Abstract
Instants are empirically inaccessible. They are like bar-lines in musical notation.1 At an instant, we cannot hear, see or measure anything. Whatever we hear, see, or measure etc., i.e. whatever we observe, we observe in periods. Nevertheless we talk about positions, pitches or temperatures at instants. We want and we need to be able to say that a continuously moving clockhand points towards the 12 o’clock mark at noon and at midnight. However, we have to justify this. We have to secure some empirical content for statements about instantaneous states by relating them to possible observations. This is a theoretical task. Instantaneous states are not observed but construed. Already, the following theory of instantaneous states has proved to be too simple:
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The state that α obtains at instant t iff t both bounds an α-phase towards the past and an a-phase towards the future.
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Notes
Musical notation agrees with the Aristotelian assumption of instants as mere boundaries. It also agrees very nicely with the Aristotelian assumption of the potentially infinite divisibility of periods: One can notate the same melody in arbitrarily many different ways by inserting as many bar-lines as one wishes without changing its audible form. Of course, one cannot actually insert infinitely many bar-lines. And, of course, no matter how many bar-lines one puts together, one will never create an aural extension (one requires at least to put a rest in between as something that has duration!).
Cf. ch.II, 2 and II, 4.
Spans have no temporal ‘direction’. However, it is not necessary to recur to instantaneous states in order to distinguish the clockhand’s passing through a 2-D-span clockwise and its passing through anti-clockwise. They can be distinguished by the different sequences of subprocesses into which they may be divided: If, for example, the clockhand is moving clockwise between 11:55 and 12:05, it passes through the span between the 11:55 and the 12:00 position first and only then through the span between the 12:00 position and the 12:05 position and if it is moving anti-clockwise, vice versa.
The position-span for the ten minutes in question is not identical with the set of all spacepoints which fall within and bound the surface through which the clockhand passes between 5 to 12 and 5 past 12. However, there is a close relationship between this set and the span: To every position-span exactly one set can be assigned which may be said to ‘describe’ it. This set describes a span in the same way as an interval may be said to describe a period.
It will be seen below that if the principle “natura non facit saltus” is assumed, this intuition can be expressed even more clearly and the distinction between lefhand side and righthand side may be dropped.
Cf. W.E. Johnson: Logic, Cambridge 1921.
I presume that the distinction has Aristotelian origin: the idea of grouping incompatible properties into ‘points of view’ (German: ‘Hinsichten’, typically by using the preposition kαià). After Johnson, Searle has (with little success) tried to revive the distinction by an attempt to formalize it. Cf. J. Searle: Determinables and the Notion of Resemblance, in: Proceedings of the Aristotelian Society Suppl. Vol. 1963, and Searle’s excellent introduction ‘Determinates and Determinables’ in the Encyclopedia of Philosophy (ed.Paul Edwards, New York 1967 ff.). Cf. also Antti Hautamäki: Points of View and their Logical Analysis, Helsinki 1986 (Acta Philosophica Fennica 41), ch.1’Determinables’. Cleland identifies determinables with sets of determinates and takes the view that determinables are the real vπtoxєiµєva of every alteration. Cf. Carol Cleland:The Individuation of Events In: Synthese, February 1991.
Cf. Aristotle’s cautious additional clause to the classical formulation of the LNC in Met. 1005b20–23.
This definition is, of course, not operational; its function is to clarify the notion of an instantaneous state. For, of course, no actual measurements from all periods which are bounded by t can be performed and compared (there are infinitely many such periods). We are, however, convinced of what these measurements would look like if we could perform them; and any measurement we do indeed perform confirms our expectations.
Details: ch. I, 3 and Appendix C.
This might again be called some kind of transcendental argument. Here we would have a typical Kantian case in which we make Nature behave according to our way of describing it, because we cannot imagine it behaving otherwise.
This example is somewhat idealized; but it is nonetheless worth considering.
Russell sidesteps the problem by restricting his definitions to a ‘material point’ as his object of motion. Cf.ch. II, 5 and Principles of Mathematics §442.
Cf. Sorabji: Time..., pp.404f.
I am grateful to Ludger Jansen for this example.
I am grateful to Bertram Kienzle for drawing my attention to this problem.
‘It will easily be seen at this point that it would be possible to develop a theory of properties of non-extended spatial entities exactly analogous to the theory of instantaneous states. The motivation for doing this would be the empirical inaccessibility of space-points and lines, which is clearly apparent, since those entitites, too, are mere boundaries in a continuum. In order to build such a theory, one needs a spatial version of modified interval semantics based on spatial intervals and with a spatial version of the ‘fall-within’ relation. A Point-prefix in the spatial version should have practically the same semantics as Inst in the temporal version.
Note that the only difference between the two-place determinable function IS-AT from ch.II, 5 and Inst-POS is that IS-AT yields space-points while Inst-POS yields spans.
It may well be the case that Inst-D(a, t1) = Inst-D(a, t2). Cf. ch.II, 5 on the problem of circular motion.
I strongly suspect that the definiens of (a) is equivalent to the following formula which does not involve the Inst-prefix and in which motion as a genuine flux is very clearly apparent: Phrase (b) would, of course, have to be analogous. The somewhat indirect procedure of working with two subperiods (c2!) is necessary because of the usual difficulty concerning circular motion (cf.II, 5). So far I have been unable to prove that the equivalence holds.
If L(t) is to be read as “t is a limiting instant between rest and motion”, we may say that L(t) is true iff either clause (i) of (c) and clause (ii) of (d) are fulfilled or clause (ii) of (c) and clause (i) of (d) are fulfilled. It is easy then to state a comparative definition of rest and motion at an instant which assigns the limiting instant to rest (with POS for ‘D’): t would have to be an instant of rest iff clause (d) of NR10) were fulfilled or L(t) were true, the definition of motion remaining clause (c) of NR10). This definition would agree with Sorabji’s proposal (ch.II, 1.1) .
For formal definitions of ‘beginning’ and ‘ending’ and for formal proofs of these claims see AppendixA.
Some changes which one would intuitively consider to be ‘discontinuous’, such as on/off-changes (cf. ch. II, 3) are C-changes which should be described by using the either/oroption: nobody except Hamblin (cf. ch.II, 4) and Priest (cf. ch. II, 3) would think that there is an instant at which an existing lamp is neither ‘on’ nor ‘off’. To describe such a change using the either/or-option, we should decide first which state is primitive and which is derived. This decision should usually be easy. One will naturally define ‘off’ as ‘not on’, and not ‘on’ as ‘not off’ (although it is, of course, true that a lamp can be seen to be on whenever it can be seen to be not off; but since instants are empirically inaccessible, this is not the whole story). If a lamp is off whenever it is not on, we simply have to ask ourselves whether it is on at the limiting instant if we want to know whether it is off or on. The limiting instant cannot be assigned the state ‘on’ (if it could it should bounded by an on-phase on either side). So, by definition, the lamp is off at it.
I am aware of the fact that these remarks are rather sketchy. But not every conceivable case needs to be discussed in detail here, if the general framework within which the discussion might take place is recognizable. Apart from that, (at least epistemic) vagueness is probably of greater importance here than can be accounted for in the present book. The topic is difficult and has an impact on ethics when abortion or brain death are concerned.
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Strobach, N. (1998). A Path to a Plausible Description of the Moment of Change. In: The Moment of Change. The New Synthese Historical Library, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9127-0_13
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