy and ancient Greek physical theory. There is little reliable evidence for situating Zeno in a broader cultural context. How much truth there may be in the ancient accounts of his brave participation in a plot to overthrow a local tyrant reported in Diogenes Laertius’s short “Life of Zeno” (D.L. 9. 25-9) cannot be determined. Plutarch’s report that Pericles heard Zeno expounding on the nature of things in the manner of Parmenides (Plu. Pericles 4.5) suggests that Zeno may indeed have visited Athens and read his famous book, as Plato’s Parmenides implies, to a group of intellectually keen Athenians. Cataldi 2005 considers the Plutarchean and related evidence in detail in an effort to get a clearer view of Zeno’s relation to Pericles. Striking evidence of the cultural impact of Zeno’s arguments is to be found in the interior of a mid-5th century BCE red-figure drinking cup discovered in the Etrurian city of Falerii, and discussed in Hoffman 2004, depicting a heroic figure racing nimbly ahead of a large tortoise, in an amusing “response” to Zeno’s Achilles paradox.

                                        • Cataldi, Silvio. 2005. Filosofi e politici nell’Atene del V secolo a.C. In Da Elea a Samo: Filosofi e politici di fronte all’impero ateniese: Atti del convegno di studi, Santa Maria Capua Vetere, 4–5 giugno 2003. Edited by Luisa Breglia and Marcello Lupi, 95–150. Naples, Italy: Arte Tipografice Editrice.

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                                          Considers at pp. 125–140 the evidence for Zeno’s Athenian sojourn and impact on Pericles amidst a wide-ranging historical discussion of the place of intellectuals in the political life of 5th century BCE. Athens.

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                                          • Dillon, John. 1986. Proclus and the Forty Logoi of Zeno. Illinois Classical Studies 11.1: 35–41.

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                                            Assembles Proclus’s references to Zeno’s treatise in his commentary on Plato’s Parmenides that do not simply depend on the dialogue. Concludes that Proclus possessed a work entitled Forty Logoi of Zeno (vel sim), that Simplicius possessed a copy as well, and that it contained at least some genuine material.

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                                            • Ebert, Teodor. 2001. Why is Evenus called a philosopher at Phaedo 61c? Classical Quarterly 51.2: 423–434.

                                              DOI: 10.1093/cq/51.2.423Save Citation »Export Citation »E-mail Citation »

                                              The final section of this article discusses the evidence in the Suda that Zeno wrote several treatises, including one entitled Against the Philosophers, and argues for the authenticity of this title and for its providing evidence that the Pythagoreans whom he supposes Zeno targeted called themselves “philosophers.”

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                                              • Hoffman, Herbert. 2004. Zeno’s tortoise. Antike Kunst 47.1: 5–9.

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                                                Contains two plates. Identifies the image of a man leaping over a tortoise in a red-figure drinking cup (Rome, Museo Villa Giulia, inv. 3591) discovered in the Etrurian city of Falerii as a depiction inspired by Zeno’s Achilles paradox. Plate 1 is an image of the cup’s interior.

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                                                • Tarrant, Harold. 1990. More on Zeno’s Forty Logoi. Illinois Classical Studies 15.1: 23–37.

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                                                  Further pursues the issues raised by Dillon 1986. Draws attention to how a passage in Proclus’s commentary not adduced by Dillon indicates that the work was already known to certain middle Platonists. Develops cases for and against its authenticity before tentatively concluding against.

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                                                  Zeno and the Mathematicians

                                                  In the first half of the 20th century, the notion, originating with Tannery 1887, that Zeno’s paradoxes were designed to target the Pythagorean mathematics of his day, became a widespread view. Booth 1957 usefully reviews the principal points of this view and argues for the more modest conclusion that Pythagorean reasoning formed an essential background to Zeno’s development of the paradoxes even if he was not specifically targeting it. The even more critical treatment in Owen 1958 should have put the view to rest, but it continues to resurface (e.g., in Caveing 1982, cited under Monographs and Collections). Although Zeno may not have taken aim against mathematicians of his own day, mathematicians ever since have responded to the challenges posed by his arguments. It is fair to say that the revival of interest in the paradoxes in the 20th century was due in large part to the development of modern set theory and mathematical models that have made possible ever more sophisticated solutions. Since the focus of this bibliography is the historical Zeno, it can provide only a small sample of the responses to the paradoxes engendered by the application of tools of modern mathematics. The articles collected in Salmon 1970, cited under Monographs and Collections, are a good place to start. For more sophisticated responses, see, for example, McLaughlin and Miller 1992 and the criticism of that work’s application of the version of nonstandard analysis known as internal set theory to the paradoxes of motion in Alper and Bridger 1997. Harrison 1996 explores the paradoxes from the perspectives of both standard and nonstandard analysis, showing how the latter model has among its surprising results the possibility that a particle can move a finite distance in a finite time while being at rest all along. There has been some backlash against the application of modern mathematical concepts in the reconstruction and analysis of the paradoxes. Papa-Grimaldi 1996 argues that mathematical resolutions miss the essential point of Zeno’s concern with the problems of the one and many relation. A similar point is developed more successfully in Hasper 2006, which argues that the problems driving dichotomy paradox and the antinomy of large and small are more properly mereological than mathematical, the main idea at work being that a whole of parts is nothing more than these parts together.

