Causation and Time Reversal

Causation and Time Reversal Abstract What would it be for a process to happen backwards in time? Would such a process involve different causal relations? It is common to understand the time-reversal invariance of a physical theory in causal terms, such that whatever can happen forwards in time (according to the theory) can also happen backwards in time. This has led many to hold that time-reversal symmetry is incompatible with the asymmetry of cause and effect. This article critiques the causal reading of time reversal. First, I argue that the causal reading requires time-reversal-related models to be understood as representing distinct possible worlds and, on such a reading, causal relations are compatible with time-reversal symmetry. Second, I argue that the former approach does, however, raise serious sceptical problems regarding the causal relations of paradigm causal processes and as a consequence there are overwhelming reasons to prefer a non-causal reading of time reversal, whereby time reversal leaves causal relations invariant. On the non-causal reading, time-reversal symmetry poses no significant conceptual nor epistemological problems for causation. 1 Introduction   1.1 The directionality argument   1.2 Time reversal 2 What Does Time Reversal Reverse?   2.1 The B- and C-theory of time   2.2 Time reversal on the C-theory   2.3 Answers 3 Does Time Reversal Reverse Causal Relations?   3.1 Causation, billiards, and snooker   3.2 The epistemology of causal direction   3.3 Answers 4 Is Time-Reversal Symmetry Compatible with Causation?   4.1 Incompatibilism   4.2 Compatibilism   4.3 Answers 5 Outlook 1 Introduction What would the world be like if run backwards in time? This question is ambiguous since it depends upon whether a ‘backwards-in-time’ world would involve an inversion of cause and effect. It is common to understand the invariance of a physical theory under time reversal—an operation that takes a motion to the temporally reversed motion—as entailing that whatever can happen forwards in time (according to the theory) can happen backwards in time, implying that causal relations are reversed under time reversal. On such a reading, the time-reversal symmetry of fundamental physics appears incompatible with the asymmetry of cause and effect, and has consequently been taken by many to motivate eliminativism about causation in physics and at the ‘fundamental level’ more generally. This so-called directionality argument traces its roots back at least as far as Bertrand Russell’s ([1912]) defence of causal scepticism. In what follows, I argue that such worries about time reversal are misplaced on two grounds. First, a causal interpretation of time reversal (whereby time reversal inverts cause and effect) requires us to understand models related by a time-reversal transformation as representing distinct possible worlds. On such a reading, time-reversal symmetry is compatible with the existence of directed causal relations since each such world preserves the asymmetry of cause and effect. However, I show that this approach does lead to major conceptual and epistemological problems regarding the direction of causation for the kinds of systems to which we typically assign unambiguous causal judgements. Second, and consequently, I demonstrate that there are overwhelming reasons to reject a causal interpretation of time reversal. Rather, causal relations should be understood to remain invariant under time reversal. On the preferred non-causal reading of time reversal, I show that time-reversal symmetry poses no major conceptual nor epistemological problems for causation. Moreover, this reading fits naturally with popular accounts of causal discovery. The article is structured as follows: The rest of the introduction covers the article’s background, by outlining the directionality argument and the concept of time reversal. Section 2 asks what time reversal reverses. I consider two rival philosophical accounts of time reversal: the ‘C-theory’, according to which pairs of models related by a time-reversal transformation represent a single possible world; and the ‘B-theory’, according to which time-reversal-related models represent distinct possible worlds. I show that a causal reading of time reversal requires the B-theory, but that the C-theory is preferable on independent grounds. Section 3 asks whether time reversal reverses causal relations. I argue that time reversal should be interpreted non-causally, and defend the epistemology of causal direction this provides. Section 4 asks whether time-reversal symmetry is compatible with causation. I show that on both causal and non-causal readings of time-reversal, compatibilist accounts of causation and time-reversal invariance are available. Finally, Section 5 considers consequences of these conclusions. 1.1 The directionality argument Philosophical consideration of causation typically concerns a relation, R, that holds between a pair of events, c and e, where R(c, e) is read as ‘c causes e’.1 This relation is standardly taken to be asymmetric, such that if event c is a cause of event e, then e is not a cause of c: R(c,e)→¬R(e,c). (1) Thus, for a pair of causally related events, a direction of causation can be defined insofar as one event causes the other and not vice versa. We can say that c is the cause and e is the effect, and there is a fact of the matter as to which is which and in which direction the causal influence propagates. The causal relation is also assumed to be time-asymmetric insofar as causes temporally precede their effects. Thus, it follows from R(c, e) that c is earlier than e: R(c,e)→E(c,e), (2) where E is the ‘earlier than’ relation.2 These features of causation are widely held to be incompatible with time symmetries of fundamental physics. In particular, time symmetry plays a central role in Russell’s ([1912]) case for causal eliminativism in (classical) physics. Russell cites time-symmetric features of the law of gravitation (taken by Russell as an exemplar of physical laws) as incompatible with both the asymmetry and time asymmetry of causation. This prima facie incompatibility has been presented as an argument in the recent literature under the name ‘the directionality argument’,3 and runs roughly as follows: If the fundamental physical theories are time-symmetric, then they are not causal. The fundamental physical theories are time-symmetric. Therefore, the fundamental physical theories are not causal. The argument trades on the intuitive incompatibility of the (time) asymmetry of causation with the time symmetry of physical theories. The notion of ‘time symmetry’ is clearly central to the argument; however, it is ambiguous. A theory can be thought to be time symmetric in at least two distinct senses: first, it can be invariant under a set of well-defined time-reversal transformations; second, its dynamical laws can be of such a form that relative to some given state of a system, they determine or give non-trivial probabilities for its possible past and future trajectories.4 This second kind of time symmetry may be termed the ‘bidirectionality’ of its laws or predictive algorithm.5 The directionality argument has been discussed in terms of time-reversal invariance by Field ([2003], p. 436), Ney ([2009], p. 747), Norton ([2009], pp. 481–2), and Frisch ([2012], p. 320).6 1.2 Time reversal Time reversal may be understood in classical terms as a set of operations that reverse a physical motion. For example, the time reverse of a ball rolling from left to right is a ball of equal mass rolling with the same speed, but from right to left. A theory is invariant under time reversal if and only if the time reverse of every motion allowed by the theory is also a motion allowed by the theory. This entails that if a theory (i) models some particular process, x, and (ii) is time-reversal invariant, then it follows that the theory also models the time reverse of x. As such, a time-reversal invariant theory can model any allowable process relative to either time direction. For convenience, call a pair of models related by a time-reversal operation ‘TR-twins’. The contention of this article is that the relationship between causation and time reversal importantly depends upon whether or not one takes TR-twins to represent distinct possible states of affairs. Intuitively, time reversal inverts the time order of a sequence of states of a system by taking the time coordinates from t to −t. In general, however, time reversal also involves an operation on the instantaneous states of systems. For example, the standard time-reversal transformation for Newtonian mechanics involves velocity reversal. Newtonian mechanics is intuitively time-reversal invariant insofar as for any motion of particles allowed by the theory (assuming the elasticity of collisions and so on), the time-reversed motion is also allowed by the theory, and so the theory admits of no irreversible processes. However, since the instantaneous state of a Newtonian system includes velocities, a time-reversal operation that merely inverts the sequence of states fails to secure the time-reversal invariance of the theory, since the velocities of particles must also be inverted in order for the new sequence of states to satisfy the equations of motion.7 As such, we may understand time reversal as taking a sequence of states Si, …, Sf to Sf*, …, Si*, where the ‘*’ superscript denotes the time-reversal operation on the instantaneous state. This feature of time reversal is common across physical theories.8 The action of time reversal upon properties of the instantaneous states of a system brings up two important points of relevance to this article, concerning the relationship between causation and time reversal. First, switching a process for its TR-twin implies not only a passive coordinate transformation—that is, a shift in perspective—but also an active transformation upon physical quantities of the system. For example, Maudlin ([2007], p. 119) holds that the need to apply time reversal to instantaneous states implies that ‘even for an instantaneous state, there is a fact about how it is oriented with respect to the direction of time’.9 On the contrary, in the next section I defend a fully passive interpretation of time reversal (the C-theory), whereby TR-twins, despite potentially differing with respect to quantities of instantaneous states, nonetheless equivalently represent a single possible world. Second, the action of time reversal upon instantaneous states might be seen as an ad hoc device designed purely to secure the time-reversal invariance of a theory.10 This brings in an important pragmatic constraint on the form of a theory’s set of time-reversal operations: the time reverse of some process as determined by this set of operations must be a reasonable candidate for how that process would ‘appear’ if ‘viewed’ relative to the opposite direction of time.11 This constraint is sufficient my purposes; the independent and complex issue of the status and justification of particular sets of time-reversal transformations for different theories is outside the scope of the article. Independent of Russell’s motivations, the relationship between time-reversal invariance and causation is interesting for its own sake. In particular, it is unclear in what sense time-reversal invariance could be incompatible with causation. As Frisch ([2014], p. 119) notes, the incompatibility of time-reversal invariance and causation is often assumed without further argument, as though the incompatibility were self-evident. Such a view is mistaken. Upon analysis, I argue that time reversal and causation have a subtler and more interesting relationship. In order for time-reversal invariance of physical theories to bear upon the metaphysics of causation, we require a philosophical account of how states of affairs are transformed under time reversal. In the next section, I outline two such accounts: the B-theory and the C-theory. We shall see how these different theories motivate different accounts of how causal relations transform under time reversal. Importantly, I argue that on both accounts time-reversal invariance and causation are compatible. 2 What Does Time Reversal Reverse? With the preliminaries out of the way, we may now turn to the compatibility of time-reversal invariance and causation. My contention is that this depends upon whether time reversal is understood as inverting causal relations. We can set out two different readings of time reversal: Causal Time Reversal (CTR): Time reversal involves inverting causal relations, taking causes to effects and vice versa. Non-causal Time Reversal (¬CTR): Time reversal does not invert causal relations; the distinction between cause and effect remains invariant under time reversal. While CTR is often assumed in the literature,12 I’ll argue that ¬CTR is preferable. Interestingly, this issue has not been directly addressed in discussions of the directionality argument. Despite this, its relevance is clear. If time reversal inverts causal relations, then we face the following problem: if the world is described by a time-reversal invariant theory, then any possible way the world could be is describable by at least two models of the theory (TR-twins) that ascribe different causal relations to the world. Conversely, if time reversal does not invert causal relations, then there is no prima facie conceptual problem of causation for a time-reversal invariant theory, since TR-twins can share the same causal structure. This brings up two central aims of the article. First, I demonstrate that assuming CTR, the exact problem time-reversal invariance poses for causation depends upon one’s preferred temporal metaphysics. Second, I defend ¬CTR over CTR, and in this way argue that the time-reversal invariance of fundamental physical theory would not warrant eliminativism about causation. This section addresses the first aim and lays the groundwork for addressing the second. 2.1 The B- and C-theory of time The issue of CTR versus ¬CTR is importantly interconnected with whether one reads time reversal as an active or passive transformation—that is, whether TR-twins represent different possible worlds or are different descriptions of a single possible world. For instance, a fully passive reading of time reversal, whereby time reversal is understood as nothing more than a redescription of a process relative to the opposite direction of time, implies ¬CTR. Such a view is outlined by Reichenbach ([1956], pp. 31–2; my emphasis): Since it is always possible to construct a converse description [of a process], positive and negative time supply equivalent descriptions, and it would be meaningless to ask which of the two descriptions is true. Reichenbach’s suggestion is offered within an analysis of the time reversibility of classical mechanics. In virtue of the lack of irreversible classical mechanical processes, Reichenbach notes that classical mechanics may describe any allowable process relative to either temporal direction, and hence it is a matter of convention to hold that classical mechanics in any sense describes or governs processes in the future direction only. As such, forwards-in-time and backwards-in-time descriptions of processes are strictly equivalent. Reichenbach relates this issue to the conventionality of geometry, where different geometries may be used to equivalently model a single state of affairs. On this account, time reversal is a passive transformation within an equivalence class of models: for any process, p, classical mechanics offers TR-twins describing p relative to each time direction, and both models should be understood as picking out one and the same possible process.13 A similar view of time reversal—as offering different but equivalent descriptions—is offered by the cosmologist Gold ([1966], p. 327): […] the description of our universe in the opposite sense of time […] sounds very strange but it has no conflict with any laws of physics. [The] strange description is not describing another universe, or how it might be but isn’t, but it is describing the very same thing. Gold, unlike Reichenbach, is here discussing a backwards-in-time macroscopic description of the world, containing putatively irreversible processes (with assumed underlying reversible mechanics). In this case, the forwards and backwards descriptions differ in that the latter describes apparently improbable behaviour (for example, the