Why mathematical solutions of Zeno's paradoxes miss the point: Zeno's one and many relation and parmenides' prohibition

Review of Metaphysics

Volumn 50, Issue 2, 1996, Pages 299-314

  Papa Grimaldi, Alba ^a  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030525037     PISSN: 00346632     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (42)

1. "Parmenides of Elea, improving upon the ruder conceptions of Xenophanes, was the first to give emphatic proclamation to the celebrated Eleatic doctrine, Absolute Ens as opposed to Relative Fientia: that is, the Cogitable, which Parmenides conceived as the One and All of reality, en kai pan, enduring and unchangeable, of which the negative was unmeaning, and the Sensible or Perceivable, which was in perpetual change"; George Grote, Aristotle, vol. 2 (London: John Murray, 1872), 243.
2. For detailed discussion see, Wesley C. Salmon, ed., Zeno's Paradoxes (Indianapolis: Bobbs-Merrill, 1970).
3. Salmon, Zeno's Paradoxes, 11.
4. John Burnet; see Bertrand Russell, "The Problem of Infinity considered Historically," in Salmon, ed., Zeno's Paradoxes, 51.
5. What I mean by "concrete multiplicity" will appear clear in the following discussion where I will point out that the multiplicity of mathematics is an abstract one insofar as it is purely a reiteration of the unit or the identical one. Mathematics is not able to show, nor is it concerned with showing, the real passage from one unit to the next which is what Zeno was concerned in pointing out with his paradoxes. As long as this passage is not conceptualized, it will be impossible to talk of concrete change or movement against the immobility and identity of each successive and reiterated unit.
6. See Mark Zangari, "Zeno, Zero and Indeterminate Forms: Instants in the Logic of Motion," Australasian Journal of Philosophy 72 (1994): 187-204.
7. William I. McLaughlin and Sylvia L. Miller, "An Epistemological Use of Nonstandard Analysis to Answer Zeno's Objections Against Motion," Synthese 92 (1992): 371-84.
8. See Zangari, "Zeno, Zero and Indeterminate Forms."
9. Ibid., 187 (author's emphasis).
10. Ibid., 191.
11. Ibid., 193.
12. Ibid.
13. Ibid., 194
14. Ibid.
15. See McLaughlin and Miller, "An Epistemological Use of Nonstandard Analysis."
16. William I. McLaughlin, "Resolving Zeno's Paradoxes," Scientific American 271, no. 5 (1994): 69.
17. Mclaughlin and Miller, "An Epistemological Use of Nonstandard Analysis," 376.
18. Ibid., 378.
19. In the Sophist we read: "You see, then, that in our disobedience to Parmenides we have trespassed far beyond the limits of his prohibition. . . . He says you remember, 'Never shall this be proved that things that are not, are, but keep back thy thought from this way of inquiry'"; Sophist 258c-d in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns (Princeton: Princeton University Press, 1961), 1005.
20. Kathleen Freeman, The Pre-Socratic Philosophers (Oxford: Blackwell, 1946), 156.
21. ". . . fo the same thing can be thought and can exist"; Parmenides, trans. Leonardo Taran (Princeton: Princeton University Press, 1965), 41.
22. ". . . but also from this, on which mortals who know nothing wander, double-headed . . . a horde incapable of judgment, by whom to be and not to be are considered the same and yet not the same, for whom the path of all things is backward turning"; ibid., 54.
23. 23 See Henri Bergson, An Introduction to Metaphysics (London Macmillan 1913), and G. W. F. Hegel, Science of Logic, trans. A. V. Miller (Atlantic Highlands, NJ: Humanities Press International, 1969).
24. See Henri Bergson, An Introduction to Metaphysics (London Macmillan 1913), and G. W. F. Hegel, Science of Logic, trans. A. V. Miller (Atlantic Highlands, NJ: Humanities Press International, 1969).
25. Parmenides 128a-c, in The Collected Dialogues of Plato, 923.
26. Parmenides 128a-c, in The Collected Dialogues of Plato, 923.
27. Hermann Diels, Vorsokratiker, in H; see D. P. Lee, Zeno of Elea (Cambridge: Cambridge University Press, 1936), 80.
28. For a detailed discussion of this point see, Alba Papa-Grimaldi, "The Paradox of Phenomenal Observation," Journal of the British Society for Phenomenology 27, no. 3 (October 1996): 294-312.
29. As Hegel pointed out many centuries later: "It is just as impossible for anything to break forth from it as to break into it; with Parmenides as with Spinoza, there is no progress from being or absolute substance to the negative, to the finite"; Hegel, Science of Logic, 94-5.