                                                  • Alper, Joseph S., and Mark Bridger. 1997. Mathematics, models and Zeno’s paradoxes. Synthese 110.1: 143–166.

                                                    DOI: 10.1023/A:1004967023017Save Citation »Export Citation »E-mail Citation »

                                                    Criticizes the solutions to the paradoxes of motion developed in McLaughlin and Miller 1992, and argues for solutions based on a model of standard mathematics wherein the ordinary real numbers are defined in terms of rational intervals.

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                                                    • Booth, N. B. 1957. Were Zeno’s arguments directed against the Pythagoreans? Phronesis 1.2: 90–103.

                                                      DOI: 10.1163/156852857X00021Save Citation »Export Citation »E-mail Citation »

                                                      Provides a skeptical review of the main tenets of the once common view that Zeno’s paradoxes were formulated as a critique of Pythagorean mathematics.

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                                                      • Harrison, Craig. 1996. The three arrows of Zeno: Cantorian and non-Cantorian concepts of the continuum and of motion. Synthese 107.2: 271–292.

                                                        DOI: 10.1007/BF00413609Save Citation »Export Citation »E-mail Citation »

                                                        An exploration of the paradoxes via both standard analysis, wherein the continuum is modeled on the real number line and motion defined as the time derivative of distance, and nonstandard analysis, which allows for infinitesimal distances between the number line’s points and thus produces surprising results for motion.

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                                                        • Hasper, Pieter Sjoerd. 2006. Zeno unlimited. Oxford studies in ancient philosophy 30:49–85.

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                                                          Argues that the mathematical approach to the paradoxes, particularly the dichotomy and the antinomy of large and small, obscures the fact that they turn on conceptual problems regarding mereological composition.

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                                                          • McLaughlin, William I., and Sylvia L. Miller. 1992. An epistemological use of nonstandard analysis to answer Zeno’s objections against motion. Synthese 92.3: 371–384.

                                                            DOI: 10.1007/BF00414288Save Citation »Export Citation »E-mail Citation »

                                                            Employs the version of nonstandard analysis known as internal set theory, interpreted within an empirical context, to argue that the paradox of the dichotomy is without force because it relies upon infinite sets, which always contain nonstandard real numbers, and that the arrow paradox applies only to a finite number of instants of time.

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                                                            • Owen, G. E. L. 1958. Zeno and the mathematicians. Proceedings of the Aristotelian society 58:199–222.

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                                                              Demonstrates that Zeno’s arguments are not aimed specifically against the Pythagoreans but against any and all accounts of how things may be divided into spatial or temporal parts. Contains perceptive discussions of the major paradoxes, though its reconstruction of a comprehensive program for Zeno’s arguments has not won acceptance. Reprinted in G. E. L. Owen, Logic, Science and Dialectic: Collected Papers in Greek Philosophy, edited by Martha Nussbaum (London: Duckworth, 1986), pp. 45–61.

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                                                              • Papa-Grimaldi, Alba. 1996. Why mathematical solutions of Zeno’s paradoxes miss the point: Zeno’s one and many relation and Parmenides’ prohibition. Review of Metaphysics 50.2: 299–314.

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                                                                Argues that mathematical solutions of the paradoxes of motion fail to touch upon the central problem that a One, qua nondivisible, can never become many and that a Many qua divisible, can never be exhausted by division in order to make of it a One.

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                                                                • Tannery, Paul. 1887. Pour l’histoire de la science Hellène: De Thalès a Empédocle. Paris: Félix Alcan.

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                                                                  Chapter 10 is devoted to Zeno and develops the once influential but now generally rejected view that Zeno’s arguments against plurality, in particular, were directed against the purportedly Pythagorean notion that the world is constructed from physical-geometrical points.

