Real Velocity and Strange Motion on Possible Worlds
2003 Southern California Philosophy Conference at UCR
Abstract
Title of submission:
Real Velocity and Strange Motion on Possible Worlds: An analysis and response to Humean objections
This paper defends the views of causation presented by Michael Tooley in his 1988 article, In Defense of the Existence of States of motion. More specifically, this paper focuses on the argument that classical Humeans cannot prima facie dismiss the possible world analysis Tooley uses to demonstrate that causation is needed in order to adequately explain velocity. Tooley's two main claims are that: (1) Russellian velocity does not logically supervene on objects since it does not supervene in all possible worlds, and (2) Tooley's causally laden version of velocity has an explanatory virtue that Russellian velocity lacks. Yet these claims are based on the analysis of a possible world, w1, of accidentally orderly movement where the laws of motion are probabilistic. The problem is getting the classical Humean to accept that this world is legitimate for philosophical analysis, i.e., that it is non-question-begging. This requires me to show that using possible worlds whose laws we know to be different from the laws of the actual world does not amount—to put it one way—to playing God. I give three reasons why the classical Humean should find it reasonable to at least accept w1 as a legitimate analysis of velocity: (i) w1 could conform to a Lewisian (i.e. an empirically motivated) similarity metric, (ii) w1 does not violate epistemological barriers once it is considered methodologically, and (iii) w1 could be construed to conform to the laws of statistical mechanics.
In 1988 Michael Tooley wrote a paper called, In Defense of the Existence of States of Motion, which offered a Realist's alternative to Russellian, supervening velocity. Velocity, according to Tooley, is a property intrinsic to objects insofar as change in position of those objects have causal explanations, i.e. are partly determined by certain relationships to positions. Tooley argued that we should think of velocity as a property that is embedded in theoretical notions of causality. This is contra to what we get in The Principles of Mathematics (1903), where velocity is some given numerical value—a limit set of quotients—that is assigned to an object's position, at a specified time. We can think of the classical Humean as embracing Russell's conception of velocity.
Tooley's "alternative" conception of velocity, his Realist version, challenges the classical Humean with a possible world analysis in which the laws of motion are probabilistic, and where there happens to be accidentally orderly movement. The initial and final positions of an object, the only input required for Russellian velocity, could not explain orderly behavior in a world that runs according to probabilistic laws of motion. What is required instead is information about present positions that minimally determine the likelihood of future positions. However, if the behavior of objects on w1 are empirically no different than the behavior of similar objects in the actual world (for example, the path of some particle moving along a curve), then how might we come to understand that the laws of one world, w1, are different from the laws of the actual world, w0? How can Tooley's argument get off of the ground? The Humean's initial response could be to claim that w1 prima facie begs the question. No well respecting Humean would consider some possible world for which the laws of motion are arbitrarily predetermined from a vantage point. My focus here will be to try and demonstrate for the classical Humean, that w1 is a legitimate analysis, and that what may initially appear to be an arbitrary stipulation of laws for w1 can conform to considerations both sensitive to empiricism and to actual world possibilities, and can be considered, in the end, methodologically useful.
Let's begin by considering w1. Imagine in w1 some particle, x, traveling along the path of a continuous curve. According to Tooley, Russell should, for every point on that curve, be able to tell us that x has such and such a velocity. But since the laws in w1 are probabilistic, and since (as a ratio) Russellian velocity has no causal bearing on position, then Russellian velocity cannot describe the motion of x at each position on the continuous path. In other words, a supervening velocity in w1 does not refer to the only relevant condition for determining the likelihood of future positions, namely, its relation to the position of an object at the present time. So, the argument goes, if Russellian velocity does not hold for all possible worlds, then it is not logically supervenient (by definition), and, furthermore, in w1 we might want to instead use a richer notion of velocity that is capable of explaining the behavior of these kinds of objects.
