Explaining causal loops

EXPLAINING CAUSAL LOOPS 259 Schaffer, J. 2010. Monism: the priority of the whole. Philosophical Review 119: 31 76. Sider, T. 2007. Parthood. Philosophical Review 116: 51 91. Tillman, C. 2011. Musical Materialism. British Journal of Aesthetics 51: 13 29. Van Inwagen, P. 1994. Composition as identity. Philosophical Perspectives 8: 207 20. Explaining causal loops ULRICH MEYER A causal loop is a sequence of events e 1,..., e n, each element of which is one of the causes of the next event, and whose last event e n is one of the causes of the first event e 1. The events that make up a loop need not be complete causes of one another, nor do they need to be complete effects of one another. In a causal loop, the arrows of causation go around in a circle, but there might be additional arrows that lead into the circle, or arrows that lead out of it. If there are no such branches then the loop is said to be causally isolated. Causal loops naturally arise in two different contexts. One is the time travel stories popularized by David Lewis (1976), in which there is backwards causation against the direction of time. The other cases involve models of the general theory of relativity first discussed by Kurt Gödel (1949) that possess closed timelike curves in which time itself loops along a particular worldline. In such models, there is no backwards causation and travelling back in time requires no particular effort; one just has to follow an appropriately chosen worldline. Both types of loop permit what looks like creation ex nihilo. In the Lewis case, suppose that you would like to travel in time, but do not have enough money to buy a time machine. Here is how you could get one. At some future time t 2, your future self uses a time machine to travel back to the present time t 1, where he hands the machine to your present self. You hold on to the time machine until t 2, when you will have become that future self. At that point, you use the time machine to go back to t 1. The time machine exists neither before t 1 nor after t 2 ; it just goes around in a circle. In the case of general relativity, there are solutions to Einstein s field equations due to Richard Gott and Li-Xin Li (1998) that do not start out with the usual big bang, but with a spacetime doughnut from which the rest of the universe and its contents branch off. (The initial part of this spacetime looks a little bit like the handle of an amphora. While there are closed paths that circle around the handle, there are also paths that continue to the rest of the amphora.) Analysis Vol 72 Number 2 April 2012 pp. 259 264 doi:10.1093/analys/ans045 ß The Author 2012. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: journals.permissions@oup.com

260 ULRICH MEYER There might well be good reasons for denying the possibility of any form of time travel. For example, Lewis only arrives at the possibility of time travel by assuming temporal parts, the psychological continuity view of personhood, and the possibility of backwards causation. All of these are assumptions one could reasonably reject. And in the case of closed timelike loops, one could follow D.H. Mellor and argue against the pernicious fallacy that...anything which is physically possible must be possible in fact (1998: 127). But these are not issues I want to address here. The question I am interested in is whether causal loops present a difficulty in addition to the perhaps problematic assumptions one needs to make in order to get the possibility of time travel in the first place. Many philosophers find causal loops deeply puzzling, and loops are widely thought to constitute a theoretical cost of any view that permits them. Perhaps they are not problematic enough to constitute a reductio of such views, but they are regarded as undesirable. A number of authors have tried to come up with time travel stories that do without causal loops altogether (Monton 2009), or at least avoid loops with creation ex nihilo (MacBeath 1982). I disagree. To say that causal loops are mysterious is to say that they are always inexplicable, and I do not think that is correct. Causal loops can admit all the explanation one could reasonably ask for. To demand that all events, including those in causal loops, must be explainable is to endorse a version of the principle of sufficient reason (PSR). There are different ways of understanding this principle, and what we say about causal loops depends on which of them we adopt. One obvious possibility is to read PSR as the principle of causation: PSR-1 Every event has a sufficient cause. George Schlesinger (1995) and David Wiggins (1996) argue that something like PSR-1 is what underwrites inferences to the best explanation. If that is right, then we might well be forced to accept PSR-1, even though there would still be room for disagreement about its epistemic status. (This is the old problem of induction in a different guise.) Enforcing PSR-1 often leads to infinitely descending chains of events in which each event is caused by the preceding one. This is familiar from standard models of classical mechanics, in which all the events at some time t 0 are caused by the events at an earlier time t 1, which in turn are caused by events at an even earlier time t 2, and so on, ad infinitum. The same can be true for causal loops. If circumstances and the laws of nature cooperate, then every event in a loop can admit of a causal explanation in terms of the events that precede it in the loop, plus events outside it if the loop is not causally isolated. What is different in the case of causal loops is that we can start out with an event e, move to its cause, and then to the cause of that cause, and then keep going until we get back to the event e that we started out with.