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                                                                  The Arguments against Plurality

                                                                  At Plato’s Parmenides 127d-128a1, Zeno is made to respond affirmatively to Socrates’ suggestion that the arguments of his book came cast in the form of antinomies all purporting to demonstrate the untenability of the commonsense presumption that the world contains many things. Two of the extant paradoxes, both reported by Simplicius in his commentary on Aristotle’s Physics, do in fact argue in this manner. In one Zeno purported to demonstrate that if there are many things, they must be both so small as to have no magnitude and so large as to have unlimited magnitude (Zeno 29B1 DK = Simp. in Ph. 140.34–141.8 & Zeno 29B2 DK = Simp. in Ph. 139.7–15). In the other he purported to demonstrate that if there are many things, they must be both limited and unlimited in number (Zeno 29B3 DK = Simp. in Ph. 140.29–33). Adequate reconstructions are provided in all of the works cited under General Overviews. The studies listed in this section provide more detailed analyses and discussions. Vlastos 1971 offers the point of departure for all subsequent reconstructions of the antinomy of large and small. Abraham 1972 ingeniously argues that the antinomy relies on a through and through division rather than a progressive division, which saddles Zeno with the false premise that the sum of an infinite set, each member of which is of finite size, must be infinitely large. The reconstruction in Prior 1978 aims to make the argument effective only against the presumption of plurality and not also against Parmenides’ One. Peterson 1978 is one of the few article-length studies of the antinomy of limited and unlimited. The arguments against plurality have stimulated less discussion than the paradoxes of motion, which are both more famous and generally regarded as raising more profound problems. However, for a far-ranging and more recent treatment of the antinomy of large and small and limited and unlimited, see Barnes, et al. 2011, cited under Monographs and Collections.

                                                                  • Abraham, William E. 1972. The nature of Zeno’s argument against plurality in DK 29B1. Phronesis 17.1: 40–53.

                                                                    DOI: 10.1163/156852872X00213Save Citation »Export Citation »E-mail Citation »

                                                                    Proposes a reconstruction of the antinomy of large and small positing a through and through division rather than a progressive division to make it proof against the common charge of mistakenly supposing that a series of the form 1/2 + 1/4 + 1/8 + 1/16, etc., does not converge.

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                                                                    • Peterson, Sandra. 1978. Zeno’s second argument against plurality. Journal of the History of Philosophy 16.3: 261–270.

                                                                      DOI: 10.1353/hph.2008.0608Save Citation »Export Citation »E-mail Citation »

                                                                      Offers a new reconstruction of Zeno’s argument that, if there are many things, they must be finitely many based on introduction of the premise that for any x, if x ≠ x + 1, then x is finite. Speculates on how Zeno might have justified using this premise.

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                                                                      • Prior, William J. 1978. Zeno’s first argument concerning plurality. Archiv für Geschichte der Philosophie 60.3: 247–256.

                                                                        DOI: 10.1515/agph.1978.60.3.247Save Citation »Export Citation »E-mail Citation »

                                                                        Seeks to improve on the reconstruction of the antimony of large and small in order to support the traditional view of Zeno as a doctrinal adherent of Parmenides against the challenge to this view posed by Solmsen 1971, cited under Zeno’s Purposes.

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                                                                        • Vlastos, Gregory. 1971. A Zenonian argument against plurality. In Essays in ancient Greek philosophy. Edited by John P. Anton and George L. Kustas, 119–144. Albany: State Univ. of New York Press.

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                                                                          Provides a careful reconstruction of the arguments in the antinomy of large and small that fills out Simplicius’s elliptical reporting of the argument for vanishing smallness while improving on previous reconstruction of the argument for infinite largeness. Reprinted in Gregory Vlastos, Studies in Greek Philosophy, vol. 1: The Presocratics, edited by Daniel W. Graham (Princeton, NJ: Princeton Univ. Press, 1993), pp. 219–240.

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                                                                          The Dichotomy and the Achilles

                                                                          In the paradox known as “the dichotomy” (also referred to as “the stadium” or “the runner”), Zeno argued that nothing moves because anything traversing a finite distance must reach its halfway point, but to do so requires reaching the point halfway point to that point, and so ad infinitum. In that known as “the Achilles,” he argued along similar lines that the slowest runner can never be overtaken by the fastest runner once given a head start. The discussion of the dichotomy and the possibility of completing an infinite series of tasks in Vlastos 1966 enters into the debate represented in the treatments of the paradox collected in Salmon 1970, cited under Monographs and Collections. The discussion in Vlastos takes as its point of departure a rejection of Aristotle’s response to the dichotomy. More thorough and sympathetic considerations of Aristotle’s response is to be found in Bostock 1972 and McKirahan 2001. Aristotle’s judgment that the Achilles is simply a variant on the dichotomy has prompted a number of modern attempts to demonstrate that it is not. In addition to that in the appendix of Vlastos 1966, see the fuller discussion in Centrone 1981.

                                                                          • Bostock, David. 1972. Aristotle, Zeno and the potential infinite. Proceedings of the Aristotelian Society 73:37–53.

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                                                                            A thorough exploration of Aristotle’s response to the dichotomy paradox.

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                                                                            • Centrone, Bruno. 1981. Un’indiretta confutazione aristotelica dell’ “Achille” di Zenone. Elenchos 2:273–289.

