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Contemporary Yugoslav Philosophy: The Analytic Approach pp 27–55Cite as

Solution of the Staccato Version of the Achilles Paradox

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Notes

  1. See Plato, Pannenides, 128 A.

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  2. Cf. Plato, Timaeus, 38 A and Aristotle, Physics, 221 b 3–7ff.

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  3. See DK 82 B 3.

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  4. Cf. G.W.F. Hegel, Vorlesungen über die Geschichte der Philosophie II (Werke 19), p. 62, Suhrkamp Verlag, Frankfurt am Main, 1971.

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  5. See F.H. Bradley, Appearance and Reality — A Metaphysical Essay, ch. XVIII, Clarendon Press, Oxford, 1966.

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  6. Ibid., p. 181.

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  7. See A.J. Ayer, Metaphysics and Common Sense, p. 66, MacMillan, 1969.

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  8. DK 29 B 1.

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  9. Cf. Aristotle, On Coming-To-Be and Passing-Away, 316 a 15ff.

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  10. See ibid., 325 a 31.

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  11. Epicurus, To Herodotus 58

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  12. Ibid., 59

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  13. Ibid., 56–57

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  14. Ibid., 61–62

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  15. Cf. P. Tannery, Pour l’histoire de la science Hellène, Paris, 1887. F. Evelin, ‘Le mouvement et les partisans des indivisibles’, Revue de metaphysique et morale 1,1893, G. Noël, ‘Le mouvement et les arguments de Zénon d’Elée’, Revue de métaphysique et morale 1, 1893. — I don’t think that this interpretation, followed by many scholars, is historically correct (see M. Arsenijević, Prostor, vreme, Zenon (Space, Time, Zeno), pp.91–92, Serbian Philosophical Society and Graphic Institute of Croatia, Belgrade-Zagreb, 1986), but, in any case, it can be used as an argument against geometric atomism.

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  16. I shall not discuss here Grunbaum’s interpretation, according to which all of the famous kinematic atomists, W. James, A.N. Whitehead, and P. Weiss, allegedly side the kinematic atomism of the former kind (cf. A. Grünbaum, ‘Relativity and the Atomicity of Becoming’, Review of Metaphysics 2, band 4, 1950 and Modern Science and Zeno’s Paradoxes, p. 48, George Allen and Unwin Ltd., London, 1968).

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  17. M. Black, Problems of Analysis, pp. 116–118, Cornell University Press, Ithaca, New York, 1954.

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  18. Berkeley was the first to claim that our faith in infinite divisibility has its origin in this extrapolation, based on the way in which the mathematicians proceed: whenever the geometer speaks of particular lines and figures, he ‘considers them abstracting from their magnitude’, which implies ‘that he cares not what the particular magnitude is, whether great or small, but looks on that as a thing indifferent to the demonstration’ (Berkeley, The Principles of Human Knowledge, §126, p. 154, George Routledge and Sons, London and New York, 1878). David Hilbert, a mathematician, was of the same opinion, and he tried to solve The Achilles on that basis (see D. Hilbert and P. Bernays, Grundlagen der Mathematik I, §1c2,p. 16, Springer Verlag, Berlin, Heidelberg, New York, 1968).

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  19. Infinitism is the religion of mathematicians. Among philosophers who used this strategy in trying to solve Zeno’s paradoxes the most influential are: B. Russell (see Our Knowledge of the External World, lectures VI, VII, The Open Court Publishing Company, Chicago and London, 1914), R. Taylor (see ‘Mr. Black on Temporal Paradoxes’, Analysis 12, 1951, and ‘Mr. Wisdom on Temporal Paradoxes’, Analysis 13, 1952), A. Grünbaum (see Modern Science and Zeno’s Paradoxes, ch.II, George Allen and Unwin Ltd., London, 1968), and W.C. Salmon (see Space, Time, and Motion, ch. II, Dickenson Publishing Co., Inc., Encino and Belmont, California, 1975).

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  20. J. Watling, ‘The Sum of an Infinite Series’, Analysis 13, 1952, p. 48.

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  21. Cf. J.F. Thomson, ‘Tasks and Super-Tasks’, Analysis 15, 1954.

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  22. See M. Black, ‘Achilles and the Tortoise’, Analysis 11, 1951, p. 97.

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  23. See above, n. 21.

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  24. P. Benacerraf, Tasks, Super-Tasks and Modern Eleatics’, The Journal of Philosophy vol. LIX 24.