The Humean could respond to this argument by saying that we lost him right at the door; the only reason Russellian velocity cannot explain the path of particle x, nor account for the relationship between the positions of objects, is because Tooley stipulated in advance that the laws of motion on w1 are probabilistic. And the only way that one might know that the laws in w1 are the way that Tooley in fact describes is if he is has a god-like vantage point. The Humean, in addition, could further claim that since there is no kind of empirical distinction to be made between particle x and its actual world twin, which we'll call particle y and which also moves along a continuous path, then there are no means by which to determine that the laws of these two worlds are indeed different. If Tooley admits that there is some kind of empirical distinction between the two worlds, then the conditions for the possibility of a supervening velocity are violated (since it is a logical or mathematical property that supervenes on a physical object, there cannot be a physical change without a logical or mathematical one). In claiming that Tooley recklessly crosses epistemic barriers, the classical Humean is able to immediately reject w1 without having to answer the challenges that it intends to pose. The trick, then, is to get the Humean past the door, or to show him that the laws of motion on w1 are not arbitrarily stipulated. Although recognizing w1 as a legitimate possible world analysis does not force the Humean to accept causally laden theories of velocity—unfortunately—it does put some pressure on him to step up to Tooley's challenge, or, at the least, to tell us something more about the relevancy criterion for these sorts of possible world analyses.
Implicit in the classical Humean's rejection of w1 is a set of criteria satisfying certain epistemic requirements about the laws of a possible world. In other words, there are certain standards that must be met if we want the Humean to take w1 seriously. Some of these standards are (i) w1 must be similar to the actual world in relevant respects—it must conform to something like a Lewisian similarity metric, (ii) should we decide to investigate w1 to some greater degree of scientific analysis, we must conform to certain acceptable scientific methodologies, and (iii) any behavior we choose to analyze on w1 must be physically possible, i.e., we must be able to the laws of physics, and applied mathematics, to describe it. Given these criteria (Φ) we can ask ourselves whether we might have any reason to believe that w1 satisfies Φ.
In its most basic form, the first criterion requires that possible worlds similar to the actual world must be capable of satisfying truth conditions for counterfactuals, and, therefore, do not "diverge", as Lewis would say, from the actual world. Similarity could mean that the possible world must share most of the same events that occur in the actual world. But it need not. For similarity could also mean that if an event is theoretically possible in the actual world, then in certain circumstances it is reasonable to consider possible worlds with instantiations of the actual world at different temporal intervals (we might also want to extend this to mere physical possibility, in which case we could admit worlds with backwards causation or time travel). The question then is: are probabilistic laws actual world possibilities at different times?
Let's briefly consider an actual world theory that employs probabilistic laws to describe kinetic phenomenon. Boltzman's statistical asymmetry, in statistical mechanics, would be such a case; here, particle distribution is a function of probability. But remember, the problem is not to explain the behavior of objects, such as particle x, in w1 according to the laws of statistical mechanics in order to make it "converge" with the actual world. We must try to account for the possible behavior of x by showing that it can be explained as an actual world possibility. This means that particle y, in the actual world must be described by classical laws or relativity laws; we need to maintain, in other words, a distinction between the laws of the two worlds and, at the same time, account for the empirically indistinct behavior of the path of the continuous curve.
Assume that w1 is a world that is an instantiation of an actual world possibility as predicted by statistical mechanics. In other words, assume that w1 is the actual world at some time in (the very distant) future. According to statistical mechanics, we know that if we are given an indefinite length of time, the universe will experience all possible distributions of particles. This means that it is inevitable that the actual world will experience highly unlikely, different fluctuations of entropy, or states of non-equilibrium distributions. During some interval of this kind, the behavior on the actual world could only be accounted for by the low probability of asymmetric particle distributions. So, we could imagine that w1 is just our world several years from now, experiencing high asymmetry. Eventually, w1 will return to equilibrium and the behavior of objects will begin to function as they now do and as entropy increases. Yet in highly asymmetrical conditions, it is hard to say how the laws of motion could be described by classical or relativistic physics. Motion would be much too random. In fact, at such low levels of entropy, we can only, at best, determine the positions of particles with various degrees of likelihood. It then seems reasonable to say that we know w1 (or the future of the actual world) has various kinds of probabilistic laws and is empirically indistinguishable from another particle with the same path, and that exists in the present, namely particle y. I should also add that we need not worry about stipulations regarding simultaneity; we need not, in other words, think that we could only consider the paths of particle x and particle y at the same time.