30. See Zangari, "Zeno, Zero and Indeterminate Forms."
31. 30 About this I do agree with Zangari that "the standard solution that seems to be currently accepted by most philosophers rests on what is often called the 'at-at' theory of motion. According to this, the 'motion' of an object does no more than correlate the position of the object to the time at which it had that position. So it is at a particular place at a particular time. If the object has the same location in the instants immediately neighbouring, then we say it is at rest; otherwise it is in motion. . . . According to the most commonly accepted view, instantaneous velocity is not an intrinsic property of the object, but a supervenient relation based on the correlation between position and time over a neighbourhood of T"; ibid., 192. This theory cannot explain dynamism as it never operates the synthesis that could intrinsically correlate different points in time and space. This was essentially Russell's solution of the paradox. As he wrote: "motion can be understood as the position occupied by an object ina continous series of points in a continous series of instants"; Bertrand Russell, I Principi della Matematica (Milano: Einaudi, 1963), 637.
32. About this I do agree with Zangari that "the standard solution that seems to be currently accepted by most philosophers rests on what is often called the 'at-at' theory of motion. According to this, the 'motion' of an object does no more than correlate the position of the object to the time at which it had that position. So it is at a particular place at a particular time. If the object has the same location in the instants immediately neighbouring, then we say it is at rest; otherwise it is in motion. . . . According to the most commonly accepted view, instantaneous velocity is not an intrinsic property of the object, but a supervenient relation based on the correlation between position and time over a neighbourhood of T"; ibid., 192. This theory cannot explain dynamism as it never operates the synthesis that could intrinsically correlate different points in time and space. This was essentially Russell's solution of the paradox. As he wrote: "motion can be understood as the position occupied by an object ina continous series of points in a continous series of instants"; Bertrand Russell, I Principi della Matematica (Milano: Einaudi, 1963), 637.
33. Or perhaps one should say "these concrete mathematical values." In fact the blunder of which Zeno's paradox is susceptible appears to be of a double nature. Either movement needs to be conceptualized in abstract, strictly logical terms for which these mathematical solutions, simply assuming the factuality of movement, fall short of providing a model, or the passage from one to many needs to be shown to yield a concrete plurality and not simply what I have called a mathematical reiteration of the unit which does not reach the concreteness of movement, change, and plurality, as all you have is a repetition of an identity. This latter expresses the shortcomings of the Pythagorean position, whereas the previous one expresses those of the Pluralists. Zeno's paradoxes are a challenge for both of them. More fundamentally I believe that these mathematical solutions contain both these ancient positions, since on the one hand they simply assume the factuality of movement in their values and formulas, and on the other their values remain purely abstract and so incapable of describing concrete plurality, insofar as they do not show the passage from one to many and vice versa, except as a reiteration of the unit (Pythagorean) or an assumption of the many as immediately intelligible (Pluralists).
34. And this can be seen as ironical since Zangari declares, with temerity, that "however, the historical facts are not the focus of my discussion. The arrow paradox, no matter how it began, has evolved into its modern form and it is with this that I am concerned"; Zangari, "Zeno, Zero and Indeterminate Forms," 190.
35. G. W. F. Hegel, The Encyclopaedia Logic (Indianapolis: Hackett, 1991), 35.
36. Ibid., 131.
37. William McLaughlin, "Resolving Zeno's Paradoxes," Scientific American 271, no. 5 (1994): 66-71.
38. Ibid., 69.
39. Ibid.
40. McLaughlin and Miller, "An Epistemological Use of Nonstandard Analysis," 382.
41. Ibid., 383.
42. Plato, Parmenides 156d-e, in The Collected Dialogues of Plato, 947.