anti-thermodynamic reforming of broken wine glasses, unmixing of coffee and milk, and so on) due to the presence of highly correlated variables. (I argue in Section 3 that such cases help to motivate the position of Reichenbach and Gold.) The key idea present in Reichenbach’s and Gold’s suggestions is that TR-twins offer distinct but equivalent descriptions of a single possible state of affairs. For convenience, I refer to this view as a ‘C-theory of time’.14 This terminology is motivated by McTaggart’s ([1908]) distinction between the B-series and the C-series. Whereas the B-series orders events in terms of the time-directed relation ‘earlier than’, the C-series is concerned only with the undirected ‘temporal betweenness’ ordering of events: […] the C series, while it determines the order, does not determine the direction. If the C series runs M, N, O, P, then the B-series […] can run either M, N, O, P (so that M is earliest and P latest) or else P, O, N, M (so that P is earliest and M latest). And there is nothing […] in the C series […] to determine which it will be. (McTaggart [1908], p. 462; my emphasis) The distinction between order and direction is key to the distinction between the B- and C-series.15 McTaggart’s usage of these terms is similar to Reichenbach’s, in delineating his position regarding time order in time-reversible physics.16,17 The C-series is contrasted by McTaggart with the B-series in terms of its lack of directionality. The C-series of a set of events does not determine their B-series: any time ordering of events in terms of temporal betweenness is compatible with two directed time orderings in terms of the ‘earlier than’ relation. This shows the difference in structure between the B- and C-series. The B- and C-theory give two different ontologies of temporal relations. On the C-theory, there are no time-directed states of affairs and, as such, no two worlds may differ solely with respect to the arrangement of ‘earlier than’ relations. We may understand time reversal as taking one B-series of events, arranged from earlier to later, to the inverse B-series by means of reversing each ‘earlier than’ relation. Such a transformation preserves the C-series of the events, since it leaves the temporal betweenness relations invariant. The adirectional ontology of time given by the C-theory entails that the two different time-directed pictures given by TR-twins differ only at the level of description—both TR-twins refer to the same time-direction-independent facts—and so the C-theory entails the Reichenbach–Gold passive interpretation of time reversal.18 Conversely, on the B-theory, there are time-directed states of affairs, and so it follows that, first, two worlds may differ solely with respect to the arrangement of ‘earlier than’ relations, and second, TR-twins describe distinct possible worlds. We can take these as necessary conditions for the B-theory that suffice to distinguish it from the C-theory.19 2.2 Time reversal on the C-theory In introducing the B- and C-theory, my aims are two-fold: first, to show that both theories offer compatibilist accounts of causation and time-reversal invariance; second, to argue that the C-theory offers the superior account of both the function of time reversal and of the epistemology of causal direction. Since I am both proposing and defending the C-theory, it is important to guard against possible objections and misunderstandings of its treatment of time reversal. 2.2.1 The C-theory doesn’t require time-reversal invariance Earman ([1974], p. 27) objects to the passive interpretation of time reversal entailed by the C-theory as it is presented here, holding that such an interpretation ‘is too powerful; for this conclusion [that time reversal amounts to a redescription of a single state of affairs] follows whether or not the laws of physics are time reversal invariant’.20 We can understand Earman’s point by noting that the following two questions concern distinct, though related, issues: Do TR-twins describe distinct possible worlds? Is some particular theory time-reversal invariant? The former question divides the B- and C-theory. The latter concerns an independent issue that might be taken to motivate either theory, though does not objectively favour either. While the latter is a broadly empirical question, the former is an a priori issue concerning the interpretation of time reversal that is conceptually independent of whether some particular theory is time-reversal invariant. Moreover, it is the former that directly concerns the relationship between causation and time reversal. Earman’s implication is that the independence of these two issues is a problem for the C-theory: the interpretation of time reversal it offers is independent of whether the relevant physics is time-reversal invariant. Importantly, the time-reversal invariance of fundamental physical theories is neither necessary nor sufficient for the C-theory. However, this is not a problem for the C-theory in itself; rather, it highlights that the C-theory primarily concerns an a priori issue that, in turn, determines one’s understanding of time-asymmetric phenomena.21 2.2.2 Time-reversal non-invariance on the C-theory Earman’s worry does, however, point to a more general problem: in reducing time reversal to a redescription of processes, the C-theory appears to trivialize time-reversal invariance since it is not immediately clear what sense can be made of time-reversal non-invariance on the C-theory. The problem is statable as a simple argument: P1. In order for a theory to be time-reversal non-invariant, a model of the theory must transform under time reversal to a non-model of theory. (Assumption) P2. On the C-theory of time, time reversal is just a redescription of a single possible world. (Definition) P3. A possible world cannot be deemed by some theory to be ‘physically possible’ relative to one description and ‘not physically possible’ relative to an equivalent description. (Assumption) C. Hence, on the C-theory, no theory can be time-reversal non-invariant. It might be thought that this implies that the difference between reversible and irreversible theories is not statable in C-theoretic terms, which would be a major weakness for the C-theory. We evidently do have a clear grasp of the differences between reversibility and irreversibility, as well as other kinds of probabilistic time asymmetries, so the C-theory had better possess the resources to account for this. However, the argument does not in fact pose such a problem for the C-theory: irreversible processes are straightforwardly describable in C-terms. Although this argument is valid, the main problem it picks out is that time reversal is insufficient to determine whether a theory describes irreversible processes when applied to entire models of a theory. This, I shall argue, is a problem for both the B- and C-theory. Let’s first establish that irreversibility is statable in C-terms. Consider a time-asymmetric law describing the behaviour of some variable x: L: The value of x increases, and never decreases, with time. This describes an ideal irreversible process: the increase in value of x.22 There are two notable features of L relevant to time reversal and the B- and C-theory. First, L describes an irreversible process. Second, L is stated in time-directed (B-theoretic) terms: if x increases relative to one temporal direction, it decreases relative to the opposite temporal direction, and thus L is not stated in a time-direction-neutral way. For the C-theorist, L is equivalent to L*: L*: The value of x decreases, and never increases, with time. If we take a model, m, that satisfies L, then its TR-twin, m*, satisfies L*. The C-theorist takes TR-twins m and m* to represent a single possible world, and so to privilege neither L nor L*. However, this does not mean that the C-theory is unable to accommodate the irreversibility described by L and L*. There is a key sense of irreversibility that is independent of time direction and so statable in C-terms: the x-process described by L (and by L*) is monotonic. For some type of process to satisfy either L or L* it must be monotonic, such that relative to a choice of positive time, it either (i) only increases or (ii) only decreases. The x-process is monotonic regardless of whether we take it to be an x-increasing or x-decreasing process. Figure 1 depicts three models to illustrate this: Figure 1a and Figure 1b depict TR-twins m and m*, respectively, and Figure 1c depicts a model, n, in which the value of x changes non-monotonically. On the C-theory, although m and m* represent a single possible world, they are structurally distinct from n. Importantly, Figure 1a and Figure 1b depict a monotonic gradient regardless of the designation of a direction of time. Given this, we can offer a C-theoretic version of L: Lc: The value of x changes monotonically in time. Such a law requires all x-processes to be coordinated such that relative to a choice of positive time, they are either all x-increasing or all x-decreasing. This key sense of irreversibility is thus statable in C-terms. Figure 1. View largeDownload slide Models of monotonic variation, m and m*, and non-monotonic variation, n, of a variable, x. (a) m represents the monotonic increase of a variable, x. (b) m* is the TR-twin of m and represents the monotonic decrease of x. (c) n represents non-monotonic variation of x. Figure 1. View largeDownload slide Models of monotonic variation, m and m*, and non-monotonic variation, n, of a variable, x. (a) m represents the monotonic increase of a variable, x. (b) m* is the TR-twin of m and represents the monotonic decrease of x. (c) n represents non-monotonic variation of x. This point may be generalized to non-idealized cases of putatively irreversible processes, such as the statistical time asymmetry of thermodynamics, and the time asymmetry of dynamical collapse theories such as the Ghirardi–Rimini–Weber theory (GRW), and also of probabilistic time asymmetries in particle physics, such as the decays of K0 and B0 mesons.23 Regarding thermodynamics, there is a clear conventional element in taking entropy to increase; for all we know, it could be that time really ‘goes’ from our future to our past, and hence a law that entropy tends to decrease rather than increase, contrary to our beliefs. What is important in accounting for the phenomena is not whether entropy ‘really’ increases or decreases, but rather that once we’ve fixed our convention about the positive direction of time, entropy either ‘increases and does not decrease’ or ‘decreases and does not increase’.24 Both time-direction-dependent descriptions pick up on the time-direction-independent monotonicity of entropy—entropy doesn’t fluctuate in both temporal directions.25 The C-theory captures this sense of irreversibility. Unlike the C-theory, the B-theory additionally legitimizes the question of whether entropy might ‘really’ increase-and-not-decrease or decrease-and-not-increase, but this is a separate issue that is not clearly epistemically accessible nor, for this reason, practically indispensable to the study of time asymmetry.26 As such, not only is law-like irreversibility and time asymmetry statable in C-terms, but it is also better understood in these terms. With this in mind, let’s return to the above argument. What is problematic regarding time reversal and irreversibility is that if some model satisfies Lc, then so does its TR-twin, regardless of whether one takes the TR-twins to represent different possible worlds, and so this is a problem for both the B- and C-theory. P1 states a requirement for a theory to be time-reversal non-invariant and this appears to be satisfied by the B-theory but not by the C-theory, since only the B-theory treats time reversal as an active transformation and so TR-twins as describing different possible worlds. On the B-theory—but not the C-theory—we can understand m and m* as representing distinct possible worlds, and so, in principle, the B-theory allows for a theory to contain m as a model without also containing m*. However, failure of time-reversal invariance in this sense would be quite odd. First, the choice of which model, m or m*, is used to represent some process is a matter of convention,27 and so a theory’s inclusion of one and not the other in its space of models would also be a matter of convention. Second, if a theory includes only one of a pair of TR-twins, it does not follow that the theory contains any law-like irreversible or probabilistically time-asymmetric processes. For law-like time asymmetry, a theory would have to satisfy a stronger condition. For example, a theory would contain a law-like irreversibility in the case that for some variable, x, the theory includes models of monotonic x-processes—such as m and m*—and excludes all models of non-monotonic x-processes—such as n. It is because of this second point that the argument as stated above is misleading. It establishes that the B-theory allows for a theory to be time-reversal non-invariant in a way that the C-theory does not, but this is only in the case when time reversal is understood as an operation upon an entire model of a theory. However, time reversal applied to an entire model cannot transform monotonic models such as m and m* to non-monotonic models such as n, and so non-invariance under such an operation is insufficient as a test for law-like irreversibility. This follows from our pragmatic constraint that time reversal functions in such a way that TR-twins represent what a process ‘looks like’ relative to the opposite time directions. Assuming this, it would be unreasonable for time reversal to fail to preserve monotonic and non-monotonic behaviour. Thus P2 in particular requires clarification; it is important to stress the context of the C-theorist’s claim that time reversal amounts to a redescription of processes. If we take a model of an ‘expanding’ gas, such that its TR-twin is a ‘contracting’ gas, the C-theory entails that these are equivalent descriptions of a gas occupying greater volume at one temporal end than the other. However, were we to embed such a system in a wider environment containing other gases displaying matching time-asymmetric behaviour, things would be different. In this case, switching one model for its TR-twin and holding the orientation of the other gases fixed would result in a physical change to the total system (for example, so that there were now a gas ‘contracting’ relative to the time direction in which the other gases were ‘expanding’). It would constitute a change to the C-series, and not only to the B-series, of the total system, since it would amount to a difference in the temporal-betweenness ordering of events, rather than only the earlier-than ordering. This sense of relative time reversal corresponds to an active change even on the C-theory. This gives us two different kinds of time reversal. First, a relative time reversal is an active transformation on both the B- and C-theory, since it changes the temporal betweenness relations, and hence constitutes a change regardless of the stipulated direction of time.28 Second, an absolute time reversal—applied to an entire model or to a total system (for example, the entire world)—is a passive transformation on the C-theory and an active transformation on the B-theory; only on the B-theory does a world identical to ours, save for the direction of time, constitute a different possible state of affairs. The argument as stated above establishes that only the B-theory allows for a theory to be non-invariant with respect to absolute time reversal. However, non-invariance under absolute time reversal is neither necessary nor sufficient for a theory to contain a law-like time asymmetry or irreversibility; thus the B- and C-theory account equally well for the existence of time asymmetries and irreversible processes. 