In addition, we could also stipulate that our similarity metric only picks out similarity in behavior and not similarity in laws; this might make some, but perhaps not all, Humeans happy. If we imagine a possible world whose laws are probabilistic, though in this instance they would specifically govern motion instead of distribution, then this world, in an indefinite length of time, will experience all possible motions of particles—including one of accidentally orderly movement, namely, the continuous curve. It then becomes a question of whether "divergence" from the actual world outweighs the empirically verified "convergence". And I can't think of any reason to say that it would since, in this particular case, our main concern is the empirical match of behavior. The crucial point, then, is to show that probabilistic laws are descriptive of real world phenomena and, as such, the probabilistic feature, or character, of the laws on w1 should not by itself rule out w1. So long as Tooley's possible world is something like the actual world's future (granted it would have to be the distant future), then we shouldn't be too worried that it carelessly crosses the boundaries of important epistemic constraints. Similarly, so long as the continuous curve, or the accidentally orderly movement, on w1 can be explained as a consequence of probabilistic laws of motion, then empirical indistinctness might be enough to establish "convergence". But the latter scenario will depend on how we choose to build the metric.
Criterion '(ii)' is intended to assure the Humean that in using w1 for philosophical analysis, we are taking basic principles of good scientific methodology into account. If we consider, for example, Occam's razor to be a basic sort of principle guiding one's methodology, then we might want to ask ourselves whether our earlier discussion of w1 conforms to it. This will really just guarantee us some degree of ontological parsimony. In other words, there is some kind of theoretical value that might be lost if we were to postulate various entities in order to explain accidentally orderly movement (i.e., the trade-off between having highly strange possible worlds operating in conformity to odd principles, for a causally laden notion of velocity that has high explanatory virtue simply wouldn't be worth it). Particularly if you think probabilistic laws of motion are the ontological equivalent of ghosts and goblins. But since it has already been argued that we could think of w1 as the actual world at a different temporal interval, then it is easy to satisfy this second criterion. At the very least, we can say that we aren't being unselective about our possible worlds, and, therefore, unselective about their ontologies. In other words, probabilistic laws, as construed above, are not wildly implausible. As such, we need not worry about failing to meet epistemic requirements with respect to how we go about postulating w1 based on what is given.
Criterion '(iii)' (which is, to reiterate, that any behavior we choose to analyze on w1 must be physically possible, i.e., we must be able to apply mathematics to it in the context of a physical theory) is a little bit more complicated since it isn't obvious that the low probability of non-equilibrium particle distributions, and the very, very distant future in which it may occur, gives the Humean a reason to take w1 seriously. We might want to amend the concern that motivates criterion '(iii)' as follows: if q is a physical possibility, then we need some independent reason telling us why (or how) we know that q is the case on w1. But if we were to look at other behaviors besides that of some particle 'x' in w1, then we would have to come up with an explanation for the strange behavior of all objects that we would find. Hence, the collective behavior of objects in w1 (if we were to take a good look around w1) is what would lead us to postulate q. This wouldn't violate any claims about logical properties supervening onto properties of locality, so long as they were exclusive. In other words, supervening properties would only attach to those positions of particles on the path of the continuous curve, but would not attach to other positions that would not maintain this same pattern. The error in this response, however, is that we don't want to be in danger of violating criterion '(ii)'; we don't want to be in the position where we have to explain why supervening properties are special to the positions of xand how we could know this was indeed the case.
Another response is then to challenge the grounds that motivate '(iii)', or to decide that if it is going to be a part of Φ, then the Humean needs to give us some reason why it should be. The third criterion might, after all, be too restrictive in critical instances; it might not allow us to consider important assumptions in order to test for the consistency of theoretical definitions. Tooley asks that we take w1 as a given such that if we were to consider a case in which q, then there would be a certain set of events that might follow. The independent reason for postulating q is to illuminate underlying relations and to test whether our theories are too restrictive, or not restrictive enough. We can call this "scientific methodology" if we think that certain thought experiments are useful tools for testing theoretical notions, and that possible world analyses are one form of thought experimentation. We can use worlds like w1 to test whether or not certain traditionally accepted theories can account for the same phenomena in different, physically possible, circumstances. What the Realist is asking of the Humean is to initially grant w1 in order to see if our best theories can handle certain kinds of tweaking, and in this particular case, to see if a supervening velocity is exhaustive. If there is any epistemic barrier left, it should not have anything to do with the postulation of q. We then might want to ask whether the Humean's agnosticism about laws in w1simply boils down to agnosticism about methodology.