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  25. A. Grünbaum, Modern Science and Zeno’s Paradoxes, pp. 94ff.

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  26. Ibid., p. 105.

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  27. Ibid., p. 100.

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  28. See ibid., p. 101.

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  29. Cf. A. Robinson, Non-Standard Analysis, pp. 55ff. and pp. 266–7, North-Holland Publishing Company, Amsterdam, 1970.

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  30. Cf. C.B. Boyer, The Concepts of the Calculus: A Critical and Historical Discussion of the Derivative and the Integral, ch. III, Dover Publications, New York and London, 1939.

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  31. The solution of The Achilles proposed by G.J. Whitrow in 1961 is in accordance with non-standard analysis, which came into being a few years later (cf. G.F. Whitrow, The Natural Philosophy of Time, p. 151, Thomas Nelson and Sons Ltd., 1961 and A. Robinson, Non-Standard Analysis, p. VIII and pp. 58ff.).

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  32. See P.T. Geach, Logic Matters, p. 238, Basil Blackwell, Oxford, 1972.

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  33. Ibid., loc. cit.

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  34. See M. Black, ‘Achilles and the Tortoise’, Analysis 11, 1951.

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  35. Cf. M. Black, ‘Achilles and the Tortoise’, Analysis 11, 1951ibid., pp. 100–101.

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  36. Cf. D.S. Schwayder, ‘Achilles Unbound’, The Journal of Philosophy vol. LII 17, pp. 454ff.

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  37. M. Black, ‘Achilles and the Tortoise’, Analysis 11, 1951, p. 101.

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  38. See Aristotle, Physics, 185 b10ff.

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  39. Cf. Aristotle, The Nicomachean Ethics, 1096 b 27, and Metaphysics, 1003 a 21.

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  40. Two parts are contiguously touching if they are physically heterogeneous or if they are not physically grown together. For Aristotle’s definition of contiguity see Physics, 227 a 6.

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  41. For Aristotle’s definition of continuity see Physics, 227 a 10. That the distinction between contiguous and continuous touchings can be inclusive see M. Arsenijevic, ‘Dodirivanje’ (Touching’), Filozofske studije (Philosophical Studies) VI, 1975.

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  42. See Aristotle, On Coming-to-Be and Passing-Away, 317 a 8ff.

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  43. See Aristotle, Physics, 231 b 15.

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  44. For the justification of my use of name indefinitism for the theory under consideration cf. G.W. Leibniz, ‘The Theory of Abstract Motion: Fundamental Principles’, Philosophical Papers and Letters, p. 139, D. Reidel Publishing Company, Dordrecht-Holland and Boston-U.S.A., 2nd edition, 2nd printing, 1976, and Kant, Kritik der reinen Vernunft (Werke III), pp. 351 ff., Druck und Verlag von Georg Reimer, Berlin, 1911.

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  45. Aristotle, Physics, 227 b 3.

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  46. Cf. ibid., 228 b 15ff.

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  47. In our century, H. Bergson proposed a solution to Zeno’s Arrow completely based on Aristotle’s analysis of the nature of the oneness of a motion (see H. Bergson, L’évolution créatrice, pp. 334ff., Librairie Felix Alcan, Paris, 1932).

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  48. See A.R. Anderson and N.D. Belnap, Jr., Entailment — The Logic of Relevance and Necessity, §§3, 4, Princeton University Press, Princeton and London, 1975.

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  49. Intensional conjuction was introduced by R.K. Meyer in ‘Some Problems No Longer Open for E and Related Ligics’, Journal of Symbolic Logic 35, 1970.

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  50. De facto, the term ‘relevant consistency’, used in relevant logics, and Nelson Goodman’s neologism ‘cotenability’ are being used synonymously (cf. A.R. Anderson and N.D. Belnap, Jr., op. cit., pp. 345–6).

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Authors
  1. Miloš Arsenijević
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  1. Macquarie University, Sydney, Australia

    Aleksandar Pavković (Lecturer and Head of Slavonic Studies) (Lecturer and Head of Slavonic Studies)

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© 1988 Kluwer Academic Publishers

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Arsenijević, M. (1988). Solution of the Staccato Version of the Achilles Paradox. In: Pavković, A. (eds) Contemporary Yugoslav Philosophy: The Analytic Approach. Nijhoff International Philosophy Series, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2821-3_3

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  • DOI: https://doi.org/10.1007/978-94-009-2821-3_3

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