Abstract
The issue whether there is any necessary connection between time and change turns, I argue, on the problem of what constitutes an accurate measurement of how much time passes. Given a plausible hypothesis about how time is measured, Shoemaker’s well known argument that time can pass without change can be seen to be unsound. But Shoemaker’s conclusion is not therefore false. The same hypothesis about time measurement supports a revised version of Shoemaker’s argument, and the revised argument does establish that the passage of a quantity of time does not require that change occur during that time. Though Shoemaker’s conclusion is vindicated, it does not follow that there is no necessary connection at all between time and change. On the contrary, there is a connection which is simply of a different kind from the one Shoemaker had in mind.
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Notes
Newton-Smith (1980, Chap. 2) advances a similar argument for the same conclusion.
A change is here understood to be a happening or process that has a certain duration and spatial location. Though there are subtle differences, I use the terms ‘change’ and ‘event’ more or less interchangeably.
Shoemaker (1969, p. 370).
I have replaced Shoemaker’s year-long freezes with 24-h freezes. The modification has no effect on the logic of the argument.
Clock ‘precision’ is determined by the number of units into which the clock divides a given chunk of time. Precision is different from accuracy, which will be discussed in the next section.
In such experiments, the gravitational time dilation effects are a consequence of the two clocks’ relative positions in a gravitational field. These effects are distinct from time dilation effects that result from relative motion predicted by special relativity.
In contexts where a where a single set of laws L is understood, I will continue to speak informally of a clock T (rather than a clock-law pair) as accurately measuring time.
(N) will hold in a Newtonian world where, specifically, constant net force implies constant acceleration and Newton’s second law of motion holds. (N) follows from these assumptions together with the principle that if an object accelerates at a constant rate a from velocity \(v_{1}\) to velocity \(v_{2}\), the time t required for acceleration is \(t = (v_{2} -v_{1}) /a\). The latter principle is a consequence of standard definitions of terms such as ‘velocity’ and ‘acceleration’.
In practice, there may be no reasonably simple clock-law system that achieves perfect agreement with observation. It might be useful in such a case to recognize a standard of approximate accuracy for cases where measurements with a clock come acceptably close to conforming to L. The details of a system of approximate accuracy need not concern us here.
For present purposes, two clocks can be considered physically identical if they are constructed in the same way from the same kinds of materials.
A full definition of F would require specifying the value of F(X) for each spatial region, not just for A and B. For the moment, we leave unresolved the problem of what should be the value of F(X) if X is in a frozen region such as C.
Though relevant, considerations of simplicity (alone) do not seem decisive in the choice between L- and L*-based systems. The function definition (F) and the associated modification of each law adds some complexity to the total set of laws. However, a proponent of laws like L* can argue that the L*-based system entails a simpler structure of time since, between any two points in history, the same number of units of time always passes regardless of one’s location or what happens at that location.
If the laws of nature are of the kind illustrated by L* in the previous section, then there will not cases where events of the same law-governed kind occur in different regions. For example, if acceleration events occur in two different regions, then the events will not be of the same law-governed kind because the laws of L* have a spatial parameter X which will have different values depending on the region where the event occurs.
In a world where gravity causes time dilation, C might be a roughly two-dimensional, spherical plane around a planet which is being bombarded by debris from elsewhere in the solar system. The increase in the planet’s mass increases the strength of the gravitational field in C and thereby increases the time dilation effect in C compared to some more distant region A where the planet’s gravity has negligible influence. \(\hbox {T}_{\mathrm{A}}\) thus measures law-governed events of all kinds that occur in C as taking longer compared to events of the same law-governed kind in A.
In this scenario, if time in X is passing faster, or more slowly, than time in A, then \(\hbox {T}_{X}\) would obviously have to run faster, or more slowly, than a clock \(\hbox {T}_{\mathrm{A}}\) (also in A) that accurately measures time in region A. Assuming that the two clocks are devices whose operations are governed by the laws of nature, \(\hbox {T}_{X}\) would not be physically identical to \(\hbox {T}_{\mathrm{A}}\) in the sense of being made in the same way from the same kinds of materials. For example, \(\hbox {T}_{X}\) might have an extra gear that causes its hands to move faster, or more slowly, than the hands of \(\hbox {T}_{\mathrm{A}}\).
