The Rate of Temporal Flow

Transcription

1 The Rate of Temporal Flow Edward Freeman There is a familiar problem with the notion of temporal passage. If time flows, it must flow at a rate, as all physical flows do. It is a fact of physical reality that rateless flow of anything, be it the flow of a river, of an atmospheric mass, of an electric current, etc., is not possible. 1 Yet, unlike ordinary physical flows, the temporal current cannot flow at a rate because such rate must be given in one temporal unit per another temporal unit, that is, time must flow at the rate of sec/sec, hour/hour, and such. Prima facie, temporal unit per the same temporal unit rate cannot possibly be a rate of temporal passage because this expression reduces the rate of temporal flow to what physicists call a dimensionless constant. However, it is the fact of physical reality that rates always have dimensions, e.g., meter/sec, gram/hour, etc. Hence, the notion of time passing at the rate of temporal unit per the same temporal unit is ill-formed. Can then the rate of temporal flow be a dimensionless quantity? Well, all other physical rates are not. An overriding argument, therefore, must be given by the friend of temporal flow showing that the rate of temporal flow should be excluded from this general rule. As far as I know, no such argument was ever attempted. For this reason alone, the concept of temporal passage should be abandoned as having no application to physical reality. The above argument should be sufficient for the rejection of the reality of temporal flow. The fluid-time theorist, however, will not be swayed so easily. Let me, therefore, grant him, if only for the sake of argument, that time flows. Because, as I have just mentioned, the rateless flow of a material substance is not possible, on the conditions that (a) time is a physical phenomenon and (b) it flows, time must flow at a rate. In addition, any type of physical flow takes place in relation to something external to it, for to flow physically is to flow from a point to a point. Therefore, if time flows, it must flow at a rate and in relation to external points of reference. Thus, in considering the phenomenon of temporal flow a question inevitably arises: At what rate does time flow and in relation to what does it flow? As far as I can tell, the only plausible answer to this question, at least provisionally, is that since time cannot flow in relation to anything non-temporal, or in relation to itself, and since it must flow at a rate, it must flow at the rate of a certain number of (static) temporal intervals per second-order temporal unit. What we in effect have here is a three-tier model of temporal flow: the first-order (fluid) time flows in relation to (static) time at the rate of a second-order time unit. Pictorially, 1 Newton famously held that Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration. Isaac Newton, Principa F. Cajori (ed.) (Berkeley: University of California Press, 1947), p.6. I shall set aside the question of absolute Newtonian time as having no bearing on the issue in hand. Besides, duration, unlike temporal flow, is measurable by the use of uniform cyclical movements of physical systems, e.g., clocks.

2 Flowing time Static time t1 t2 t3 t4 t5 t6 t7 t8 tn earlier Figure 1 later On this picture of temporal passage, a question naturally arises at what rate a second-order time passes. The only answer, as far as I can tell, is that it passes at the rate of third-order temporal intervals per fourth-order temporal unit and so ad infinitum. We thus must reject this picture of temporality. There arises an additional difficulty. On the supposition that time flows over static distances, it must flow at the rate given by a number of static time units per second-order fluid time unit. Over what static distances then does the second-order fluid time flow? If it flows over first-order static distances, then both first-order and second-order fluid times flow over the same static temporal spans, in fact, an infinite number of fluid temporal series flow over these static spans since the second-order fluid time must flow at the rate of a third-order fluid time unit and so ad infinitum. If, on the other hand, a second-order fluid time flows over second-order static spans, then in addition to the infinite number of fluid temporal series we have an infinite number of static temporal series. This picture of temporal passage is too cumbersome to be plausible. Furthermore, this notion of the rate of temporal passage is too metaphorical to have any analytical value, for A-time is conceived as a fluid object, of a sort, and B-time is conceived as a static space-like expanse over which A-time flows at the rate of a certain number of B- points per second order A-time unit. Ned Markosian takes issue with renderings of the notion of the rate of temporal passage like the present one. 2 He rejects the hypothesis that If time flows or passes, then there is some second time-dimension, with respect to which the passage of normal time is to be measured. 3 Markosian thinks that it is sensible to compare the pure passage of time by which, in effect, he means absolute Newtonian time to time itself. 4 Accordingly, on his account, the question How fast does time pass? is a sensible question with a sensible answer: time passes at the rate of one hour per hour. 5 In arguing his case, Markosian appeals to the authority of Arthur Prior who in fact did hold the notion of time passing at the rate of sec/sec to be perfectly cogent and did not need to be supplemented by the concept of a second-order time. 6 To demonstrate this, Prior draws a parallel between accelerated motion, expressed as meter/sec/sec, and the rate of temporal passage expressed as sec/sec. Evidently, Prior is mislead by the superficial similarity of the two concepts and does not realize that unlike the sec/sec rate, the meter/sec/sec rate does express a certain magnitude. More importantly, the dimension of acceleration is not meter/sec/sec, but [meter/sec]/sec, which is a standard fraction. With a little algebra, this fraction becomes meter/sec x sec. The important point is that we have 2 Markosian, N. How Fast Does Time Pass? Philosophy and Phenomenological Research 53 (1993), pp Ibid, p How Fast Does Time Pass? p Ibid, p Prior, A. Papers on Time and Tense (new edition) (Oxford: Oxford University Press, 2003), p.9. 2