2.3 Answers To recap, both the B- and C-theory can support physical laws that describe irreversible or probabilistically time-asymmetric processes. The central point at issue between the two theories is whether the application of a set of time-reversal transformations to a model of some theory takes us to a model that describes a different possible world. Time reversing an entire model amounts to a redescription of a single possible world according to the C-theory, but amounts to a description of a second, distinct possible world according to the B-theory. On the C-theory, time reversal is a purely passive, coordinative transformation, meaning TR-twins differ only in terms of notation: they represent a single possible world, and hence notation that varies under time reversal (such as the direction of velocities) should not be taken to represent a property of the target system. On the B-theory, time reversal involves altering fundamental temporal relations and hence takes us from one logically possible world to a distinct logically possible world.29 I have argued that the extra structure postulated by the B-theory is not required to account for temporally asymmetric phenomena. With regard to the relationship between causation and time reversal, we can now clarify the problem for causation posed by CTR. According to CTR, TR-twins represent different sets of causal relations. On the C-theory, since TR-twins represent a single possible world, the only way for TR-twins to agree on causes and effects is for there to be no causes or effects, given the assumption of CTR, is for there to be no causes or effects. Hence, this motivates causal eliminativism. The B-theory, conversely, contains the logical space for TR-twins to disagree over causal direction facts insofar as they represent different possible processes in different possible worlds, and thus the cause–effect asymmetry can be preserved in each possible world. However, in the next section I show that the combination of the B-theory and CTR leads to major problems in paradigm cases of causal processes, and as such I propose that ¬CTR should be preferred. 3 Does Time Reversal Reverse Causal Relations? We may now ask whether time reversal inverts causal relations. This section examines the relative appeal of CTR and ¬CTR in the context of (i) a time-symmetric process and (ii) a time-asymmetric process. 3.1 Causation, billiards, and snooker 3.1.1 Causation and time-symmetric processes Figure 2 depicts the time-symmetric process of a collision of two idealized billiard balls of equal mass on a frictionless plane. In Figure 2a, ball L has non-zero momentum and ball R is at rest. In Figure 2b, there is a perfectly elastic collision, upon which the total momentum of one ball is transferred to the other. Figure 2c depicts ball L at rest and ball R with non-zero momentum. Viewed from left to right—Figure 2a to Figure 2c—(call this the ‘left-to-right’ description), L's movement appears to cause R’s movement, and viewed from right to left—Figure 2c to Figure 2a—(call this the ‘right-to-left’ description), R's movement appears to cause L’s movement. Assuming CTR, if the left-to-right description represents a causal process in which L’s momentum causes R to move, then the right-to-left description represents the distinct causal process in which R’s momentum causes L to move. To fill in these distinct accounts, we can imagine a right-pointing arrow from L to R in Figure 2a on the left-to-right description, and a left-pointing arrow from R to L in Figure 2c in the right-to-left description. Alternatively, if we assume ¬CTR, both the left-to-right and right-to-left descriptions represent the same causal process. Figure 2. View largeDownload slide The time-symmetric process of a collision of two idealized billiard balls of equal mass on a frictionless plane. (a) L has non-zero momentum and R is at rest. (b) A perfectly elastic collision where momentum is completely transfered from one ball to the other. (c) L is at rest and R has non-zero momentum. Figure 2. View largeDownload slide The time-symmetric process of a collision of two idealized billiard balls of equal mass on a frictionless plane. (a) L has non-zero momentum and R is at rest. (b) A perfectly elastic collision where momentum is completely transfered from one ball to the other. (c) L is at rest and R has non-zero momentum. As we’ve seen, CTR and ¬CTR relate differently to the B- and C-theory. On the C-theory, since time reversal amounts to a redescription of a single possible process, CTR is untenable since it requires Figure 2a–Figure 2c and Figure 2c–Figure 2a to represent distinct possible processes. Hence, the C-theory requires ¬CTR and CTR requires the B-theory. The combination of the C-theory and ¬CTR applied to the billiards example suggests that if there is a causal process described here, the direction of causation is at best ambiguous. (I examine this issue in Section 3.2.) On the B-theory, since the left-to-right and right-to-left descriptions represent different possible processes, it is natural to read them as distinct causal processes. The combination of the B-theory and CTR fits with microphysical accounts of causation, such as dispositional accounts, in which causation is understood as some kind of unidirectional influence from one object to another. On such an account, the left-to-right and right-to-left descriptions refer to fundamentally different processes. In the left-to-right process, the initial non-zero left-to-right momentum of L is a cause of the collision. In the right-to-left process, the ‘initial’ non-zero right-to-left momentum of R is a cause of the collision.30 3.1.2 Causation and time-asymmetric processes Figure 3 depicts the time-asymmetric collision of two realistic snooker balls of equal mass on a frictional snooker table. In this case, an element of agential control is introduced: there is a snooker cue that interacts with the white cue ball. Furthermore, the presence of a non-conservative force—friction—brings in an important explanatory asymmetry between the left-to-right (Figure 3a to Figure 3c) and right-to-left (Figure 3c to Figure 3a) descriptions, and it is more convenient to describe the process in time-directed terms, unlike in the time-symmetric case. Figure 3. View largeDownload slide The time-asymmetric collision of two realistic snooker balls of equal mass on a frictional snooker table. (a) The snooker cue interacts with cue ball and the object ball is at rest. (b) The cue ball and the object ball collide, with a partial transfer of momentum. (c) The cue ball and object ball are at rest, and the surface temperature of the table is slightly raised. Figure 3. View largeDownload slide The time-asymmetric collision of two realistic snooker balls of equal mass on a frictional snooker table. (a) The snooker cue interacts with cue ball and the object ball is at rest. (b) The cue ball and the object ball collide, with a partial transfer of momentum. (c) The cue ball and object ball are at rest, and the surface temperature of the table is slightly raised. In the conventional left-to-right description (Figure 3a to Figure 3c), the cue strikes the cue ball, setting it in motion; next, the cue ball collides with the object ball, transferring most of its momentum to the object ball. The object ball then loses momentum due to the frictional force of the baize on the snooker table until it is at rest, as depicted in Figure 3c. This left-to-right description contains a number of causal terms, implying the following: the cue movement causes the cue ball’s movement; the cue ball’s movement causes the object ball’s movement; the baize causes the object ball to lose momentum. In the unconventional right-to-left description in Figure 3c–Figure 3a, an anomalous series of causal processes is implied. First, heat in the baize together with incoming air molecules conspire to set the object ball in motion. Next, the object ball’s motion in synchrony with inverse, concentrating soundwaves jointly imparts a gain in momentum in the collision of the object ball into the cue ball. Finally, the cue ball’s momentum is absorbed in a collision with the cue. As a candidate causal process, the right-to-left description shown in Figure 3c–Figure 3a is highly unsatisfactory. Two issues in particular stand out: (i) there are several points that imply a violation of the causal Markov condition (CMC), and (ii) the snooker player apparently loses her agential control over the balls’ motion. These imply both a causal and explanatory asymmetry between the two available time-directed descriptions (Figure 3a–Figure 3c and Figure 3c–Figure 3a), which, as I shall next argue, motivates ¬CTR. 3.2 The epistemology of causal direction If the unconventional right-to-left description of the scenario is understood as a causal process, this implies the existence of causally independent variables that nonetheless exhibit coordinated behaviour and hence are not statistically independent, in violation of the CMC.31 In simpler terms, there are a number of coincidences that can’t be explained away with reference to some common interactions in the causal past. As such, there is good reason to think that this does not represent a genuine causal process: it does not meet a standard criterion widely taken to be characteristic of causal relations, and central to the explanatory asymmetry of causes and effects.32 Calling such a process ‘causal’ is to insist on using the term quite outside its standard linguistic context and is thus heuristically unhelpful. After all, in order for the concept of causation to be useful in philosophical discourse, there ought to be reasonable restrictions on its domain of application so to exclude processes that violate standard causal criteria such as the CMC and its variants.33 For this reason, it is useful to defer to the patterns of conditional dependencies and independencies of variables to ascertain causal direction, as is characteristic of causal modelling.34 Furthermore, the introduction of agential control brings in a pragmatic constraint on causal inference: it is natural to stipulate that the snooker player has causal control over the cue and of the cue ball, and not vice versa. One can entertain a causal process such as that depicted from Figure 3c–Figure 3a, whereby the cue’s movements are (at least in part) caused by the motion of the cue ball, but it detaches various causal intuitions we have about snooker players from the causal relations described in the account. Reichenbach ([1956], p. 47) considers a similar problem regarding, in his case, tennis players and time-reversed ‘causal’ processes: It would be a strange experience indeed to see [tennis] players run backward. Such a motion, although compatible with the laws of mechanics, is unusual because we are safer if our steps are controlled by our eyes. This element of control is important in that it can be appealed to in order to privilege one of the two possible causal stories given relative to the opposite directions of time. Regardless of any underlying time symmetry, and regardless of any freedom to describe some process relative to either time direction, it is desirable to hold that we are not mistaken in such control judgements. This is because the appeal to control plays an explanatory role: it is reasonable to take the snooker player’s actions to explain the subsequent motion of the snooker balls and not vice versa. The notion of control and manipulation is central to agency and interventionist theories of causation, such as those of Menzies and Price ([1993]), Pearl ([2000]), and Woodward ([2003]). These provide a deflationist epistemology of the direction of causation, whereby the direction of causation is determined by the kind of patterns of correlations to which causal discovery algorithms are sensitive. In the case of Figure 3, we can appeal to the CMC, or more prosaically appeal to beliefs about the snooker player’s agential control, to ground a direction of causation.35 A deflationist account of causal direction holds that there is a direction of causation only in the presence of the right kind of probabilistic asymmetries (for example, irreversible processes, time-asymmetric screening-off conditions, and so on). Although the deflationist approach is applicable to our agential snooker case, it leaves open the status of causation in our idealized billiards case, in which there are insufficient asymmetries to ground a direction of causation. One option is that there just is no direction of causation intrinsic to such time-symmetric systems, but if one can refer to a wider system containing (for example) irreversible processes, then this can be used to define a direction of causation in the time-symmetric system. Such problem cases need not worry us in practice, since in general we do have sufficient asymmetric processes (for example, ourselves) to which to refer.36 In the idealized case of a world consisting solely of our idealized billiards example, the deflationist may hold that there is no fact about causal direction.37 Such an attitude towards idealized time-symmetric systems does not entail eliminativism nor scepticism about the direction of causation in worlds containing sufficient time asymmetries to determine a direction of causation. As such, the compatibility of such worlds with physical theories that are empirically adequate with respect to our world does not motivate eliminativism about causal direction with respect to our world. Whereas ¬CTR aligns with a deflationist account of causal direction, CTR aligns with a hyperrealist account of causal direction, whereby there is a causal direction that both outruns and is independent of the physical facts.38 This is because in order for time reversal to invert the direction of causation, the direction of causation must be independent of time-independent causal algorithms that ground the deflationist account of causal direction.39 To point to an example of hyperrealism, Maudlin ([2007], p. 172) takes the direction of causation to be determined by the ‘passage of time’, which he regards as ‘an ontological primitive [that] accounts for the basic distinction between what is to the future of an event and what is to its past’. As such, the direction of causation is independent of any particular probabilistically time-asymmetric processes in the world: ‘[causal] production [is] built on the foundational temporal asymmetry that would obtain even if the world were always in thermal equilibrium (even then, later states would arise out of earlier ones)’ (Maudlin [2007], p. 177). A hyperrealist account might seem preferable in the idealized billiards case, since the deflationist approach is silent about causal direction. However, the hyperrealist is still faced with the epistemic problem faced by the deflationist: there is no clear causal direction to be derived from the physical facts.40 Rather, the hyperrealist approach here is to stipulate a preferred causal arrow to artificially break the symmetry. While this may be innocuous in the billiards case, it creates a significant problem in cases like the agential snooker example where we have objective physical grounds for determining a preferred arrow of causation. Since the hyperrealist approach is by its nature insensitive to the kinds of factors that inform causal judgements, it gives up the explanatory benefits of the deflationist approach. Taking the direction of causation to be an ontological primitive licences worries about whether the snooker player’s action ‘really’ causes the movement of the snooker balls or vice versa, which is not a legitimate worry on the deflationist approach. In the kinds of cases where we naturally make unambiguous causal judgments, such as the snooker case, the deflationist epistemology of causation of ¬CTR is preferable to the hyperrealism of CTR. As such, causal relations should not be taken to reverse under time reversal. 3.3 Answers If we are to consider archetypal causal processes, namely, those that satisfy standard algorithms for causal discovery, then we ought to hold that causal relations do not invert under time reversal and so prefer ¬CTR to CTR. Though it may be intuitively plausible for causal relations to reverse under time reversal, such a view is reasonable only with respect to suitably time-symmetric cases—like the idealized billiards case of Figure 2—where there is no clearly preferred direction of causation. I have argued that in such cases it is better to be neutral with respect to causal direction than to adopt a hyperrealist account of causal direction. ¬CTR fits naturally with a C-theory of time. Combining the two, one may consider the two temporally opposed descriptions of the agential snooker example as equivalent descriptions of a single possible causal process. The Figure 3c–Figure 3a description, though unconventional in its form, may be taken to represent the same causal relations that are naturally read from the Figure 3a–Figure 3c description. The asymmetry between the two descriptions is not due to any important link between causation and time, but rather to time-independent factors that inform causal judgements. The issues of agency and the CMC lead to the same judgements about causal direction regardless of what one takes to be the underlying direction of time. This entails that any underlying time-reversal invariance of the microphysical description is beside the point; one may hold that there is a clear causal direction—a natural criterion for distinguishing between causes and effects in the example—which is invariant under time reversal. 