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                                                                              Argues that the Achilles paradox is distinct from the dichotomy in that it brings into play temporal as well as spatial continuity and thus that Aristotle’s treatment of the Achilles as simply a variant on the dichotomy makes his response to it in Physics 6.9 inadequate and necessitates the return to the paradox in Physics 8.

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                                                                              • McKirahan, Richard. 2001. Zeno’s dichotomy in Aristotle. Philosophical Inquiry 23.1–2: 1–24.

                                                                                DOI: 10.5840/philinquiry2001231/216Save Citation »Export Citation »E-mail Citation »

                                                                                Following some general remarks on how Aristotle treats Zeno apart from Parmenides and Melissus, focuses in detail on how Aristotle in Physics 6.2, 6.9, and 8.8 addresses not only Zeno’s original paradox but two more-sophisticated versions.

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                                                                                • Vlastos, Gregory. 1966. Zeno’s race course. Journal of the History of Philosophy 4.2: 95–108.

                                                                                  DOI: 10.1353/hph.2008.1416Save Citation »Export Citation »E-mail Citation »

                                                                                  Reconstructs the dichotomy as arguing the runner can never reach his goal. Understands the argument as ultimately resting on the idea that it is impossible to complete an infinite sequence of discrete acts. Argues in an appendix that the Achilles is in fact a distinct argument.

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                                                                                  The Arrow

                                                                                  Aristotle’s cryptically concise report of Zeno’s argument that the moving arrow is standing still given that a thing is at rest when it is against what is equal (Physics 6.9.239b5–7) has provoked diverse efforts at reconstructing the original. There has been disagreement about whether Aristotle’s objections that Zeno falsely supposes that time is composed of “nows” or durationless instants (Physics 6.9.239b8-9, 30-33) and that it is a mistake to conceive of an object as either moving or at rest in a now (Physics 6.3.234a24-31) are to the point. While Owen 1958 (cited under Zeno and the Mathematicians) argues against that Aristotle’s objections were irrelevant, Vlastos 1966 is more tolerant. Lear 1981 gives the debate new impetus by suggesting that Zeno might reject the point adduced in both Owen and Vlastos that an object may be moving at an instant even if not in an instant by pointing out that the notion of instantaneous velocity presumes motion over time, the very phenomenon Zeno is arguing against. Lear develops a more robust reconstruction based on understanding the arrow as purportedly moving in the present, designed to make the paradox immune to resolution by standard calculus. White 1982 accepts this reconstruction and proposes that the paradox may be resolved with the tools of nonstandard analysis. The understanding of Aristotle’s analysis of the paradox that figures in Vlastos 1966 and Lear 1981 reconstructions is criticized by Magidor 2008. Diogenes Laertius 9.72 attributes to Zeno the argument that “what moves moves neither in the place it is nor in a place it is not.” Opinion is divided on whether this statement is reliable evidence for the larger antinomy to which the arrow paradox originally belonged. That Sextus Empiricus assigns the argument Diogenes attributes to Zeno to Diodorus Cronus instead (PH 2.245, 3.71, M. 10.86-9) tells against its reliability. Still, Vlastos 1966 argues in favor of situating the arrow paradox within the larger context suggested by Diogenes’ report. More recently, Arsenijević, et al. 2008 has developed a reconstruction according to which Zeno’s argument proceeded via branching premises that allow for the possibility that the arrow is sometimes in a time interval as well as its always being at an instant.

                                                                                  • Arsenijević, Miloš, Sandra Šćepanović, and Gerald J. Massey. 2008. A new reconstruction of Zeno’s “flying arrow.” Apeiron 41.1: 1–43.

                                                                                    DOI: 10.1515/APEIRON.2008.41.1.1Save Citation »Export Citation »E-mail Citation »

                                                                                    Usefully surveys opinions on the interpretative issues affecting reconstruction of the paradox. Proposes understanding the argument as applying to rigid bodies and as proceeding via two branches, and explains how both the dynamic and static theories of motion can be understood as developing via rejection of certain of its premises.

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                                                                                    • Lear, Jonathan. 1981. A note on Zeno’s arrow. Phronesis 26.2: 91–104.

                                                                                      DOI: 10.1163/156852881X00196Save Citation »Export Citation »E-mail Citation »

                                                                                      Argues that the concept of the present instant is crucial to Zeno’s argument. Because the arrow is always in the present instant, when it occupies a space equal to itself and so is at rest, it is never moving.

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                                                                                      • Magidor, Ofra. 2008. Another note on Zeno’s arrow. Phronesis 53.4–5: 359–372.

                                                                                        DOI: 10.1163/156852808X338328Save Citation »Export Citation »E-mail Citation »

                                                                                        Proposes a new interpretation of Aristotle’s solution to the arrow paradox that challenges the common view, shared by Vlastos and Lear, that Aristotle held that nothing can be in motion or at rest at an instant.

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                                                                                        • Vlastos, Gregory. 1966. A note on Zeno’s arrow. Phronesis 11.1: 3–18.