EXPLAINING CAUSAL LOOPS 261 By endorsing PSR-1, we would already have denied the existence of uncaused first causes that could serve as the ultimate explanation of events. All that PSR-1 requires is that every event has a causal explanation, not that the chain of explanations comes to an end somewhere. One might worry that explanations in a causal loop would be unacceptably circular, but I do not think this is a serious concern. For one, the composite explanation that we obtain by tracing the causal ancestry of an event e around a causal loop is not just that the event e occurred because it occurred. The composite explanation would also make reference to events outside the loop (if the loop is not causally isolated) and to the laws of nature. For them to permit loops at all, the laws must have a very peculiar structure, and that alone provides at least some explanation of why there was a causal loop to begin with, and why e was part of it. Moreover, it is in any case unclear why we must combine the local explanations into a composite explanation of the event e in terms of itself. Consider the closely related problem of defining an earlier-than relation < in a cyclical time series. If we tried to retain all the principles that are familiar from linear time then we quickly run into trouble. For example, if we stipulate that < is transitive then every instant in such a cyclical time is earlier than itself, which prohibits < from being either antisymmetric or irreflexive, as it is in linear time. Bas van Fraassen (1970: s. 3.1) and W.H. Newton-Smith (1980: s. 3.2) conclude that a cyclical time series cannot be described in terms of a binary earlier-than relation. I think a better way of dealing with the issue is to abandon the transitivity of <, and to use it as a local ordering relation, as suggested by Mark Reynolds (1994). Something similar would be a natural way of understanding PSR-1 in a closed causal loop. There are non-trivial, local explanations of events in the loop in terms of their immediate predecessors, but we cannot always combine these local explanations without losing explanatory strength. That is, there might be a good explanation of event e in terms of events f 1,..., f m, and all the fs might have good explanations in terms of events g 1,..., g n, but there is no good explanation of e in terms of g 1,..., g n. (To put this in logical terms, explanation need not have the cut property, which is a generalized version of transitivity.) There can be good local explanations that do not get undermined by the fact that, by stringing them together, we would obtain a weaker explanation of e in terms of itself. In this sense, causal loops are amenable to complete causal explanations, and are compatible with PSR-1. Having said this, there is reason to think that PSR-1 is not quite what G.W. Leibniz had in mind when he endorsed the principle of sufficient reason. In De rerum originatione radicali (1697), he writes: Let us imagine the book on the Elements of Geometry to have been eternal, one copy always having been made from another; then it is clear that, though we can give a reason for the present book based on the

262 ULRICH MEYER preceding book from which it was copied, we can never arrive at a complete reason, no matter how many books we may assume in the past, for one can always wonder why such books should have existed at all times; why there should be books at all, and why they should be written in this way. What is true of books is true also of the different states of the world; every subsequent state is somehow copied from the preceding one (although according to certain laws of change). No matter how far we may have gone back to earlier states, therefore, we will never discover in them a full reason why there should be a world at all, and why it should be such as it is. (Leibniz 1956: vol. II: 790) Requiring there to be a full reason in this sense is to ask for an explanation of why the entire sequence of events is the way it is, rather than otherwise. By its very nature, a full reason could not be a causal reason, and would thus go beyond what is at issue in PSR-1. This yields a second interpretation of the principle of sufficient reason: PSR-2 There is sufficient reason why the entire world is the way it is. Something like this seems to be behind the worry about generation ex nihilo. In the example discussed earlier, we might be able to explain the existence of a time machine at t 1 in terms of the existence of a time machine at t 2, but this does not appear to explain why there is a time machine at all. But if we take this worry seriously then we should also worry about, say, why it is that there are electrons. We can easily explain this causally, in terms of the laws of nature and the fact that there were electrons 5 minutes ago. But then the question arises why those earlier electrons existed, and we are quickly led into an infinite regress of causal explanations that never succeed in giving a full reason for why there are any electrons at all. Michael Dummett (1986) suggests that infinitely descending causal chains and causal loops are equally mysterious. I think the real culprit is PSR-2, which has the untenable consequence of ruling out contingent truths. To prove this, let C be the set of all contingently true propositions, and suppose that, as postulated by PSR-2, some proposition p is a sufficient reason for why the actual world is the way it is. Then p is sufficient reason for why C contains the elements that it does contain. But in order for p to be a sufficient reason, it has to be true, and if it is true then it is either contingently or necessarily true. It cannot be contingently true, though, because then it would be an element of C. Without trivializing PSR-2, no contingent proposition can count as sufficient reason for itself. So p has to be necessary. But if p is necessary, then any element of C, for which p is a sufficient reason, would have to be necessary as well. Since C contains only contingent propositions, this means that C is empty. So if PSR-2 is true then there are no contingent propositions. (Christopher Hill (1982) and Quentin Smith (1995) both present versions of this argument; I do not know who ultimately deserves credit for it.)