4 Is Time-Reversal Symmetry Compatible with Causation? We are now in a position to evaluate the central question of the article: is time-reversal symmetry compatible with causation? In the previous sections, we considered the following questions: Do TR-twins represent distinct possible worlds? Does time reversal invert causal relations? These present four options, as listed in Table 1. It follows from our considerations that Options 2–4 give us compatibilism about causation and time-reversal symmetry, and that of these, Option 3 (¬CTR and the C-theory) is the preferred option. Before reviewing the compatibilist options, we can first look at the incompatibilism of Option 1. Table 1. The options available for each theory. C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 Table 1. The options available for each theory. C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 4.1 Incompatibilism 4.1.1 Option 1: Causal time reversal + C-theory = incompatibilism I have suggested that, assuming the C-theory, if there are directed causal relations between events, then these cannot be flipped under time reversal. I have taken this to show that the C-theory requires a non-causal understanding of time reversal. Interestingly, Gold ([1966], p. 327) appears to go in the opposite direction and take his passive (C-theoretic) interpretation of time reversal to entail a Russellian causal eliminativism, holding that ‘the idea of a cause and effect relationship now becomes meaningless’. Gold’s ([1966], pp. 327–8) contention is based on a causal interpretation of time reversal: You may see relationships within [a time-direction-neutral description] which are of the kind that in the conventional description one would be called the cause and the other the effect. In the description with the opposite sense of time you would just have to reverse these roles. Given that the C-theory lacks the structure to commit to two such worlds with distinct causal relations, applying CTR does indeed entail eliminativism: the only way for TR-twins to agree on causes and effects, assuming that these are flipped by time reversal, is simply for there to be no causes or effects. Conversely, if a C-theorist wants to commit to directed causal relations, then these must be fixed by properties of the C-theoretic model expressible in time-direction-neutral terms, and thus left invariant under time reversal. Seen in this way, the C-theorist is committed to no causal relations being flipped by time reversal and thus to ¬CTR, contra Gold. The central problem is that the following three claims form an inconsistent triad: There are directed causal relations between events. CTR: Time reversal reverses causal relations. C-Theory: TR-twins describe the same possible world. Though each statement is independently plausible, the three jointly entail a contradiction. However, as we have seen, we may reject any one of these claims and avoid inconsistency. As should be clear, I take the second claim to be the one to reject. What is most important though is that either the second or the third claim may be rejected so as to save the first. The mutual incompatibility does not mark out the first as being the problematic claim. 4.2 Compatibilism 4.2.1 Option 2: Causal time reversal + B-theory = compatibilism Option 2 avoids incompatibility by rejecting Claim 3 of the triad (the C-theory). The B-theory holds that TR-twins describe distinct possible worlds, and this provides the logical space for there to exist directed causal relations that are flipped by time reversal without engendering a contradiction: in each B-theoretic world, the asymmetry of cause and effect is preserved. In place of the direct incompatibility of Option 1, Option 2 gives us practical and epistemological problems concerning directed causal relations in the kinds of cases in which we routinely make unambiguous causal judgements, such as in the snooker example (Figure 3). In allowing the sequences of Figure 3a–Figure 3c and Figure 3c–Figure 3a to represent distinct causal processes, Option 2: (i) leads to a problem of underdetermination, since both ‘causal’ processes are consistent with the same sets of data; and, more importantly, (ii) fails to account for why Figure 3a–Figure 3c and Figure 3c–Figure 3a are asymmetric with respect to explanation, in that only the former satisfies standard algorithms for causal discovery. These problems are unique to this approach. It is only by committing to CTR that the causal realist can entertain the possibility of processes whose causal direction is the opposite of that given by causal discovery algorithms. 4.2.2 Option 3: Non-causal time reversal + C-theory = compatibilism I have argued that Option 3 is the preferred account: by holding that TR-twins represent the same possible world (C-theory), and that cause and effect are invariant under time reversal (¬CTR), one can hold the time-reversal invariance of a theory to pose no conceptual or epistemological problems for the direction of causation. We’ve seen that Option 3 has several key benefits. First, the key sense of law-like time-asymmetry that is satisfied by irreversible or probabilistically time-asymmetric processes is captured by the C-theory. Second, combining the C-theory with ¬CTR allows for a deflationist epistemology of causal direction that (i) preserves causal direction judgements as determined by standard causal discovery algorithms, and (ii) dissolves scepticism as to whether the direction of causation matches our standard causal direction judgements. 4.2.3 Option 4: Non-causal time reversal + B-theory = compatibilism The final option, which I have not discussed up to this point, is to combine the non-causal account of time reversal with the B-theory. In principle, there are a couple of ways to do this: (i) defend a primitivist account of the direction of causation and stipulate that this should not be inverted by time reversal; or (ii) defend the same epistemology of causal direction as that of the C-theorist, but in addition, hold that TR-twins describe distinct worlds with different time-direction facts. (ii) preserves the epistemic advantages of my preferred option—Option 3—but additionally allows for two worlds to differ solely in terms of ‘earlier than’ relations. In this sense, the B-theorist can avoid the epistemological problems faced in Option 2. However, this then requires that the direction of time is wholly independent of the direction of causation. This kind of realism about the direction of time may have independent motivations and benefits that are outside the scope of this article. However, in terms of the cases we’ve considered, I take the C-theory to be the natural metaphysics of time for a non-causal interpretation of time reversal. 4.3 Answers Time-reversal symmetry is compatible with the existence of directed causal relations. However, realism about causal direction comes with restrictions, as shown in our inconsistent triad: either CTR or the C-theory must be rejected, as summarized in Table 2. Moreover, I have argued that the most reasonable resolution of the triad is to reject CTR: time reversal should not be understood as inverting causal relations. Table 2. Is time-reversal symmetry compatible with causation? C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ Table 2. Is time-reversal symmetry compatible with causation? C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ Crucially, the compatibility of time-reversal symmetry and causation depends upon the interpretation of time reversal itself, and is independent of whether any particular physical theory is invariant under time reversal, contrary to standard presentations of the directionality argument. In our incompatibilist option—Option 1—the incompatibility is due to the combination of the CTR and the C-theory. For the compatibilist options, compatibility is due either to holding that time reversal does not invert causal relations (¬CTR) or to holding that TR-twins represent distinct possible worlds (B-theory). Each option is consistent with both time-reversal invariant and time-reversal non-invariant theories. 5 Outlook Causation and time-reversal invariance are not straightforwardly incompatible. Rather, the relationship between the two depends upon one’s interpretation of time reversal. I’ve shown that there are several compatibilist options available to the causal realist. Moreover, I have argued in favour of both the C-theory and ¬CTR: time reversal should be understood as a passive transformation that redescribes a single possible world, and so time reversal does not invert causal relations. This entails a suggestion about how to think of properties of instantaneous states that are acted upon by time-reversal operations: Such properties (such as velocity and momentum) are either (i) not causal, or (ii) not genuine properties of instantaneous states. That is to say, we cannot take a naïve view of velocities or momenta as telling us something about the direction of causal propagation or information flow. We can either take velocities to be non-causal in nature, so that velocities do not amount to something like causal dispositions—they just point one way or the other without contributing to the causal structure of a system—or we can take the direction of the velocity of a particle to be fixed by its position in a wider causal environment in which causal relations can be determined relative to causal discovery algorithms. This suggests a certain contextuality of such quantities—x has some velocity only relative to causal model Y.41 If we take CTR and the C-theory to both be appealing, which is reasonable, then we might think that causation is eliminated. However, this tacitly presupposes that causal facts are to be found in the microdynamics in the first place. I think this is to start off on the wrong foot. We should think that insofar as we do make causal direction judgements and wish to ascribe them to physical systems, these judgements derive from higher-level statistical observations and agential presuppositions that are themselves neutral regarding any microdynamical arrows of time or causation. Thus the time-reversal symmetry of the underlying dynamics need not require us to doubt whether there really are directed causal relations. This is welcome, since it is quite reasonable to be ambivalent about whether fundamental physics is time-reversal invariant. After all, the world of our experience accords to time-reversal non-invariant laws (for example, thermodynamics), which are underpinned by the time-reversal invariant laws of classical physics, which themselves are an approximation of quantum mechanics, which is, on many popular formulations, time-reversal non-invariant. It is desirable to avoid such worries when considering the status of causation. Footnotes 1 I assume the causal relation to hold between pairs of events for reasons of simplicity. What I say can be extended to more complicated cases, such as where an effect has multiple causes, and vice versa, and also where the relation holds between type events or possible values of variables. 2 Though these two features fall far short of a full ‘folk theory’ of causation, they suffice for the aim of this article, which is to assess whether such an account of causation is compatible with time-reversal symmetry. 3 See (Field [2003]; Ney [2009]; Frisch [2012]; Farr and Reutlinger [2013]) for versions of the directionality argument. 4 Farr and Reutlinger ([2013]) argue that Russell’s discussion of time symmetry appears to refer not to time-reversal invariance per se, but rather to the bidirectionality of the law of gravitation, in that it nomically entails the past and future trajectories of a given state—Russell ([1912], p. 15) holds that ‘the law [of gravitation] makes no difference between past and future: the future “determines” the past in exactly the same sense in which the past “determines” the future’. 5 By ‘predictive algorithm’ I have in mind the Born rule in quantum mechanics. In the case that such an algorithm is bidirectional, ‘predictive algorithm’ is a misnomer—such an algorithm would also be retrodictive. 6 Of these, only Norton explicitly endorses the claim that time-reversal invariance is incompatible with the asymmetry and time asymmetry of causation. Frisch rejects such an argument. Field and Ney both make implicit reference to both time-reversal invariance and the predictive/retrodictive symmetry of classical theories in discussing Russell’s claim. 7 In classical Hamiltonian mechanics, a state is given by the three-dimensional position and momentum values of the particles. Here, the momenta are vectorial properties—they are the product of velocity and mass. As such, everything I say about the direction of velocities can be translated to talk about the direction of momenta should the reader wish. This distinction makes no difference to the points made about time reversal and causation. 8 For instance, the standard set of time-reversal transformations in electrodynamics inverts the magnetic field, in quantum mechanics inverts spin, and so on. See (Sachs [1987]) for a detailed account of time-reversal operators across physics. 9 The idea of an instantaneous state being time directed has itself been taken to be conceptually problematic. Albert ([2000], p. 18) rhetorically asks ‘what can it possibly mean for a single instantaneous physical situation to be happening “backward”?’. Callender ([2000], Footnote 4) objects that ‘it just does not make sense to time-reverse a truly instantaneous state of a system’. 10 For instance, Arntzenius and Greaves ([2009], p. 563) note that ‘any theory, including ones that are (intuitively!) not time reversal invariant, can be made to come out “time reversal invariant” if we place no constraints on what counts as the “time reversal operation” on instantaneous states’. 11 Insofar as time-reversal operations may be applied to in-principle unobservable processes (such as the quantum-mechanical evolution of a system between measurements), the idea of a time-reversed state ‘appearing’ a certain way or being ‘viewed’ backwards in time is a heuristic metaphor. 12 For instance, from a recent article in Nature Physics: ‘Under time reversal […], states should become effects and vice versa’ (Oreshkov and Cerf [2015], p. 3). 13 In cases in which a model is its own time reverse—for example, a stationary particle—the same model describes both ‘forwards’ and ‘backwards’ versions of the relevant process. 14 The C-theory of time is presented and defended in (Farr [unpublished]). The claim that TR-twins are equivalent descriptions of a single state of affairs is entailed by the C-theory but not exhaustive of it. For the aims of the present article, this claim is the relevant feature of the C-theory. 15 Although McTaggart ([1908], [1927]) consistently refers to the C-series as ‘non-temporal’, this is due to precisely the same reasoning for which he takes the B-series to be non-temporal, that is, that neither series contains ‘real’ (A-series) change—in neither series is there a division between past, present, and future that changes. Farr ([unpublished]) argues that a C-theory of time is defensible once we relax the assumption that time requires A-series change. 16 Max Black ([1959]) similarly distinguishes the ‘order’ and ‘arrangement’ of a series of events, claiming that only the former is observable and hence fundamental. 17 See (Reichenbach [1956], Chapters 2–6). 18 In particular, any quantities that differ between TR-twins (such as instantaneous velocity, spin, and so on, as discussed above) can be considered descriptive artefacts that equally correspond to a single time-direction-independent (C-theoretic) state of affairs. 