                                                                                          DOI: 10.1163/156852866X00094Save Citation »Export Citation »E-mail Citation »

                                                                                          Argues that the arrow paradox was part of the argument against motion attributed to Zeno in Diogenes Laertius 9.72 and Epiphanius, Against the Heretics 3.11. Endorses and expands on Aristotle’s objection that it is senseless to speak of an object as either moving or at rest in an instant.

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                                                                                          • White, Michael J. 1982. Zeno’s arrow, divisible infinitesimals, and Chrysippus. Phronesis 27.3: 239–254.

                                                                                            DOI: 10.1163/156852882X00168Save Citation »Export Citation »E-mail Citation »

                                                                                            Proposes a resolution of the paradox as reconstructed by Lear 1981 rooted in nonstandard analysis and tentatively suggests that its concept of nontrivial divisible infinitesimals is to be found in the evidence regarding Chrysippus’s doctrines of time, space, and motion.

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                                                                                            The Moving Rows

                                                                                            The fourth paradox of motion Aristotle describes as “the one about the things in the stadium moving from opposite directions, being of equal bulk, alongside things of equal size, with some moving from the end of the stadium and some from the middle, at equal speed, in which case he supposes it turns out that half the time is equal to its double” (Ph. 6.9.239b33–240a1). He goes on to describe a scenario in which two groups of bodies move in opposite directions past a third group of stationary bodies. Reconstruction of Zeno’s original paradox is made even more difficult than usual by the underdetermined and critical reporting of Aristotle, who thinks the argument depends upon a transparent falsehood. Once efforts originating with Tannery 1887, cited under Zeno and the Mathematicians, to sharpen the paradox by positing that the magnitudes involved are meant to be atomic or infinitesimal were rightly rejected as having no basis in the evidence, the challenge remained to understand this paradox as having anything like the depth of the others. Mansfeld 1982 discounts Simplicius’s commentary on Aristotle’s presentation in order to offer a reconstruction according to which there are only two rows and a genuinely paradoxical scenario, noting that the stationary third row is introduced only as part of Aristotle’s effort at refutation. The reconstruction offered in Ferber 1995, cited under Monographs and Collections, likewise posits only two moving rows and proposes that Zeno’s argument depended on the supplied premise that each period of time contains an infinite and therefore equal number of points of time. This reconstruction is critiqued at length in Knorr 1983. Osborne 2001 seeks light on the paradox by considering it in relation to the arrow paradox. Davey 2007 attempts a reconstruction according to which the paradox may be seen to have the same sort of importance for physical theory generally accorded the other paradoxes of motion.

                                                                                            • Davey, Kevin. 2007. Aristotle, Zeno, and the stadium paradox. History of Philosophy Quarterly 24.2: 127–146.

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                                                                                              After a critical review of previous interpretations, offers a new reconstruction of the paradox such that it can be seen as raising important questions about whether the passage of time should be quantified in the measure theoretic or set theoretic sense.

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                                                                                              • Knorr, Wilbur. 1983. Zeno’s paradoxes still in motion. Ancient Philosophy 3.1: 55–66.

                                                                                                DOI: 10.5840/ancientphil19833119Save Citation »Export Citation »E-mail Citation »

                                                                                                A discussion review of the 1981 first edition of Ferber 1995 (cited under Monographs and Collections) critiquing its reconstruction of the moving rows paradox.

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                                                                                                • Mansfeld, Jaap. 1982. Digging up a paradox: A philological note on Zeno’s stadium. Rheinisches Museum für Philologie 125.1: 1–24.

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                                                                                                  By attending to the topographical indications in Aristotle’s account of the paradox and scrupulously distinguishing (unlike Simplicius and those since relying on him) between Zeno’s argument and Aristotle’s refutation of it, offers a reconstruction of the paradox according to which it involves only two rows moving past one another. Reprinted in Jaap Mansfeld, Studies in the Historiography of Greek Philosophy (Assen, The Netherlands: Van Gorcum, 1990), pp. 319–342.

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                                                                                                  • Osborne, Catherine. 2001. Comment mesurer le mouvement dans le vide? Quelques remarques sur deux paradoxes de Zénon d’Élée. In Les anciens savants: Études sur les philosophies préplatoniciennes. Edited by Pierre-Marie Morel and Jean-François Pradeau, 157–165. Les Cahiers philosophiques de Strasbourg 12. Strasbourg, France: Université Marc Bloch.

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                                                                                                    Reviews efforts to give point to the paradox before developing a new understanding of the paradox via comparison with essential features of the arrow paradox.

