Kodikas/Code, Volume 19 (1-2), 1996 (Special issue Erleben und Repräsentation von Zeit, edited by Dieter Münch), pages 101-119. Gunter Narr Verlag, Tübingen, ISBN 3-8233-9944-6
Abstract
Both ancient and medieval philosophers were much exercised by the notion of continuity, in particular the problem of the instant of change. Aristotle held that when an object starts moving, there is neither a last instant at which it is at rest, nor a first instant when it is in motion, since being at rest and being in motion are properties which can hold of an object only over an interval, not at an instant. Modern mathematical analysis provides a different picture, based on instantaneous rates of change. The velocity of an object at an instant is the limit of its average velocity over an interval containing that instant as the length of the interval tends to zero. By the mathematical definition of continuity, when an object starts moving, there is a last moment when it is at rest, but no first moment when it is in motion.
The mathematical picture is useful in a wide range of applications, but many philosophers have found it disquieting. C. L. Hamblin suggested that for qualitative discourse we dispense with the notion of an instant altogether, and adopt a model of time in which intervals exist as entities in their own right rather than as aggregates of instants. Hamblin's approach has been duplicated in Artificial Intelligence by J. F. Allen, whose interval-based temporal framework has been widely influential. Allen's main reason for abolishing instants was that if we allow them then we have the problem of deciding whether our intervals are open or closed.
I argue that Allen was mistaken both in thinking that we can dispense with instants and in thinking that if we allow them then we have to consider open and closed intervals. A more satisfactory picture is to regard an instant as arising from the division of an interval into two parts which jointly exhaust the interval without either of them containing the instant which separates them. I present a model of time which includes both instants and intervals, without the question of closure arising, and argue that this model is the right one for the kind of qualitative simulation which has become an important desideratum of the Artificial Intelligence enterprise and which is closely related to the kinds of qualitative descriptions which Hamblin was interested in.
This paper was originally presented at the 7th International Conference of the Deutsche Gesellschaft für Semiotik, Tübingen, October 4-7, 1993.
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