19 This is a non-standard way of presenting the commitments of a B-theory of time. This is due to the fact that the B-series is standardly presented in negative terms—in that it does not commit to the A-series’ properties of ‘pastness’, ‘presentness’, and ‘futurity’, nor an objective passage of time. However, the B-series is characterized by the inclusion of ‘earlier than’ relations that are not present in the C-series. Farr ([unpublished]) argues that the standard focus on the negative and not positive aspects of the B-series is due to the historical prominence of the debate over temporal passage, which separates the A-series from the B- and C-series. The separate issue of the directionality of time, which separates the B- and C-series, has occupied far less literature. 20 Earman’s criticism here is specifically aimed at the passive interpretation of time reversal defended by Black ([1962]), but also applies to his other targets, Reichenbach and Gold. Black claims that it would follow from time-reversal invariance of fundamental physics that ‘earlier than’ is a three-place relation (such that x is earlier than y only relative to some third term z—for example, an observer, some process, and so on). Earman rightly notes that Black’s conclusion is actually a consequence of the passive interpretation of time reversal, and follows regardless of the time-reversal invariance. 21 Indeed, Earman ([1974], p. 24) points to this distinction: […] the Reichenbach–Gold position [that is, the C-theory] cannot be based solely on time reversal invariance, but must rely on specialized assumptions about the nature of time reversal invariance. These assumptions have never been explicitly stated, much less justified. I should add that these specialized assumptions concern time reversal and not invariance under time reversal per se. 22 Though I use irreversibility as an illustrative example, the following line of reasoning equally well applies to probabilistic time asymmetries that are weaker than strict irreversibility. 23 The experimental violation of the combination of charge and parity symmetry (CP symmetry) in particle is well documented. For a discussion of CP violation in K0 meson decay, see (Sachs [1987], Chapters 8–9); for B0 mesons, see (Abe et al. [2001]). 24 This point is made at length by Price ([1996b]), particularly Chapters 2 and 7. 25 With respect to GRW, the key content of the irreversibility of collapses is that the set of collapses is co-oriented with respect to time such that in each GRW model there is, relative to a choice of time direction, either: (i) only collapses; or (ii) only ‘anti-collapses’. 26 Even supposing there is a privileged direction of time along which processes ‘really’ occur, we evidently do not need knowledge of this to collectively prefer to say that entropy ‘increases’ rather than ‘decreases’. 27 In other words, in principle, we might have preferred to describe processes in our world from future-to-past rather than from past-to-future without getting anything ‘wrong’. 28 Note, however, that on the B-theory, relative time reversal (change of the C-series) can, in principle, be carried out in two different ways: first, by holding the laboratory’s time orientation fixed and time reversing the experimental system; second, by holding the experimental system’s time orientation fixed and time reversing the laboratory. 29 Whether or not these logically possible worlds are deemed physically possible depends upon whether the relevant physical theory is time-reversal invariant. 30 It is also tenable for the B-theorist to adopt ¬CTR, as is discussed in Section 4.2. 31 The statistical dependence here is merely implicit. Given that the example depicts only a single (though abstract) run of the process, there is merely an apparent coincidence in that the initial conditions are highly improbable—they appear fine-tuned to entail coordinated behaviour. On multiple runs of this exact scenario, the statistics produced would provide a straightforward violation of the CMC, which holds that causally independent variables (relative to their causal pasts) are statistically independent (cf. Hausman and Woodward [1999]). 32 This point can be quite easily restated in terms of Lewis’s ([1979]) counterfactual theory of causation: according to Lewis's possible world semantics, the coincidences in the processes in Figure 3c–Figure 3a entail that in such a case, the past ‘overdetermines’ the future, and thus there is counterfactual dependence of earlier events upon later events but not vice versa. 33 For example, common-cause principles and screening-off conditions (cf. Arntzenius [2010]). 34 cf. (Pearl [2000]; Spirtes [2001]; Woodward [2003]). 35 Stipulations about agency and control play a key constitutive role in causal modelling. In general, multiple causal models will be compatible with the statistical data concerning relationships between variables of a system, and designating certain variables as ‘exogenous’ (that is, ‘free’ variables that are not effects of other variables in the system) narrows down the set of viable causal models for the system. 36 See (Farr [2016]) for a discussion of this issue in the context of the debate between John Norton ([2009]) and Mathias Frisch ([2009], [2014]) about causal reasoning in time symmetric systems. 37 The C-theorist could instead commit to a symmetric notion of ‘causal betweenness’, which provides an ordering that is invariant across TR-twins. This route appears to be taken by (Reichenbach [1956], p. 191). 38 See (Price and Weslake [2010]) for a critique of hyperrealist accounts of the direction of causation. 39 The patterns of statistical (in)dependence with which causal discovery algorithms are concerned are themselves neutral with respect to the direction of time. For instance, retrocausality—whereby a pair of cause-and-effect events are such that the cause event is later relative to clock time (for example, of a laboratory) than the effect event—is conceptually possible relative to such algorithms. 40 As Price and Weslake ([2010]) argue, hyperrealism about causation requires a denial of physicalism. 41 Price ([1996a]) suggests this kind of case as a problem for reducing causal direction to the fork asymmetry of causal models in microphysics: the direction of a causal process will ultimately be determined by which variable one includes in one’s model. This is suggestive of an arbitrariness of causal direction as determined by causal models. In particular, an open issue for this approach is what to make of ‘post-selected’ causal models, whereby the data are chosen in such a way to reveal patterns of correlations that suggest the opposite causal direction to that which we take to hold in the world. Whether we can reject the significance of such apparently causal relations on grounds of being artificial or unnatural is interesting; but this is an issue for another article. Acknowledgements This work was funded by the Templeton World Charity Foundation (TWCF 0064/AB38). Thanks to Jeremy Butterfield, Fabio Costa, Natalja Deng, Sam Fletcher, Huw Price, Sally Shrapnel, Magdalena Zych, and two anonymous referees for helpful comments. I dedicate the paper to the memory of my snooker buddy and dad, Peter Robert Farr. References Abe K. , Abe R. , Adachi I. , Ahn B. S. , Aihara H. , Akatsu M. , Alimonti G. , Asai K. , Asai M. , Asano Y. , Aso, T., Aulchenko, V., Aushev, T, Bakich, A. M., Banas, E, Behari, S., Behera, P. K., Beiline, D., Bondar, A., Bozek, A., Browder, T. E., Casey, B. C., Chang, P., Chao, Y., Chen, K. F., Cheon, B. G., Chistov, R., Choi, S. K., Choi, Y., Dong, L. Y., Dragic, J., Drutskoy, A., Eidelman, S., Eiges, V., Enari, Y., Enomoto, R., Everton, C. W., Fang, F., Fujii, H., Fukunaga, C., Fukushima, M., Gabyshev, N., Garmash, A., Gershon, T. J., Gordon, A., Gotow, K., Guler, H., Guo, R., Haba, J., Hamasaki, H., Hanagaki, K., Handa, F., Hara, K., Hara, T., Hastings, N. C., Hayashii, H., Hazumi, M., Heenan, E. M., Higasino, Y., Higuchi, I., Higuchi, T., Hirai, T., Hirano, H., Hojo, T., Hokuue, T., Hoshi, Y., Hoshina, K., Hou, S. R., Hou, W. S., Hsu, S. C., Huang, H. C., Igarashi, Y., Iijima, T., Ikeda, H., Ikeda, K., Inami, K., Ishikawa, A., Ishino, H., Itoh, R., Iwai, G., Iwasaki, H., Iwasaki, Y., Jackson, D. 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W., Natkaniec, Z., Neichi, K., Nishida, S., Nitoh, O., Noguchi, S., Nozaki, T., Ogawa, S., Ohshima, T., Ohshima, Y., Okabe, T., Okazaki, T., Okuno, S., Olsen, S. L., Ozaki, H., Pakhlov, P., Palka, H., Park, C. S., Park, C. W., Park, H., Peak, L. S., Peters, M., Piilonen, L. E., Prebys, E., Rodriguez, J. L., Root, N., Rozanska, M., Rybicki, K., Ryuko, J., Sagawa, H., Sakai, Y., Sakamoto, H., Satapathy, M., Satpathy, A., Schrenk, S., Semenov, S., Senyo, K., Settai, Y., Sevior, M. E., Shibuya, H., Shwartz, B., Sidorov, A., Stanic, S., Sugi, A., Sugiyama, A., Sumisawa, K., Sumiyoshi, T., Suzuki, J., Suzuki, K., Suzuki, S., Suzuki, S. Y., Swain, S. K., Tajima, H., Takahashi, T., Takasaki, F., Takita, M., Tamai, K., Tamura, N., Tanaka, J., Tanaka, M., Taylor, G. N., Teramoto, Y., Tomoto, M., Tomura, T., Tovey, S. N., Trabelsi, K., Tsuboyama, T., Tsukamoto, T., Uehara, S., Ueno, K., Unno, Y., Uno, S., Ushiroda, Y., Vahsen, S. E., Varvell, K. E., Wang, C. C., Wang, C. H., Wang, J. G., Wang, M. Z., Watanabe, Y., Won, E., Yabsley, B. D., Yamada, Y., Yamaga, M., Yamaguchi, A., Yamamoto, H., Yamanaka, T., Yamashita, Y., Yamauchi, M., Yanaka, S., Yashima, J., Yokoyama, M., Yoshida, K., Yusa, Y., Yuta, H., Zhang, C. C., Zhang, J., Zhao, H. W., Zheng, Y., Zhilich, V., Zontar, D. and Belle Collaboration. [ 2001 ]: ‘ Observation of Large CP Violation in the Neutral B Meson System ’, Physical Review Letters , 87 , p. 091802 . Google Scholar CrossRef Search ADS PubMed Albert D. Z. [ 2000 ]: Time and Chance , Harvard, MA : Harvard University Press . Arntzenius F. [ 2010 ]: ‘Reichenbach’s Common Cause Principle’, in Zalta E. N. (ed.), The Stanford Encyclopedia of Philosophy . Arntzenius F. , Greaves H. [ 2009 ]: ‘ Time Reversal in Classical Electromagnetism ’, British Journal for the Philosophy of Science , 60 , p. 557 – 84 . Google Scholar CrossRef Search ADS Black M. [ 1959 ]: ‘ The “Direction” of Time ’, Analysis , 19 , pp. 54 – 63 . Google Scholar CrossRef Search ADS Black M. [ 1962 ]: Models and Metaphors: Studies in Language and Philosophy , Ithaca, NY : Cornell University Press . Callender C. [ 2000 ]: ‘ Is Time “Handed” in a Quantum World? ’, Proceedings of the Aristotelian Society , 100 , pp. 247 – 69 . Earman J. [ 1974 ]: ‘ An Attempt to Add a Little Direction to “The Problem of the Direction of Time” ’, Philosophy of Science , 41 , pp. 15 – 47 . Google Scholar CrossRef Search ADS Farr M. [ 2016 ]: ‘ Mathias Frisch: Causal Reasoning in Physics ’, British Journal for the Philosophy of Science , 64 , pp. 1207 – 13 . Google Scholar CrossRef Search ADS Farr M. [unpublished]: ‘The C Theory of Time’. Farr M. , Reutlinger A. [ 2013 ]: ‘ A Relic of a Bygone Age? Causation, Time Symmetry, and the Directionality Argument ’, Erkenntnis , 78 , pp. 215 – 35 . Google Scholar CrossRef Search ADS Field H. [ 2003 ]: ‘Causation in a Physical World’, in Loux M. , Zimmerman D. (eds), Oxford Handbook of Metaphysics , Oxford : Oxford University Press , pp. 4354 – 60 . Frisch M. [ 2009 ]: ‘ Causality and Dispersion: A Reply to John Norton ’, British Journal for the Philosophy of Science , 60 , pp. 487 – 95 . Google Scholar CrossRef Search ADS Frisch M. [ 2012 ]: ‘ No Place for Causes? Causal Skepticism in Physics ’, European Journal for Philosophy of Science , 2 , pp. 313 – 36 . Google Scholar CrossRef Search ADS Frisch M. [ 2014 ]: Causal Reasoning in Physics , Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Gold T. [ 1966 ]: ‘Cosmic Processes and the Nature of Time’, in Colodny R. G. (ed.), Mind and Cosmos , Pittsburgh, PA : University of Pittsburgh Press , pp. 3 – 29 . Hausman D. , Woodward J. [ 1999 ]: ‘ Independence, Invariance, and the Causal Markov Condition ’, British Journal for the Philosophy of Science , 50 , pp. 521 – 83 . Google Scholar CrossRef Search ADS Lewis D. [ 1979 ]: ‘ Counterfactual Dependence and Time’s Arrow ’, Noûs , 13 , pp. 455 – 76 . Google Scholar CrossRef Search ADS Maudlin T. [ 2007 ]: The Metaphysics within Physics , Oxford : Oxford University Press . Google Scholar CrossRef Search ADS McTaggart J. M. E. [ 1908 ]: ‘ The Unreality of Time ’, Mind , 17 , pp. 457 – 74 . Google Scholar CrossRef Search ADS McTaggart J. M. E. [ 1927 ]: The Nature of Existence, Volume II, Cambridge : Cambridge University Press . Menzies P. , Price H. [ 1993 ]: ‘ Causation as a Secondary Quality ’, British Journal for the Philosophy of Science , 44 , pp. 187 – 203 . Google Scholar CrossRef Search ADS Ney A. [ 2009 ]: ‘ Physical Causation and Difference-Making ’, British Journal for the Philosophy of Science , 60 , p. 737 . Google Scholar CrossRef Search ADS Norton J. D. [ 2009 ]: ‘ Is There an Independent Principle of Causality in Physics? ’, British Journal for the Philosophy of Science , 60 , pp. 475 – 86 . 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Google Scholar CrossRef Search ADS Sachs R. G. [ 1987 ]: The Physics of Time Reversal , Chicago, IL : University of Chicago Press . Spirtes P. , Glymour C. , Scheines R. [ 2001 ]: Causation, Prediction, and Search , Cambridge, MA : MIT Press . Woodward J. [ 2003 ]: Making Things Happen: A Theory of Causal Explanation , Oxford : Oxford University Press . © The Author(s) 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The British Journal for the Philosophy of Science Oxford University Press

Causation and Time Reversal

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Oxford University Press
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© The Author(s) 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com
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Abstract

Abstract What would it be for a process to happen backwards in time? Would such a process involve different causal relations? It is common to understand the time-reversal invariance of a physical theory in causal terms, such that whatever can happen forwards in time (according to the theory) can also happen backwards in time. This has led many to hold that time-reversal symmetry is incompatible with the asymmetry of cause and effect. This article critiques the causal reading of time reversal. First, I argue that the causal reading requires time-reversal-related models to be understood as representing distinct possible worlds and, on such a reading, causal relations are compatible with time-reversal symmetry. Second, I argue that the former approach does, however, raise serious sceptical problems regarding the causal relations of paradigm causal processes and as a consequence there are overwhelming reasons to prefer a non-causal reading of time reversal, whereby time reversal leaves causal relations invariant. On the non-causal reading, time-reversal symmetry poses no significant conceptual nor epistemological problems for causation. 1 Introduction   1.1 The directionality argument   1.2 Time reversal 2 What Does Time Reversal Reverse?   2.1 The B- and C-theory of time   2.2 Time reversal on the C-theory   2.3 Answers 3 Does Time Reversal Reverse Causal Relations?   3.1 Causation, billiards, and snooker   3.2 The epistemology of causal direction   3.3 Answers 4 Is Time-Reversal Symmetry Compatible with Causation?   