3 the dimension of time in the denominator and something else in the numerator. The coherence of that sort of fraction does not show the coherence of a fraction with time in the numerator and denominator. Markosian also contemplates an antirealist reply to the rate of temporal passage argument, namely that the passage of time is a change whose rate simply cannot be measured, so that there is no need to posit any second time-dimension with respect to which the passage of normal time is to be measured. 7 I find this suggestion unconvincing for the following reasons. To begin with, its introduction at this point of Markosian s discussion is non sequitur, for whether or not the rate of temporal passage is measurable in principle is not in question here. Let us therefore turn to his other suggestion. Markosian contends that time passes at the rate of a temporal unit per temporal unit, where both temporal units are of the same temporal dimension. But how could this be? There are two types of rate of change: one is the rate either relational (extrinsic) change and the other is the rate of or non-relational (intrinsic) change. An example of a relational rate is the speed of a moving car with respect to the ground, an example of an intrinsic rate is the rate of crop growth. 8 It is a basic feature of physics that rates are measured by chronometers together with devices which measure something other than time, such as tapes, scales, or the like. Accordingly, in order for us to determine a rate of either qualitative or quantitative change at least two distinct coordinates, one of which is temporal, are needed. For instance, in figure 2, the rate of crop growth is expressed using the distinct dimensions of time and weight. g hr. Figure 2 We could hardly express the rate of this particular qualitative change using either just time units or just weight units; both parameters are needed. Gram per gram, or meter per meter do not give us the rate of change. Why should the rate of temporal passage be somehow exempt from this elementary requirement of physics? Temporal unit per (the same type) temporal unit is no more a rate of change than gram per gram or meter per meter is. What then can be said about such a peculiar change as temporal flow? Well, if time flows, it flows at a rate that is expressible as sec/sec* where the second occurrence 7 Ibid, p Not all changes go at a rate. Being a presidential candidate and then being a president, for instance, is an example of such a change. We can ignore this complication, however, as having no bearing on the issue in hand. 3

4 of second refers to a second-order time unit. There seems to be no way around this conclusion. But this, as I have argued, leads to infinite regress. It might be argued that the rate of temporal flow can be expressed without an appeal to second-order time by using prima facie two distinct temporal units of measurement, namely B-intervals and A-seconds. If so, the rate of temporal flow would be expressible as Rate t = B-interval/A-second. Although this conception can indeed be expressed using coordinates, as shown in Figure 3, or, for that matter using /, its ostensible legitimacy is misleading. On this analysis, A-time is conceived as moving over the B-dimension at a rate of a certain number of B-points, or a certain B-distances per A-second. But time is not an object. Time is itself is a dimension; to say that it moves over the B-dimension, without some further explanation of what it is that rolls over the B-series, lacks clear sense. A-seconds Figure 3 B-points Let us nonetheless suppose, again for the sake of argument, that time does pass at the rate of temporal unit per temporal unit where the two units are of the same dimension. Obviously, we cannot express this alleged rate using standard coordinates, since we have only one such coordinate. Can we then express it as a ratio, say as Rate t = sec/sec? Again, if the flow of time is a sort of change, then, since time cannot be measured with respect to anything non-temporal, there must be two distinct magnitudes of this peculiar change, both of which are temporal; that is, there must be a second-order time at which first-order time flows and so ad infinitum. Peter van Inwagen has another argument against the idea that time flows at the rate of temporal unit per (the same type) temporal unit. He points out that second per second is not really a rate because one second divided by one second gives us 1 and 1 is not a rate: 9 Sixty seconds per minute is not an answer to this question, [How fast does time move?] for sixty seconds is one minute and 1 is not, and 9 Unless, of course, the second occurrence of second in sec/sec refers to a second-order second. But this option, as we have seen, leads to vicious infinite regress. 4