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                                                                                                    The Minor Paradoxes

                                                                                                    Aristotle briefly introduces two further arguments by Zeno—the paradox of place and the millet seed—that are not straightforwardly construed as arguments against plurality or motion. Aristotle twice mentions a problem posed by Zeno to the effect that since everything that is in a place, if place is, then it must be in a place, and so ad infinitum (Physics 4.1.209a23-5 and 4.3.210b22-3, cf. Simp. in Ph. 562.3-6). His mention of the millet seed paradox is even more allusive: “Zeno’s argument is not correct,” he says, “that any portion of millet seed whatsoever makes a sound” (Ph. 7.5.250a20-1, cf. Simp. in Ph. 1108.18-28). The fuller version reported in Simplicius’s commentary has been taken to anticipate the sorites paradox, development of which is normally credited to the Megarian philosopher Eubulides. What Aristotle and Simplicius report of these paradoxes appears to derive in both cases from some fuller argument by Zeno for which we no longer have any evidence. Discussions have accordingly focused in these instances more on Aristotle’s treatment than on Zeno’s own argumentation. Such is the case with both the treatment of the paradox of place in Morison 2002 and that of the millet seed paradox in Wardy 1990.

                                                                                                    • Morison, Benjamin. 2002. On location: Aristotle’s concept of place. Oxford: Clarendon.

                                                                                                      DOI: 10.1093/0199247919.001.0001Save Citation »Export Citation »E-mail Citation »

                                                                                                      Chapter 3 (pp. 81–102) is devoted entirely to reconstruction and discussion of Zeno’s paradox of place as well as Aristotle’s solution.

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                                                                                                      • Wardy, Robert. 1990. The chain of change: A study of Aristotle’s Physics VII. Cambridge, UK: Cambridge Univ. Press.

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                                                                                                        The commentary on Physics 7.5.250a9-25 at pp. 317–327 discusses the version of the millet seed paradox found in Simplicius, where Zeno is suspiciously represented as engaged in an argument with Protagoras, and explores Aristotle’s response.

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                                                                                                        Zeno’s Purposes

                                                                                                        What led Zeno to develop the paradoxes? The opening of Plato’s Parmenides depicts Zeno on a visit to Athens in the company of Parmenides. Zeno is questioned by Socrates after a reading from his famous book, and Plato has Zeno endorse Socrates’ description of his arguments as combating the natural presumption that there are many things. Socrates also characterizes Zeno as having tried to conceal the fact that in so arguing he has said essentially the same thing as Parmenides. Although some had doubted the reliability of this dialogue’s evidently fictionalized account as a guide to the purposes of the historical Zeno, many have nonetheless seen it as licensing the view that Zeno was a monist like his Eleatic mentor Parmenides who sought to demonstrate in his own fashion that we are misled by sensory evidence if we believe that the world is populated by numerous entities capable of moving from place to place. Solmsen 1971 is a landmark study in that it argued forcefully against reliance on the Parmenides as any sort of guide to the historical Zeno’s purposes. Von Fritz 1974 and Vlastos 1975 both seek to reestablish the basic reliability of the Parmenides’ story in the wake of the Solmsen critique. Prior 1978, cited under Arguments against Plurality, and Makin 1982 both reconsider Zeno’s own arguments in the wake of this controversy in order to stress the fundamental agreement between Parmenides’ monism and Zeno’s rejection of plurality. Others have found a better guide to Zeno’s purposes in Aristotle, who does not associate him with Parmenides but understands his provocative challenges to commonsensical views as making him an important precursor to sophistic antilogic and eristic disputation. Such an Aristotelian view of Zeno as a philosopher without a philosophy is ably advocated in Barnes 1982, cited under General Overviews, and Barnes, et al. 2011, cited under Monographs and Collections. Cordero 1988 resists the view in Barnes 1982 but nevertheless finds Zeno to be a nihilist on a par with the sophist Gorgias. Recent years have also seen a move away from the once dominant view of Parmenides as a strict monist, and this has had important consequences for understanding Zeno’s relation to him. Barnes argues that Melissus, not Parmenides, was the originator of strict monism. In light of its view of Parmenides as a “predicational monist,” Curd 1993 reconsiders Zeno’s relation to Parmenides. Rapp 2006 likewise reconsiders Zeno’s purposes and relation to Parmenides in light of a recognition that there are different types or monism. Palmer 2009 argues that Plato’s testimony does not support the conventional view that Zeno’s arguments aimed to support a Parmenidean strict monism, as part of its broader argument that Parmenides was such a monist at all, and develops the Aristotelian view of Zeno as a proto-sophist.

                                                                                                        • Cordero, Nestor-Luis. 1988. Zénon d’Élée, moniste ou nihiliste? La Parola del Passato 43:100–126.

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                                                                                                          Argues that Zeno must be regarded as a nihilist on a par with the sophist Gorgias given his inability to identify the principle of individuation required by any adequate ontology.

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                                                                                                          • Curd, Patricia Kenig. 1993. Eleatic monism in Zeno and Melissus. Ancient Philosophy 13.1: 1–22.