4.1 Incompatibilism   4.2 Compatibilism   4.3 Answers 5 Outlook 1 Introduction What would the world be like if run backwards in time? This question is ambiguous since it depends upon whether a ‘backwards-in-time’ world would involve an inversion of cause and effect. It is common to understand the invariance of a physical theory under time reversal—an operation that takes a motion to the temporally reversed motion—as entailing that whatever can happen forwards in time (according to the theory) can happen backwards in time, implying that causal relations are reversed under time reversal. On such a reading, the time-reversal symmetry of fundamental physics appears incompatible with the asymmetry of cause and effect, and has consequently been taken by many to motivate eliminativism about causation in physics and at the ‘fundamental level’ more generally. This so-called directionality argument traces its roots back at least as far as Bertrand Russell’s ([1912]) defence of causal scepticism. In what follows, I argue that such worries about time reversal are misplaced on two grounds. First, a causal interpretation of time reversal (whereby time reversal inverts cause and effect) requires us to understand models related by a time-reversal transformation as representing distinct possible worlds. On such a reading, time-reversal symmetry is compatible with the existence of directed causal relations since each such world preserves the asymmetry of cause and effect. However, I show that this approach does lead to major conceptual and epistemological problems regarding the direction of causation for the kinds of systems to which we typically assign unambiguous causal judgements. Second, and consequently, I demonstrate that there are overwhelming reasons to reject a causal interpretation of time reversal. Rather, causal relations should be understood to remain invariant under time reversal. On the preferred non-causal reading of time reversal, I show that time-reversal symmetry poses no major conceptual nor epistemological problems for causation. Moreover, this reading fits naturally with popular accounts of causal discovery. The article is structured as follows: The rest of the introduction covers the article’s background, by outlining the directionality argument and the concept of time reversal. Section 2 asks what time reversal reverses. I consider two rival philosophical accounts of time reversal: the ‘C-theory’, according to which pairs of models related by a time-reversal transformation represent a single possible world; and the ‘B-theory’, according to which time-reversal-related models represent distinct possible worlds. I show that a causal reading of time reversal requires the B-theory, but that the C-theory is preferable on independent grounds. Section 3 asks whether time reversal reverses causal relations. I argue that time reversal should be interpreted non-causally, and defend the epistemology of causal direction this provides. Section 4 asks whether time-reversal symmetry is compatible with causation. I show that on both causal and non-causal readings of time-reversal, compatibilist accounts of causation and time-reversal invariance are available. Finally, Section 5 considers consequences of these conclusions. 1.1 The directionality argument Philosophical consideration of causation typically concerns a relation, R, that holds between a pair of events, c and e, where R(c, e) is read as ‘c causes e’.1 This relation is standardly taken to be asymmetric, such that if event c is a cause of event e, then e is not a cause of c: R(c,e)→¬R(e,c). (1) Thus, for a pair of causally related events, a direction of causation can be defined insofar as one event causes the other and not vice versa. We can say that c is the cause and e is the effect, and there is a fact of the matter as to which is which and in which direction the causal influence propagates. The causal relation is also assumed to be time-asymmetric insofar as causes temporally precede their effects. Thus, it follows from R(c, e) that c is earlier than e: R(c,e)→E(c,e), (2) where E is the ‘earlier than’ relation.2 These features of causation are widely held to be incompatible with time symmetries of fundamental physics. In particular, time symmetry plays a central role in Russell’s ([1912]) case for causal eliminativism in (classical) physics. Russell cites time-symmetric features of the law of gravitation (taken by Russell as an exemplar of physical laws) as incompatible with both the asymmetry and time asymmetry of causation. This prima facie incompatibility has been presented as an argument in the recent literature under the name ‘the directionality argument’,3 and runs roughly as follows: If the fundamental physical theories are time-symmetric, then they are not causal. The fundamental physical theories are time-symmetric. Therefore, the fundamental physical theories are not causal. The argument trades on the intuitive incompatibility of the (time) asymmetry of causation with the time symmetry of physical theories. The notion of ‘time symmetry’ is clearly central to the argument; however, it is ambiguous. A theory can be thought to be time symmetric in at least two distinct senses: first, it can be invariant under a set of well-defined time-reversal transformations; second, its dynamical laws can be of such a form that relative to some given state of a system, they determine or give non-trivial probabilities for its possible past and future trajectories.4 This second kind of time symmetry may be termed the ‘bidirectionality’ of its laws or predictive algorithm.5 The directionality argument has been discussed in terms of time-reversal invariance by Field ([2003], p. 436), Ney ([2009], p. 747), Norton ([2009], pp. 481–2), and Frisch ([2012], p. 320).6 1.2 Time reversal Time reversal may be understood in classical terms as a set of operations that reverse a physical motion. For example, the time reverse of a ball rolling from left to right is a ball of equal mass rolling with the same speed, but from right to left. A theory is invariant under time reversal if and only if the time reverse of every motion allowed by the theory is also a motion allowed by the theory. This entails that if a theory (i) models some particular process, x, and (ii) is time-reversal invariant, then it follows that the theory also models the time reverse of x. As such, a time-reversal invariant theory can model any allowable process relative to either time direction. For convenience, call a pair of models related by a time-reversal operation ‘TR-twins’. The contention of this article is that the relationship between causation and time reversal importantly depends upon whether or not one takes TR-twins to represent distinct possible states of affairs. Intuitively, time reversal inverts the time order of a sequence of states of a system by taking the time coordinates from t to −t. In general, however, time reversal also involves an operation on the instantaneous states of systems. For example, the standard time-reversal transformation for Newtonian mechanics involves velocity reversal. Newtonian mechanics is intuitively time-reversal invariant insofar as for any motion of particles allowed by the theory (assuming the elasticity of collisions and so on), the time-reversed motion is also allowed by the theory, and so the theory admits of no irreversible processes. However, since the instantaneous state of a Newtonian system includes velocities, a time-reversal operation that merely inverts the sequence of states fails to secure the time-reversal invariance of the theory, since the velocities of particles must also be inverted in order for the new sequence of states to satisfy the equations of motion.7 As such, we may understand time reversal as taking a sequence of states Si, …, Sf to Sf*, …, Si*, where the ‘*’ superscript denotes the time-reversal operation on the instantaneous state. This feature of time reversal is common across physical theories.8 The action of time reversal upon properties of the instantaneous states of a system brings up two important points of relevance to this article, concerning the relationship between causation and time reversal. First, switching a process for its TR-twin implies not only a passive coordinate transformation—that is, a shift in perspective—but also an active transformation upon physical quantities of the system. For example, Maudlin ([2007], p. 119) holds that the need to apply time reversal to instantaneous states implies that ‘even for an instantaneous state, there is a fact about how it is oriented with respect to the direction of time’.9 On the contrary, in the next section I defend a fully passive interpretation of time reversal (the C-theory), whereby TR-twins, despite potentially differing with respect to quantities of instantaneous states, nonetheless equivalently represent a single possible world. Second, the action of time reversal upon instantaneous states might be seen as an ad hoc device designed purely to secure the time-reversal invariance of a theory.10 This brings in an important pragmatic constraint on the form of a theory’s set of time-reversal operations: the time reverse of some process as determined by this set of operations must be a reasonable candidate for how that process would ‘appear’ if ‘viewed’ relative to the opposite direction of time.11 This constraint is sufficient my purposes; the independent and complex issue of the status and justification of particular sets of time-reversal transformations for different theories is outside the scope of the article. Independent of Russell’s motivations, the relationship between time-reversal invariance and causation is interesting for its own sake. In particular, it is unclear in what sense time-reversal invariance could be incompatible with causation. As Frisch ([2014], p. 119) notes, the incompatibility of time-reversal invariance and causation is often assumed without further argument, as though the incompatibility were self-evident. Such a view is mistaken. Upon analysis, I argue that time reversal and causation have a subtler and more interesting relationship. In order for time-reversal invariance of physical theories to bear upon the metaphysics of causation, we require a philosophical account of how states of affairs are transformed under time reversal. In the next section, I outline two such accounts: the B-theory and the C-theory. We shall see how these different theories motivate different accounts of how causal relations transform under time reversal. Importantly, I argue that on both accounts time-reversal invariance and causation are compatible. 2 What Does Time Reversal Reverse? With the preliminaries out of the way, we may now turn to the compatibility of time-reversal invariance and causation. My contention is that this depends upon whether time reversal is understood as inverting causal relations. We can set out two different readings of time reversal: Causal Time Reversal (CTR): Time reversal involves inverting causal relations, taking causes to effects and vice versa. Non-causal Time Reversal (¬CTR): Time reversal does not invert causal relations; the distinction between cause and effect remains invariant under time reversal. While CTR is often assumed in the literature,12 I’ll argue that ¬CTR is preferable. Interestingly, this issue has not been directly addressed in discussions of the directionality argument. Despite this, its relevance is clear. If time reversal inverts causal relations, then we face the following problem: if the world is described by a time-reversal invariant theory, then any possible way the world could be is describable by at least two models of the theory (TR-twins) that ascribe different causal relations to the world. Conversely, if time reversal does not invert causal relations, then there is no prima facie conceptual problem of causation for a time-reversal invariant theory, since TR-twins can share the same causal structure. This brings up two central aims of the article. First, I demonstrate that assuming CTR, the exact problem time-reversal invariance poses for causation depends upon one’s preferred temporal metaphysics. Second, I defend ¬CTR over CTR, and in this way argue that the time-reversal invariance of fundamental physical theory would not warrant eliminativism about causation. This section addresses the first aim and lays the groundwork for addressing the second. 2.1 The B- and C-theory of time The issue of CTR versus ¬CTR is importantly interconnected with whether one reads time reversal as an active or passive transformation—that is, whether TR-twins represent different possible worlds or are different descriptions of a single possible world. For instance, a fully passive reading of time reversal, whereby time reversal is understood as nothing more than a redescription of a process relative to the opposite direction of time, implies ¬CTR. Such a view is outlined by Reichenbach ([1956], pp. 31–2; my emphasis): Since it is always possible to construct a converse description [of a process], positive and negative time supply equivalent descriptions, and it would be meaningless to ask which of the two descriptions is true. Reichenbach’s suggestion is offered within an analysis of the time reversibility of classical mechanics. In virtue of the lack of irreversible classical mechanical processes, Reichenbach notes that classical mechanics may describe any allowable process relative to either temporal direction, and hence it is a matter of convention to hold that classical mechanics in any sense describes or governs processes in the future direction only. As such, forwards-in-time and backwards-in-time descriptions of processes are strictly equivalent. Reichenbach relates this issue to the conventionality of geometry, where different geometries may be used to equivalently model a single state of affairs. On this account, time reversal is a passive transformation within an equivalence class of models: for any process, p, classical mechanics offers TR-twins describing p relative to each time direction, and both models should be understood as picking out one and the same possible process.13 A similar view of time reversal—as offering different but equivalent descriptions—is offered by the cosmologist Gold ([1966], p. 327): […] the description of our universe in the opposite sense of time […] sounds very strange but it has no conflict with any laws of physics. [The] strange description is not describing another universe, or how it might be but isn’t, but it is describing the very same thing. Gold, unlike Reichenbach, is here discussing a backwards-in-time macroscopic description of the world, containing putatively irreversible processes (with assumed underlying reversible mechanics). In this case, the forwards and backwards descriptions differ in that the latter describes apparently improbable behaviour (for example, the anti-thermodynamic reforming of broken wine glasses, unmixing of coffee and milk, and so on) due to the presence of highly correlated variables. (I argue in Section 3 that such cases help to motivate the position of Reichenbach and Gold.) The key idea present in Reichenbach’s and Gold’s suggestions is that TR-twins offer distinct but equivalent descriptions of a single possible state of affairs. For convenience, I refer to this view as a ‘C-theory of time’.14 This terminology is motivated by McTaggart’s ([1908]) distinction between the B-series and the C-series. Whereas the B-series orders events in terms of the time-directed relation ‘earlier than’, the C-series is concerned only with the undirected ‘temporal betweenness’ ordering of events: […] the C series, while it determines the order, does not determine the direction. If the C series runs M, N, O, P, then the B-series […] can run either M, N, O, P (so that M is earliest and P latest) or else P, O, N, M (so that P is earliest and M latest). And there is nothing […] in the C series […] to determine which it will be. (McTaggart [1908], p. 462; my emphasis) The distinction between order and direction is key to the distinction between the B- and C-series.15 McTaggart’s usage of these terms is similar to Reichenbach’s, in delineating his position regarding time order in time-reversible physics.16,17 The C-series is contrasted by McTaggart with the B-series in terms of its lack of directionality. The C-series of a set of events does not determine their B-series: any time ordering of events in terms of temporal betweenness is compatible with two directed time orderings in terms of the ‘earlier than’ relation. This shows the difference in structure between the B- and C-series. The B- and C-theory give two different ontologies of temporal relations. On the C-theory, there are no time-directed states of affairs and, as such, no two worlds may differ solely with respect to the arrangement of ‘earlier than’ relations. We may understand time reversal as taking one B-series of events, arranged from earlier to later, to the inverse B-series by means of reversing each ‘earlier than’ relation. Such a transformation preserves the C-series of the events, since it leaves the temporal betweenness relations invariant. The adirectional ontology of time given by the C-theory entails that the two different time-directed pictures given by TR-twins differ only at the level of description—both TR-twins refer to the same time-direction-independent facts—and so the C-theory entails the Reichenbach–Gold passive interpretation of time reversal.18 Conversely, on the B-theory, there are time-directed states of affairs, and so it follows that, first, two worlds may differ solely with respect to the arrangement of ‘earlier than’ relations, and second, TR-twins describe distinct possible worlds. We can take these as necessary conditions for the B-theory that suffice to distinguish it from the C-theory.19 2.2 Time reversal on the C-theory In introducing the B- and C-theory, my aims are two-fold: first, to show that both theories offer compatibilist accounts of causation and time-reversal invariance; second, to argue that the C-theory offers the superior account of both the function of time reversal and of the epistemology of causal direction. Since I am both proposing and defending the C-theory, it is important to guard against possible objections and misunderstandings of its treatment of time reversal. 