So is Tooley begging the question by asking that the Humean grant this assumption? Do worlds like w1 presuppose theoretical notions of causality, even if it is only minimal? This type of objection is important because it forces us to ask ourselves whether we want to be bothered with having to explain probabilistically determined motion for testing theoretical notions. In other words, the only way that w1 is question begging is if we do not think it is important to account for the behavior of objects when they are probabilistically determined. If Tooley's argument is only negative (if it is just trying to establish that velocity is not logically supervening), then perhaps his argument may be question begging. On the other hand, if Tooley's argument has a positive conclusion, if he wants to establish that causally laden notions of velocity are high in explanatory virtue, and that a possible world like w1 is a good intuition pump for this idea, then Humeans might want to take w1 seriously. Those of you familiar with Tooley's paper know that the possible world of accidentally orderly movement is but one of several possible worlds which he considers. There are worlds of discontinuous motion (or instantaneous velocity) and continuous motion. If the Humean wishes to take the further step and claim that none of Tooley's worlds are acceptable, then it might seem as if the charge of question begging might be reversed. What then is to count as a legitimate possible world analysis, or legitimate thought experiment? While it is true that as scientifically minded philosophers, we will want to enforce scientific constraints for the kinds of possible worlds that we postulate; we do not want to be too restrictive so as to render interesting and important thought experiments completely moot.
Then what are the results of Tooley's analysis? In worlds with probabilistic motion something more than a ratio of initial to final position is required to determine velocity. What 'something more than' means is, I think, open for speculation. Tooley, of course, is going to argue that 'something more than' is some kind of a causal story that fixes certain relations between states of one object to other states of the same object. The important thing to emphasize, however, is that this is the more interesting philosophical question. In fact, this is an issue that the Humean might very well want to take up with Tooley if he were to initially accept w1. One question the Humean might want to ask Tooley is: how do relevant causal principles connect to the minimal degrees of prediction for future positions of objects? How does one establish causality from mere relations? In any case, the point I am trying to push here is that the Humean should give Tooley, and other Realists, some kind of reason why probabilistic motion is uninteresting that is independent of the fact that it might not cohere with a logical/mathematical supervening theory of velocity.
I have tried to show why the Humean has no obvious reason to reject, prima facie, a possible world analysis of worlds governed by probabilistic laws of motion. If there are any reasons why we should not take these worlds that experience accidentally orderly movement seriously, then it shouldn't have anything to do with problems concerning how we might know about the laws of such worlds. In fact, if there are reasons for rejecting w1, those reasons ought to be independent of any tension that the laws of w1 might create for traditionally accepted theories so long as w1 is philosophically interesting. Of course, if the Humean wants to claim that worlds like w1 are not philosophically interesting, and that we need not take them seriously, then it is up to him to provide us with reasons why we ought to think the same thing. It is also up to him to tell us what might serve as adequate criteria for deciding what possible worlds count as worthy of consideration, if he thinks that there are any. Perhaps the Humean prefers to defer to the Lewisian similarity metric. In such a case, w1 still remains on safe ground. But whether or not the Humean can defend himself against Tooley's argument once he grants w1 is something I will leave to him to try and do. The task, I'm sure, will not be easy.
BIBLIOGRAPHY
Callender, Craig. "Humean Supervenience and Rotating Homogenous Matter," Mind, Vol. 110, 431, January 2001.
Callender, Craig. Introducing Time. Totem Books: 2001.
Lewis, David. Counterfactuals. Cambridge, Mass.: Harvard University Press, 1973.
McLaughlin, Brian. "Varieties of Supervenience," Supervenience: New Essays. Savellos, Elias (ed), Cambridge: Needham Heights, 1995.
Tooley, Michael. "In Defense of the Existence of States of Motion," Philosophical Topics, Vol. XVI, No. 1, Spring 1988.
Tooley, Michael. Time, Tense, & Causation. Clarendon Press, Oxford: 1997.
Russell, Bertrand. The Principles of Mathematics. W.W. Norton & Co., New York: 1996. First published in 1903.