                                                                                                            DOI: 10.5840/ancientphil199313132Save Citation »Export Citation »E-mail Citation »

                                                                                                            Based on her previously developed account of Parmenides as maintaining that anything that is must be a “predicational unity,” proposes that Zeno’s paradoxes were designed to show that any pluralist attempt to use the Parmenidean account in the service of perceptibles must fail.

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                                                                                                            • Makin, Stephen. 1982. Zeno on plurality. Phronesis 27.3: 223–238.

                                                                                                              DOI: 10.1163/156852882X00159Save Citation »Export Citation »E-mail Citation »

                                                                                                              Develops a reading of the paradoxes as based on the Parmenidean notion that whatever is must be homogenous, as opposed to the notion that whatever is extended is divisible, in order to maintain thesis of agreement between Parmenides’ monism and Zeno’s arguments against plurality.

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                                                                                                              • Palmer, John. 2009. Parmenides and Presocratic philosophy. Oxford and New York: Oxford Univ. Press.

                                                                                                                DOI: 10.1093/acprof:oso/9780199567904.001.0001Save Citation »Export Citation »E-mail Citation »

                                                                                                                Amidst Chapter 5’s critical examination of the notion that Parmenides, Zeno, and Melissus together constitute a doctrinally unified “Eleatic school,” argues that his efforts to demonstrate how the commonsense view of the world is fraught with contradiction made him an influential precursor of sophistic antilogic and eristic disputation.

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                                                                                                                • Rapp, Christof. 2006. Zeno and the Eleatic anti-pluralism. In La costruzione del discorso filosofico nell’età dei presocratici. Edited by Maria M. Sassi, 161–182. Pisa, Italy: Edizioni della Normale.

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                                                                                                                  After distinguishing various species of monism and comparing Parmenides and Melissus’s views on several relevant issues, identifies a number of assumptions underlying Zeno’s argumentation relevant to reassessing his relation to Parmenides.

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                                                                                                                  • Solmsen, Friedrich. 1971. The tradition about Zeno of Elea re-examined. Phronesis 16.2: 116–141.

                                                                                                                    DOI: 10.1163/156852871X00089Save Citation »Export Citation »E-mail Citation »

                                                                                                                    Rejects the reliability of the account of Zeno’s purposes in Plato’s Parmenides and looks instead to Simplicius’s reports of comments by Eudemus and Alexander of Aphrodisias for guidance before concluding that we cannot be positive about Zeno’s intentions.

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                                                                                                                    • Vlastos, Gregory. 1975. Plato’s testimony concerning Zeno of Elea. Journal of Hellenic Studies 95: 136–162.

                                                                                                                      DOI: 10.2307/630875Save Citation »Export Citation »E-mail Citation »

                                                                                                                      Takes Socrates’ remarks in Plato’s Parmenides as the basis for identifying six fundamental points regarding Zeno’s purposes and considers them at some length, though without taking into account Zeno’s corrections in the dialogue of Socrates’ view that he simply advocated Parmenides’ monism by different means.

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                                                                                                                      • Von Fritz, Kurt. 1974. Zeno of Elea in Plato’s Parmenides. In Serta Turyniana: Studies in Greek literature and palaeography in honour of A. Turyn. Edited by J. L. Heller and J. K. Newman, 329–341. Urbana: Univ. of Illinois Press.

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                                                                                                                        While acknowledging that Plato’s representation of Zeno’s relation to Parmenides was not meant to be historically accurate, argues that it nevertheless should not be regarded as pure invention.

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                                                                                                                        Influence in Antiquity

                                                                                                                        Zeno’s arguments inspired natural philosophers to develop theories of the structure of matter, space, and time capable of eluding their paradoxical conclusions. Presocratic atomism is standardly understood as having been developed partially in response to Zeno’s paradoxes of infinite divisibility. Furley 1967 provides a nuanced articulation of this standard view, for which see also Chapter 20 of Barnes 1982, cited under General Overviews, and Chapter 20 of Kirk, et al. 1983, cited under Texts, Translations, and Commentaries. Lewis 1999 accepts the general thesis that Presocratic atomism was developed in response to Zenonian and other Eleatic arguments but criticizes the standard view in order to suggest the Eleatic problems to which the atomists were responding concern division rather than divisibility. A more skeptical perspective on the traditional view is provided by Sedley 2008, where the analysis of Aristotle’s reconstruction of Democritus’s reasoning in On Generation and Corruption 1.2 leads to a downplaying of Zeno’s influence. Chapter 6 of Palmer 2009, cited under Zeno’s Purposes, makes the case for understanding Zeno as having also influenced the altogether different physical theory of Anaxagoras. Aristotle’s responses to the paradoxes of motion are indicative of Zeno’s influence on the development of his own understanding of space, time, motion, and material structure, and Zeno’s continuing influence can be seen in later ancient physical theories as well. Sorabji 1983 and White 1992 do not focus specifically on Zeno but should be consulted by those interested in how later ancient philosophers and natural scientists dealt with problems he had raised. Sorabji devotes a chapter to the paradoxes of motion at the opening of its section on theories regarding the divisibility of matter and time. Those seeking a deeper understanding of how Aristotelian and Hellenistic physical theories dealt with the problems Zeno raised should proceed to White 1992. In his lost dialogue Sophist, Aristotle reportedly identified Zeno as the inventor of dialectic. The reasons and justification for doing so are thoroughly explored in Berti 1988. Paradoxes in the style of Zeno continued to be developed among the Greek philosophers. Aristotle reports post-Zenonian variants of the dichotomy, on which see McKirahan 2001, cited under Dichotomy and the Achilles. For Zeno’s influence on the paradoxes of the Megarian philosopher Eubulides and on Diodorus Cronus, who developed his own paradoxes of motion, see, respectively, Wheeler 1983 and Sedley 1977.

