Some points about discontinuity

Abstract

At the close of his paper 'Infinity' (Proc. Arist. Soc, Suppl. vol. XXXVIII, 1964, pp. 57-68), Mr. R. Harr? considers "what license might be per mitted in making up kinematic concepts with different combinations of continuous and discontinuous space and time" (p. 66). Harr? enumerates the possibilities : (1) space and time continuous, (2) space discontinuous and time continuous, (3) space continuous and time discontinuous, and (4) time and space discontinuous. He then argues (pp. 66-68) that "the only unobjectionable combination is continuous space and continuous time" (p. 67). In this paper I am concerned only to examine the arguments by which Harr? tries to show that (2) space discontinuous, time continuous and (3) space continuous, time discontinuous are not "unobjectionable combinations". These arguments are as follows: Consider the combination of continuous time with discontinuous space. The arrow will then either be at rest at its places while time ticks on, or nowhere at all 'between' them, since for discontinuous space there is no between. So, in these circumstances motion would be impossible. Consider next the combination of discontinuous time with continuous space. But now the racecourse can never be crossed since for each time we must be some place and so there will be places on the journey which we have not time to be in, though they must be traversed for us to get to the other side. Mr. J. Lucas has pointed out to me that prohibition of (2) is equivalent to the well-known requirement that one thing cannot be in two places at once, while the prohibition of (3) amounts to the equally well-known requirement that two different instants cannot be contemporaneous at the same place. This would seem to show that the prohibitions of (2) and (3) are so deeply embedded in our conceptual scheme as to amount to necessary truths and thus to make the continuous-discontinuous combinations virtually self-contradictory, (p. 67)