2.2.1 The C-theory doesn’t require time-reversal invariance Earman ([1974], p. 27) objects to the passive interpretation of time reversal entailed by the C-theory as it is presented here, holding that such an interpretation ‘is too powerful; for this conclusion [that time reversal amounts to a redescription of a single state of affairs] follows whether or not the laws of physics are time reversal invariant’.20 We can understand Earman’s point by noting that the following two questions concern distinct, though related, issues: Do TR-twins describe distinct possible worlds? Is some particular theory time-reversal invariant? The former question divides the B- and C-theory. The latter concerns an independent issue that might be taken to motivate either theory, though does not objectively favour either. While the latter is a broadly empirical question, the former is an a priori issue concerning the interpretation of time reversal that is conceptually independent of whether some particular theory is time-reversal invariant. Moreover, it is the former that directly concerns the relationship between causation and time reversal. Earman’s implication is that the independence of these two issues is a problem for the C-theory: the interpretation of time reversal it offers is independent of whether the relevant physics is time-reversal invariant. Importantly, the time-reversal invariance of fundamental physical theories is neither necessary nor sufficient for the C-theory. However, this is not a problem for the C-theory in itself; rather, it highlights that the C-theory primarily concerns an a priori issue that, in turn, determines one’s understanding of time-asymmetric phenomena.21 2.2.2 Time-reversal non-invariance on the C-theory Earman’s worry does, however, point to a more general problem: in reducing time reversal to a redescription of processes, the C-theory appears to trivialize time-reversal invariance since it is not immediately clear what sense can be made of time-reversal non-invariance on the C-theory. The problem is statable as a simple argument: P1. In order for a theory to be time-reversal non-invariant, a model of the theory must transform under time reversal to a non-model of theory. (Assumption) P2. On the C-theory of time, time reversal is just a redescription of a single possible world. (Definition) P3. A possible world cannot be deemed by some theory to be ‘physically possible’ relative to one description and ‘not physically possible’ relative to an equivalent description. (Assumption) C. Hence, on the C-theory, no theory can be time-reversal non-invariant. It might be thought that this implies that the difference between reversible and irreversible theories is not statable in C-theoretic terms, which would be a major weakness for the C-theory. We evidently do have a clear grasp of the differences between reversibility and irreversibility, as well as other kinds of probabilistic time asymmetries, so the C-theory had better possess the resources to account for this. However, the argument does not in fact pose such a problem for the C-theory: irreversible processes are straightforwardly describable in C-terms. Although this argument is valid, the main problem it picks out is that time reversal is insufficient to determine whether a theory describes irreversible processes when applied to entire models of a theory. This, I shall argue, is a problem for both the B- and C-theory. Let’s first establish that irreversibility is statable in C-terms. Consider a time-asymmetric law describing the behaviour of some variable x: L: The value of x increases, and never decreases, with time. This describes an ideal irreversible process: the increase in value of x.22 There are two notable features of L relevant to time reversal and the B- and C-theory. First, L describes an irreversible process. Second, L is stated in time-directed (B-theoretic) terms: if x increases relative to one temporal direction, it decreases relative to the opposite temporal direction, and thus L is not stated in a time-direction-neutral way. For the C-theorist, L is equivalent to L*: L*: The value of x decreases, and never increases, with time. If we take a model, m, that satisfies L, then its TR-twin, m*, satisfies L*. The C-theorist takes TR-twins m and m* to represent a single possible world, and so to privilege neither L nor L*. However, this does not mean that the C-theory is unable to accommodate the irreversibility described by L and L*. There is a key sense of irreversibility that is independent of time direction and so statable in C-terms: the x-process described by L (and by L*) is monotonic. For some type of process to satisfy either L or L* it must be monotonic, such that relative to a choice of positive time, it either (i) only increases or (ii) only decreases. The x-process is monotonic regardless of whether we take it to be an x-increasing or x-decreasing process. Figure 1 depicts three models to illustrate this: Figure 1a and Figure 1b depict TR-twins m and m*, respectively, and Figure 1c depicts a model, n, in which the value of x changes non-monotonically. On the C-theory, although m and m* represent a single possible world, they are structurally distinct from n. Importantly, Figure 1a and Figure 1b depict a monotonic gradient regardless of the designation of a direction of time. Given this, we can offer a C-theoretic version of L: Lc: The value of x changes monotonically in time. Such a law requires all x-processes to be coordinated such that relative to a choice of positive time, they are either all x-increasing or all x-decreasing. This key sense of irreversibility is thus statable in C-terms. Figure 1. View largeDownload slide Models of monotonic variation, m and m*, and non-monotonic variation, n, of a variable, x. (a) m represents the monotonic increase of a variable, x. (b) m* is the TR-twin of m and represents the monotonic decrease of x. (c) n represents non-monotonic variation of x. Figure 1. View largeDownload slide Models of monotonic variation, m and m*, and non-monotonic variation, n, of a variable, x. (a) m represents the monotonic increase of a variable, x. (b) m* is the TR-twin of m and represents the monotonic decrease of x. (c) n represents non-monotonic variation of x. This point may be generalized to non-idealized cases of putatively irreversible processes, such as the statistical time asymmetry of thermodynamics, and the time asymmetry of dynamical collapse theories such as the Ghirardi–Rimini–Weber theory (GRW), and also of probabilistic time asymmetries in particle physics, such as the decays of K0 and B0 mesons.23 Regarding thermodynamics, there is a clear conventional element in taking entropy to increase; for all we know, it could be that time really ‘goes’ from our future to our past, and hence a law that entropy tends to decrease rather than increase, contrary to our beliefs. What is important in accounting for the phenomena is not whether entropy ‘really’ increases or decreases, but rather that once we’ve fixed our convention about the positive direction of time, entropy either ‘increases and does not decrease’ or ‘decreases and does not increase’.24 Both time-direction-dependent descriptions pick up on the time-direction-independent monotonicity of entropy—entropy doesn’t fluctuate in both temporal directions.25 The C-theory captures this sense of irreversibility. Unlike the C-theory, the B-theory additionally legitimizes the question of whether entropy might ‘really’ increase-and-not-decrease or decrease-and-not-increase, but this is a separate issue that is not clearly epistemically accessible nor, for this reason, practically indispensable to the study of time asymmetry.26 As such, not only is law-like irreversibility and time asymmetry statable in C-terms, but it is also better understood in these terms. With this in mind, let’s return to the above argument. What is problematic regarding time reversal and irreversibility is that if some model satisfies Lc, then so does its TR-twin, regardless of whether one takes the TR-twins to represent different possible worlds, and so this is a problem for both the B- and C-theory. P1 states a requirement for a theory to be time-reversal non-invariant and this appears to be satisfied by the B-theory but not by the C-theory, since only the B-theory treats time reversal as an active transformation and so TR-twins as describing different possible worlds. On the B-theory—but not the C-theory—we can understand m and m* as representing distinct possible worlds, and so, in principle, the B-theory allows for a theory to contain m as a model without also containing m*. However, failure of time-reversal invariance in this sense would be quite odd. First, the choice of which model, m or m*, is used to represent some process is a matter of convention,27 and so a theory’s inclusion of one and not the other in its space of models would also be a matter of convention. Second, if a theory includes only one of a pair of TR-twins, it does not follow that the theory contains any law-like irreversible or probabilistically time-asymmetric processes. For law-like time asymmetry, a theory would have to satisfy a stronger condition. For example, a theory would contain a law-like irreversibility in the case that for some variable, x, the theory includes models of monotonic x-processes—such as m and m*—and excludes all models of non-monotonic x-processes—such as n. It is because of this second point that the argument as stated above is misleading. It establishes that the B-theory allows for a theory to be time-reversal non-invariant in a way that the C-theory does not, but this is only in the case when time reversal is understood as an operation upon an entire model of a theory. However, time reversal applied to an entire model cannot transform monotonic models such as m and m* to non-monotonic models such as n, and so non-invariance under such an operation is insufficient as a test for law-like irreversibility. This follows from our pragmatic constraint that time reversal functions in such a way that TR-twins represent what a process ‘looks like’ relative to the opposite time directions. Assuming this, it would be unreasonable for time reversal to fail to preserve monotonic and non-monotonic behaviour. Thus P2 in particular requires clarification; it is important to stress the context of the C-theorist’s claim that time reversal amounts to a redescription of processes. If we take a model of an ‘expanding’ gas, such that its TR-twin is a ‘contracting’ gas, the C-theory entails that these are equivalent descriptions of a gas occupying greater volume at one temporal end than the other. However, were we to embed such a system in a wider environment containing other gases displaying matching time-asymmetric behaviour, things would be different. In this case, switching one model for its TR-twin and holding the orientation of the other gases fixed would result in a physical change to the total system (for example, so that there were now a gas ‘contracting’ relative to the time direction in which the other gases were ‘expanding’). It would constitute a change to the C-series, and not only to the B-series, of the total system, since it would amount to a difference in the temporal-betweenness ordering of events, rather than only the earlier-than ordering. This sense of relative time reversal corresponds to an active change even on the C-theory. This gives us two different kinds of time reversal. First, a relative time reversal is an active transformation on both the B- and C-theory, since it changes the temporal betweenness relations, and hence constitutes a change regardless of the stipulated direction of time.28 Second, an absolute time reversal—applied to an entire model or to a total system (for example, the entire world)—is a passive transformation on the C-theory and an active transformation on the B-theory; only on the B-theory does a world identical to ours, save for the direction of time, constitute a different possible state of affairs. The argument as stated above establishes that only the B-theory allows for a theory to be non-invariant with respect to absolute time reversal. However, non-invariance under absolute time reversal is neither necessary nor sufficient for a theory to contain a law-like time asymmetry or irreversibility; thus the B- and C-theory account equally well for the existence of time asymmetries and irreversible processes. 2.3 Answers To recap, both the B- and C-theory can support physical laws that describe irreversible or probabilistically time-asymmetric processes. The central point at issue between the two theories is whether the application of a set of time-reversal transformations to a model of some theory takes us to a model that describes a different possible world. Time reversing an entire model amounts to a redescription of a single possible world according to the C-theory, but amounts to a description of a second, distinct possible world according to the B-theory. On the C-theory, time reversal is a purely passive, coordinative transformation, meaning TR-twins differ only in terms of notation: they represent a single possible world, and hence notation that varies under time reversal (such as the direction of velocities) should not be taken to represent a property of the target system. On the B-theory, time reversal involves altering fundamental temporal relations and hence takes us from one logically possible world to a distinct logically possible world.29 I have argued that the extra structure postulated by the B-theory is not required to account for temporally asymmetric phenomena. With regard to the relationship between causation and time reversal, we can now clarify the problem for causation posed by CTR. According to CTR, TR-twins represent different sets of causal relations. On the C-theory, since TR-twins represent a single possible world, the only way for TR-twins to agree on causes and effects is for there to be no causes or effects, given the assumption of CTR, is for there to be no causes or effects. Hence, this motivates causal eliminativism. The B-theory, conversely, contains the logical space for TR-twins to disagree over causal direction facts insofar as they represent different possible processes in different possible worlds, and thus the cause–effect asymmetry can be preserved in each possible world. However, in the next section I show that the combination of the B-theory and CTR leads to major problems in paradigm cases of causal processes, and as such I propose that ¬CTR should be preferred. 3 Does Time Reversal Reverse Causal Relations? We may now ask whether time reversal inverts causal relations. This section examines the relative appeal of CTR and ¬CTR in the context of (i) a time-symmetric process and (ii) a time-asymmetric process. 