                                                                                                                        • Berti, Enrico. 1988. Zenone di Elea: Inventore della dialettica? La Parola del Passato 43:19–41.

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                                                                                                                          Assesses Aristotle’s reported identification of Zeno as the inventor of dialectic in relation to the other relevant testimony in Aristotle and Plato before undertaking a characterization of Zeno’s dialectical methods vis-à-vis its characterization by Kant and Hegel.

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                                                                                                                          • Furley, David J. 1967. Two studies in the Greek atomists. Princeton, NJ: Princeton Univ. Press.

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                                                                                                                            Considers, as part of its treatment of indivisible magnitudes, the Zenonian arguments most relevant for the formulation of the early atomist hypothesis regarding material structure and discusses how Leucippus and Democritus developed their theory in response to the Eleatic arguments regarding divisibility.

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                                                                                                                            • Lewis, Eric. 1999. The dogmas of indivisibility: On the origins of ancient atomism. In Proceedings of the Boston Area Colloquium in Ancient Philosophy. Vol. 14. Edited by John J. Cleary and Gary M. Gurtler, S. J., 1–21. Leiden, The Netherlands: Brill.

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                                                                                                                              Argues that Presocratic atomism was motivated by Zenonian and other Eleatic arguments problematizing, not divisibility, but the results of the processes of dividing everywhere or dividing infinitely.

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                                                                                                                              • Sedley, David. 1977. Diodorus Cronus and Hellenistic philosophy. Cambridge Classical Journal 23:74–120.

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                                                                                                                                Amidst its thorough overview of Diodorus Cronus’s thought and his place in the history of Greek philosophy, identifies his affiliation to Parmenides and Zeno in terms of their importance in the development of dialectic rather than any doctrinal attachment.

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                                                                                                                                • Sedley, David. 2008. Atomism’s Eleatic roots. In The Oxford handbook of Presocratic philosophy. Edited by Patricia Curd and Daniel W. Graham, 305–332. Oxford and New York: Oxford Univ. Press.

                                                                                                                                  DOI: 10.1093/oxfordhb/9780195146875.001.0001Save Citation »Export Citation »E-mail Citation »

                                                                                                                                  Isolates Aristotle’s historicizing reconstruction of Democritus’s reasoning from his own “neo-Democritean” argument in GC1.2 to suggest that Democritus’s atoms were designed less in response to Zeno than as a plurality of essentially Parmenidean entities.

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                                                                                                                                  • Sorabji, Richard. 1983. Time, Creation, and the Continuum: Theories in antiquity and the early middle ages. Ithaca, NY: Cornell Univ. Press.

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                                                                                                                                    Chapter 21 treats Zeno’s paradoxes of motion as a preface to a comprehensive survey of the development of the notions of atoms, time-atoms, and the continuum in physical theory throughout ancient Greek and Islamic philosophy and in medieval philosophy in the Latin west down to the 14th century.

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                                                                                                                                    • Wheeler, Samuel C. 1983. Megarian paradoxes as Eleatic arguments. American Philosophical Quarterly 20.3: 287–296.

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                                                                                                                                      Discusses the paradoxes of Eubulides, among which are the famous paradoxes of the Liar and the Heap, as evidence for the Eleaticism of the Megarian philosophers.

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                                                                                                                                      • White, Michael J. 1992. The continuous and the discrete: Ancient physical theories from a contemporary perspective. Oxford: Clarendon.

                                                                                                                                        DOI: 10.1093/acprof:oso/9780198239529.001.0001Save Citation »Export Citation »E-mail Citation »

                                                                                                                                        Provides a solid discussion of the philosophical and mathematical ideas underpinning the Aristotelian continuum theory of spatial magnitude, time, and motion and also discusses alternative theories developed in the Hellenistic period. Zeno’s paradoxes, particularly the dichotomy and the arrow, figure prominently throughout.

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