Published online by Cambridge University Press: 01 January 2020
Zeno's challenge to the usual mathematical characterization of extension is still with us. Butchvarov, considering the limits of ontological analysis, writes, “I shall not explore [the decision to accept the infinite regress in which the pursuit of the analytical ideal is involved], beyond noting that the infinite divisibility of space is the reductio ad absurdum of any attempt to understand space in terms of its ultimate, simple parts.” Grünbaum states this problem, commonly known as the Measure Paradox, concisely, “[How can one conceive] of an extended continuum as an aggregate of unextended elements ?”
1 Butchvarov, Panayot, “The Limits of Ontological Analysis,” in The Ontological Turn, ed. Gram, M.S. and Klemke, E.D. (Iowa City: The University of Iowa Press, 1974), p. 15.Google Scholar
2 Grlinbaum, Adoph, “Zeno's Metrical Paradox of Extension.” in Zeno's Paradoxes, ed. Salmon, W. (Indianapolis: Bobbs-Merrill, 1970), p. 176.Google Scholar
3 Ibid., p. 178.
4 Ibid., p. 192.
5 Ibid., p. 197.
6 One ought not to counter that there are more sets than set conditions, for that counter is based on the assumption that notation is at most countably infinite. If one allows uncountably many locations in a micron of space, he should permit the possibility of uncountably many notation types. In the case of set conditions we need not worry about any space. There is plenty of room for them in Quine's unlovely slum.
7 Hume, David, A Treatise of Human Nature, ed. Selby-Bigge, L.A. (Oxford: The Clarendon Press, 1888), p. 4.Google Scholar
8 Ibid., p. 638.
9 Aristotle, Physics, 212a 20-21.
10 Aristotle, Physics, 207b 4.
11 Aristotle, Physics, 211 b 15·20.
12 Aristotle, Physics, 263b 8-9.
13 Aristotle, Physics, 204b 8-10.
15 Newton, Isaac, Mathematical Principles of Natural Philosophy, Definition VIII, Scholium.Google Scholar
16 Plato, Timaeus, 48E-49A.
17 See Sklar's, Lawrence Space, Time, and Spacetime (Berkeley: University of California Press, 1974), pp. 169-71.Google Scholar
18 Plato, Timaeus, 528.
19 The point is convincingly argued by Paul Benacerraf in “Tasks, Super-Tasks, and the Modern Eleatics, “in Zeno's Paradoxes, ed. W. Salmon (Indianapolis: Bobbs-Merrill, 1970), pp. 103-29.
19 The point is convincingly argued by Benacerraf, Paul in “Tasks, Super-Tasks, and the Modern Eleatics, “in Zeno's Paradoxes, ed. Salmon, W. (Indianapolis: Bobbs-Merrill, 1970), pp. 103-29.Google Scholar