5 cannot ever be, an answer to a question of the form, How fast is suchand-such moving? no matter what such-and-such may be. 10 Eric Olson echoes van Inwagen: The real problem with saying that time passes at one second per second is not that this is a funny sort of rate, but that it is no rate of change at all (van Inwagen 2002:59). One second per second is one second divided by one second. And when you divide one second by one second, you get one. Not one of anything, just one. Dividing anything by itself, unless it is zero, gives you one. Sixty seconds per minute and twenty-four hours per day are also one, because sixty seconds is equal to one minute and twentyfour hours is one day. And one is not a rate of change. A thing can change at a rate of one mile per hour or one degree per minute, but not at a rate of one. 11 Hud Hudson, Ned Markosian, Ryan Wasserman, and Dennis Whitcomb, have recently attempted to reply to this criticism. 12 They write Suppose that a car is passing at a constant rate of 1 kilometer per minute. Letting C abbreviate the rate of the car s passage and k/m abbreviate kilometers per minute, we have C = 1 k/m`. Now consider the following principle: The Inverse Rate Equivalence Principle (IREP): n x/y = 1_ n y/x. IREP tells us that if the xs pass at a rate of n per y, then the ys pass at a rate of 1_ n per x. So, for example, if Montana completes passes at a rate of 20 per game, then the games go by at a rate of.05 per completion. Applying IREP to the case of the car gives us 1 k/m = 1 m/k. In other words, if the car passes at a rate of one kilometer per minute, then time passes at a rate of one minute per kilometer covered by the car. This would simply be an alternative way of expressing the rate of time s passage: 1 m/k = R. 13 Their analysis might be faulted for wrongly equating the rate of change in physical systems, in their case, chronometers, with the rate of temporal passage itself. One might contend that it is not time that passes at the rate of one minute per kilometer, it is the large hand of the clock that moves a one-minute-notch on the dial while the car covers one kilometer. Hudson et al might reply that this criticism is overly operationalistic. Let us thus grant that it is time itself passes at one minute per kilometer as the car goes one kilometer per minute. There is still a difficulty. While IREP allows us to say that time passes at 1 min/1km, this definition of the rate of temporal passage requires that the speed of the car is given as 1km/1min. Unless the speed of the car is given, the definition is ill-formed. However, speed is distance covered per unit time. Hence, defining time in terms of distance and speed assumes what needs to be defined. Additionally, on the above example and IREP, we allowed to say that time passes at the rate of one minute per one kilometer. But what about a different car that passes at 10 van Inwagen, P. Metaphysics (Bolder, Colorado: Westview Press, 2002), p Olson, E. The Rate of Time s Passage, Analysis 69 (2009), pp.3-9, p.5 12 Hudson, H. Markosian, N. Wasserman, R. & Whitcomb, D. The Rate of Passage: Reply to van Inwagen published online, The Rate of Passage: Reply to van Inwagen p. 3 5

6 the very same highway at the rate of two kilometers per minute? Should not we say that in this case time passes at the rate of two minutes per kilometer? It seems to me that it is exactly what IREP demands. Yet, it cannot be true that the rate of passage of time varies form one moving object to another. What in fact varies is the speed of moving objects, not the speed of temporal passage. Surely, it is not time that passes at the rate of one minute per kilometer, it is the large hand of the clock that moves a one-minute-notch on the dial while the car covers one kilometer. But let us grant, if only for the sake of argument, that it is time itself that passes as the car goes one kilometer per minute. On the IREP, we are allowed to say that time passes at the rate of one minute per one kilometer. But consider now a three-lane highway such that there is a car in each lane and they travel at different speed: one car is cruising at ½ kilometer per minute, the second car passes at one kilometer per minute, and third car zooms at two kilometers per minute. Should we not then say that on the IREP time passes at three different rates? It seems to me that it is exactly what the IREP demands. Yet, it cannot be true that the rate of passage of time varies from one moving object to another. What in fact varies is the speed of moving objects, not of the rate of temporal flow. It seems to me there is no way that we can endow the notion of the rate of temporal passage with a stable sense and harder we try the more obvious it becomes that all such attempts are futile, Hudson et al is case in point. We thus must conclude that the notion of rate of temporal passage is incoherent, and since there can be no rate at which time flows, there can be no temporal flow as such. 14 Edward Freeman efreeman@gradcenter.cuny.edu Works cited: Hudson, H., Markosian, N., Wasserman, R., & Whitcomb, D. The Rate of Passage: Reply to van Inwagen (published online, 2009.) Markosian, N., How Fast Does Time Pass? Philosophy and Phenomenological Research 53 (1993), pp Olson, E., The Rate of Time s Passage, Analysis 69 (2009), pp.3-9. Prior, A., Papers on Time and Tense (new edition) (Oxford: Oxford University Press, 2003). van Inwagen, P., Metaphysics (Bolder, Colorado: Westview Press, 2002). 14 I am indebted to Michael Levin for his extensive comments on earlier versions of this paper which led to its numerous and substantial revisions. 6