The supposition is similar to that of Newton–Smith’s (1980, Chap. 2) variation on Shoemaker’s argument. But there is an important difference: Newton–Smith’s argument depended on an assumption analogous to Shoemaker’s (1) that if an area of space is empty, the amount of time that passes in the empty area will be whatever is indicated by clocks in non-empty, non-changeless areas. Such an assumption would be unjustified for reasons given earlier, and no such assumption is required for the present argument.
I argued in (2004) that Shoemaker’s inductive evidence might be discounted by a counter-induction: inductive evidence that a region is changeless for a certain period of time will also support the conclusion that at least one region is not changeless during that period. While that counter-induction can still be run, I think that its effect is now significantly blunted by evidence that time-determining physical properties such as \(\hbox {P}_{1}\), \(\hbox {P}_{2}\), etc. exist and remain in place when changing objects are removed from a region.
Lawlike necessity is the minimal kind of necessity that seems justified for (CL). Since (CL) seems to be a formulation of our understanding of what it is to accurately measure time, some might argue for a stronger claim of necessity such as a priori or conceptual necessity of some kind. But present concerns do not appear to require resolving this issue.
These objections and Smart’s considered next may have been intended to apply only to claims about the rate of “movement” of the Now (or the Present, or the Spotlight) formulated in terms of an A-series of instants of time. The position argued here is officially neutral both about whether there is any such entity as the Now and about whether, if there is such an entity, it has a rate of motion that can be identified with the relative rate of passage of time.
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I am grateful to the anonymous referees for Synthese for encouraging and helpful comments on an earlier version of this paper.
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Appendix: Proof of (TC)
Appendix: Proof of (TC)
(TC): For any two spatial regions X and Y, time in X passes n times as fast as time in Y if and only if law-governed events in Y take n times as long as otherwise identical law-governed events in X.
For simplicity, suppose that n = 2. The case for (TC) is best argued in one direction at a time. Suppose first that time in region X passes twice as fast as time in region Y. Let K be any law-governed event kind, and assume that the laws of nature require that K-events take \(t_{K}\) units of time. Then, if there are two clocks \(\hbox {T}_{X}\) and \(\hbox {T}_{Y}\) that accurately measure time in X and Y, respectively, (CL) implies that \(\hbox {T}_{X}\) measures any K-events in X as taking \(t_{K}\) units of time, and \(\hbox {T}_{Y}\) measures K-events that occur in Y as taking \(t_{K}\) units. But since time passes twice as fast in X as in Y, \(\hbox {T}_{X}\) shows \(2~\times ~t_{\mathrm{K}}\) units of time passing whenever \(\hbox {T}_{Y}\) shows \(t_{\mathrm{K}}\) units passing. That is, \(\hbox {T}_{X}\) shows exactly enough time for two K-events in sequence in X whenever \(\hbox {T}_{Y}\) shows exactly enough time for only one K-event in Y. It follows that K-events in Y take twice as long as K-events in X.
For the right-to-left direction, suppose that I am in a place X with a clock that accurately measures time in X. I want to know how much time passes in another place Y, though there is no clock that I can assume to accurately measure time in Y. Nevertheless, suppose I have good grounds (theoretical and/or experimental) for thinking that the right side of (TC) is true. That is, if any law-governed event were to occur in Y and its duration were measured with my clock, the clock would measure it as taking twice as long as an otherwise identical law-governed event occurring in X. Provided that the same holds for all law-governed event kinds, then, under (CL), I can conclude that if there were a clock that showed the correct time in Y, it would show time in Y as passing half as fast as time in X. It follows that time in X passes twice as fast as time in Y.
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Warmbrod, K. Time, change and time without change. Synthese 194, 3047–3067 (2017). https://doi.org/10.1007/s11229-016-1090-4
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DOI: https://doi.org/10.1007/s11229-016-1090-4