EXPLAINING CAUSAL LOOPS 263 Leibniz would have been happy to accept this conclusion. If ours is the best of all possible worlds, as he thinks it is, then an omnipotent, omniscient and supremely benevolent God could not have failed to create it, rather than some other world. Thus if God exists necessarily then it is necessary that the actual world is the way it is. But this defence does not help the rest of us, who do not believe in the necessary existence of God, and who reject the Panglossian claim that ours is the best of all possible worlds. The world could have been different from what it actually is. This shows that PSR-2 is false, and that requests for full explanations are misguided. Causal loops thus admit all the explanation one can reasonably ask for. If the laws of nature cooperate then the events that make up a loop can be explained causally. To ask for more, and to request a full explanation of causal loops, is to ask for something that is impossible. In this case, the blame would fall on the person asking the question, not on our inability to answer it. Causal loops are no more mysterious than infinitely descending causal chains, and Gott-Li universes are no more mysterious than universes in which there have always been electrons. 1 Colgate University Hamilton, NY 13346, USA umeyer@colgate.edu References Dummett, M. 1986. Causal loops. In The Nature of Time, eds. R. Flood and M. Lockwood, 135 69. Oxford: Blackwell. Gödel, K. 1949. A remark about the relationship between relativity theory and idealistic philosophy. In Albert Einstein: Philosopher Scientist, ed. P. Schilpp, 557 62. Evanston, Ill.: Library of Living Philosophers. Gott, J.R. and L.-X. Li. 1998. Can the universe create itself? Physical Review D 58: 023501. Hill, C. 1982. On a revised version of the principle of sufficient reason. Pacific Philosophical Quarterly 63: 236 42. Leibniz, G.W. 1956. Philosophical Papers and Letters, tr. L.E. Loemker. Chicago: University of Chicago Press. Lewis, D. 1976. The paradoxes of time travel. American Philosophical Quarterly 13: 145 52. MacBeath, M. 1982. Who was Dr who s father? Synthese 51: 397 430. Mellor, D.H. 1998. Real Time II. London: Routledge. Monton, B. 2009. Time travel without causal loops. Philosophical Quarterly 59: 54 67. Newton-Smith, W.H. 1980. The Structure of Time. London: Routledge. 1 This paper was inspired by a conference on time travel held at North Carolina State University in April 2011. I would like to thank Professor John Carroll and his colleagues for hosting this conference, and all the participants for very stimulating discussions.

264 JONATHAN TALLANT AND DAVID INGRAM Reynolds, M. 1994. Axiomatisation and decidability of F and P in cyclical time. Journal of Philosophical Logic 23: 197 224. Schlesinger, G. 1995. A pragmatic version of the principle of sufficient reason. Philosophical Quarterly 45: 439 59. Smith, Q. 1995. A defense of a principle of sufficient reason. Metaphilosophy 26: 97 106. van Fraassen, B. 1970. An Introduction to the Philosophy of Time and Space. New York: Random House. Wiggins, D. 1996. Sufficient reason. Acta Philosophica Fennica 61: 117 32. Time for distribution? JONATHAN TALLANT AND DAVID INGRAM 1. Introduction The presentist one who believes that only present objects exist faces a familiar problem. If only present objects exist, then what makes true our true claims about the past? What is the truth-maker for the proposition <Ross was a child>? 1 It can t be Ross s past self; the presentist claims that Ross s past self does not exist. According to Ross Cameron (2011), the truth-makers for past and future tensed propositions are presently instantiated Temporal Distributional Properties (TDPs). These properties are of the form: being-achild-and-then-being-an-adult-and-then-being-a-senior-citizen. (TDPs are more complex than this. The example is simplified and designed to describe the lifespan of Ross at only three instants, existing only in three states.) Ross s instantiation of this property makes it true that <Ross was a child>. We present an argument against Cameron s view. 2 There are two ways that we might understand the term distribute as it appears in TDP. On one reading the resulting TDPs are not up to the task of playing the truth-maker role; on the other, TDPs are incompatible with presentism. Before turning to this, we explain the putative nature of a TDP and try to offer some motivations for Cameron s view. 2. Spatial distribution and temporal distribution We begin with the spatial analogue of TDPs, Spatial Distributional Properties (SDPs), as they appear in Parsons (2004). Consider an object, O, which is 1 We borrow the example from Cameron (2011: 61). 2 We do not discuss Cameron s (2011: n. 17) fall-back position as this is, it seems to us, a wholly distinct view. Analysis Vol 72 Number 2 April 2012 pp. 264 270 doi:10.1093/analys/ans041 ß The Authors 2012. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: journals.permissions@oup.com