3.1 Causation, billiards, and snooker 3.1.1 Causation and time-symmetric processes Figure 2 depicts the time-symmetric process of a collision of two idealized billiard balls of equal mass on a frictionless plane. In Figure 2a, ball L has non-zero momentum and ball R is at rest. In Figure 2b, there is a perfectly elastic collision, upon which the total momentum of one ball is transferred to the other. Figure 2c depicts ball L at rest and ball R with non-zero momentum. Viewed from left to right—Figure 2a to Figure 2c—(call this the ‘left-to-right’ description), L's movement appears to cause R’s movement, and viewed from right to left—Figure 2c to Figure 2a—(call this the ‘right-to-left’ description), R's movement appears to cause L’s movement. Assuming CTR, if the left-to-right description represents a causal process in which L’s momentum causes R to move, then the right-to-left description represents the distinct causal process in which R’s momentum causes L to move. To fill in these distinct accounts, we can imagine a right-pointing arrow from L to R in Figure 2a on the left-to-right description, and a left-pointing arrow from R to L in Figure 2c in the right-to-left description. Alternatively, if we assume ¬CTR, both the left-to-right and right-to-left descriptions represent the same causal process. Figure 2. View largeDownload slide The time-symmetric process of a collision of two idealized billiard balls of equal mass on a frictionless plane. (a) L has non-zero momentum and R is at rest. (b) A perfectly elastic collision where momentum is completely transfered from one ball to the other. (c) L is at rest and R has non-zero momentum. Figure 2. View largeDownload slide The time-symmetric process of a collision of two idealized billiard balls of equal mass on a frictionless plane. (a) L has non-zero momentum and R is at rest. (b) A perfectly elastic collision where momentum is completely transfered from one ball to the other. (c) L is at rest and R has non-zero momentum. As we’ve seen, CTR and ¬CTR relate differently to the B- and C-theory. On the C-theory, since time reversal amounts to a redescription of a single possible process, CTR is untenable since it requires Figure 2a–Figure 2c and Figure 2c–Figure 2a to represent distinct possible processes. Hence, the C-theory requires ¬CTR and CTR requires the B-theory. The combination of the C-theory and ¬CTR applied to the billiards example suggests that if there is a causal process described here, the direction of causation is at best ambiguous. (I examine this issue in Section 3.2.) On the B-theory, since the left-to-right and right-to-left descriptions represent different possible processes, it is natural to read them as distinct causal processes. The combination of the B-theory and CTR fits with microphysical accounts of causation, such as dispositional accounts, in which causation is understood as some kind of unidirectional influence from one object to another. On such an account, the left-to-right and right-to-left descriptions refer to fundamentally different processes. In the left-to-right process, the initial non-zero left-to-right momentum of L is a cause of the collision. In the right-to-left process, the ‘initial’ non-zero right-to-left momentum of R is a cause of the collision.30 3.1.2 Causation and time-asymmetric processes Figure 3 depicts the time-asymmetric collision of two realistic snooker balls of equal mass on a frictional snooker table. In this case, an element of agential control is introduced: there is a snooker cue that interacts with the white cue ball. Furthermore, the presence of a non-conservative force—friction—brings in an important explanatory asymmetry between the left-to-right (Figure 3a to Figure 3c) and right-to-left (Figure 3c to Figure 3a) descriptions, and it is more convenient to describe the process in time-directed terms, unlike in the time-symmetric case. Figure 3. View largeDownload slide The time-asymmetric collision of two realistic snooker balls of equal mass on a frictional snooker table. (a) The snooker cue interacts with cue ball and the object ball is at rest. (b) The cue ball and the object ball collide, with a partial transfer of momentum. (c) The cue ball and object ball are at rest, and the surface temperature of the table is slightly raised. Figure 3. View largeDownload slide The time-asymmetric collision of two realistic snooker balls of equal mass on a frictional snooker table. (a) The snooker cue interacts with cue ball and the object ball is at rest. (b) The cue ball and the object ball collide, with a partial transfer of momentum. (c) The cue ball and object ball are at rest, and the surface temperature of the table is slightly raised. In the conventional left-to-right description (Figure 3a to Figure 3c), the cue strikes the cue ball, setting it in motion; next, the cue ball collides with the object ball, transferring most of its momentum to the object ball. The object ball then loses momentum due to the frictional force of the baize on the snooker table until it is at rest, as depicted in Figure 3c. This left-to-right description contains a number of causal terms, implying the following: the cue movement causes the cue ball’s movement; the cue ball’s movement causes the object ball’s movement; the baize causes the object ball to lose momentum. In the unconventional right-to-left description in Figure 3c–Figure 3a, an anomalous series of causal processes is implied. First, heat in the baize together with incoming air molecules conspire to set the object ball in motion. Next, the object ball’s motion in synchrony with inverse, concentrating soundwaves jointly imparts a gain in momentum in the collision of the object ball into the cue ball. Finally, the cue ball’s momentum is absorbed in a collision with the cue. As a candidate causal process, the right-to-left description shown in Figure 3c–Figure 3a is highly unsatisfactory. Two issues in particular stand out: (i) there are several points that imply a violation of the causal Markov condition (CMC), and (ii) the snooker player apparently loses her agential control over the balls’ motion. These imply both a causal and explanatory asymmetry between the two available time-directed descriptions (Figure 3a–Figure 3c and Figure 3c–Figure 3a), which, as I shall next argue, motivates ¬CTR. 3.2 The epistemology of causal direction If the unconventional right-to-left description of the scenario is understood as a causal process, this implies the existence of causally independent variables that nonetheless exhibit coordinated behaviour and hence are not statistically independent, in violation of the CMC.31 In simpler terms, there are a number of coincidences that can’t be explained away with reference to some common interactions in the causal past. As such, there is good reason to think that this does not represent a genuine causal process: it does not meet a standard criterion widely taken to be characteristic of causal relations, and central to the explanatory asymmetry of causes and effects.32 Calling such a process ‘causal’ is to insist on using the term quite outside its standard linguistic context and is thus heuristically unhelpful. After all, in order for the concept of causation to be useful in philosophical discourse, there ought to be reasonable restrictions on its domain of application so to exclude processes that violate standard causal criteria such as the CMC and its variants.33 For this reason, it is useful to defer to the patterns of conditional dependencies and independencies of variables to ascertain causal direction, as is characteristic of causal modelling.34 Furthermore, the introduction of agential control brings in a pragmatic constraint on causal inference: it is natural to stipulate that the snooker player has causal control over the cue and of the cue ball, and not vice versa. One can entertain a causal process such as that depicted from Figure 3c–Figure 3a, whereby the cue’s movements are (at least in part) caused by the motion of the cue ball, but it detaches various causal intuitions we have about snooker players from the causal relations described in the account. Reichenbach ([1956], p. 47) considers a similar problem regarding, in his case, tennis players and time-reversed ‘causal’ processes: It would be a strange experience indeed to see [tennis] players run backward. Such a motion, although compatible with the laws of mechanics, is unusual because we are safer if our steps are controlled by our eyes. This element of control is important in that it can be appealed to in order to privilege one of the two possible causal stories given relative to the opposite directions of time. Regardless of any underlying time symmetry, and regardless of any freedom to describe some process relative to either time direction, it is desirable to hold that we are not mistaken in such control judgements. This is because the appeal to control plays an explanatory role: it is reasonable to take the snooker player’s actions to explain the subsequent motion of the snooker balls and not vice versa. The notion of control and manipulation is central to agency and interventionist theories of causation, such as those of Menzies and Price ([1993]), Pearl ([2000]), and Woodward ([2003]). These provide a deflationist epistemology of the direction of causation, whereby the direction of causation is determined by the kind of patterns of correlations to which causal discovery algorithms are sensitive. In the case of Figure 3, we can appeal to the CMC, or more prosaically appeal to beliefs about the snooker player’s agential control, to ground a direction of causation.35 A deflationist account of causal direction holds that there is a direction of causation only in the presence of the right kind of probabilistic asymmetries (for example, irreversible processes, time-asymmetric screening-off conditions, and so on). Although the deflationist approach is applicable to our agential snooker case, it leaves open the status of causation in our idealized billiards case, in which there are insufficient asymmetries to ground a direction of causation. One option is that there just is no direction of causation intrinsic to such time-symmetric systems, but if one can refer to a wider system containing (for example) irreversible processes, then this can be used to define a direction of causation in the time-symmetric system. Such problem cases need not worry us in practice, since in general we do have sufficient asymmetric processes (for example, ourselves) to which to refer.36 In the idealized case of a world consisting solely of our idealized billiards example, the deflationist may hold that there is no fact about causal direction.37 Such an attitude towards idealized time-symmetric systems does not entail eliminativism nor scepticism about the direction of causation in worlds containing sufficient time asymmetries to determine a direction of causation. As such, the compatibility of such worlds with physical theories that are empirically adequate with respect to our world does not motivate eliminativism about causal direction with respect to our world. Whereas ¬CTR aligns with a deflationist account of causal direction, CTR aligns with a hyperrealist account of causal direction, whereby there is a causal direction that both outruns and is independent of the physical facts.38 This is because in order for time reversal to invert the direction of causation, the direction of causation must be independent of time-independent causal algorithms that ground the deflationist account of causal direction.39 To point to an example of hyperrealism, Maudlin ([2007], p. 172) takes the direction of causation to be determined by the ‘passage of time’, which he regards as ‘an ontological primitive [that] accounts for the basic distinction between what is to the future of an event and what is to its past’. As such, the direction of causation is independent of any particular probabilistically time-asymmetric processes in the world: ‘[causal] production [is] built on the foundational temporal asymmetry that would obtain even if the world were always in thermal equilibrium (even then, later states would arise out of earlier ones)’ (Maudlin [2007], p. 177). A hyperrealist account might seem preferable in the idealized billiards case, since the deflationist approach is silent about causal direction. However, the hyperrealist is still faced with the epistemic problem faced by the deflationist: there is no clear causal direction to be derived from the physical facts.40 Rather, the hyperrealist approach here is to stipulate a preferred causal arrow to artificially break the symmetry. While this may be innocuous in the billiards case, it creates a significant problem in cases like the agential snooker example where we have objective physical grounds for determining a preferred arrow of causation. Since the hyperrealist approach is by its nature insensitive to the kinds of factors that inform causal judgements, it gives up the explanatory benefits of the deflationist approach. Taking the direction of causation to be an ontological primitive licences worries about whether the snooker player’s action ‘really’ causes the movement of the snooker balls or vice versa, which is not a legitimate worry on the deflationist approach. In the kinds of cases where we naturally make unambiguous causal judgments, such as the snooker case, the deflationist epistemology of causation of ¬CTR is preferable to the hyperrealism of CTR. As such, causal relations should not be taken to reverse under time reversal. 3.3 Answers If we are to consider archetypal causal processes, namely, those that satisfy standard algorithms for causal discovery, then we ought to hold that causal relations do not invert under time reversal and so prefer ¬CTR to CTR. Though it may be intuitively plausible for causal relations to reverse under time reversal, such a view is reasonable only with respect to suitably time-symmetric cases—like the idealized billiards case of Figure 2—where there is no clearly preferred direction of causation. I have argued that in such cases it is better to be neutral with respect to causal direction than to adopt a hyperrealist account of causal direction. ¬CTR fits naturally with a C-theory of time. Combining the two, one may consider the two temporally opposed descriptions of the agential snooker example as equivalent descriptions of a single possible causal process. The Figure 3c–Figure 3a description, though unconventional in its form, may be taken to represent the same causal relations that are naturally read from the Figure 3a–Figure 3c description. The asymmetry between the two descriptions is not due to any important link between causation and time, but rather to time-independent factors that inform causal judgements. The issues of agency and the CMC lead to the same judgements about causal direction regardless of what one takes to be the underlying direction of time. This entails that any underlying time-reversal invariance of the microphysical description is beside the point; one may hold that there is a clear causal direction—a natural criterion for distinguishing between causes and effects in the example—which is invariant under time reversal. 4 Is Time-Reversal Symmetry Compatible with Causation? We are now in a position to evaluate the central question of the article: is time-reversal symmetry compatible with causation? In the previous sections, we considered the following questions: Do TR-twins represent distinct possible worlds? Does time reversal invert causal relations? These present four options, as listed in Table 1. It follows from our considerations that Options 2–4 give us compatibilism about causation and time-reversal symmetry, and that of these, Option 3 (¬CTR and the C-theory) is the preferred option. Before reviewing the compatibilist options, we can first look at the incompatibilism of Option 1. Table 1. The options available for each theory. C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 Table 1. The options available for each theory. C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 C-theory B-theory CTR Option 1 Option 2 ¬CTR Option 3 Option 4 4.1 Incompatibilism 4.1.1 Option 1: Causal time reversal + C-theory = incompatibilism I have suggested that, assuming the C-theory, if there are directed causal relations between events, then these cannot be flipped under time reversal. I have taken this to show that the C-theory requires a non-causal understanding of time reversal. Interestingly, Gold ([1966], p. 327) appears to go in the opposite direction and take his passive (C-theoretic) interpretation of time reversal to entail a Russellian causal eliminativism, holding that ‘the idea of a cause and effect relationship now becomes meaningless’. Gold’s ([1966], pp. 327–8) contention is based on a causal interpretation of time reversal: You may see relationships within [a time-direction-neutral description] which are of the kind that in the conventional description one would be called the cause and the other the effect. In the description with the opposite sense of time you would just have to reverse these roles. Given that the C-theory lacks the structure to commit to two such worlds with distinct causal relations, applying CTR does indeed entail eliminativism: the only way for TR-twins to agree on causes and effects, assuming that these are flipped by time reversal, is simply for there to be no causes or effects. Conversely, if a C-theorist wants to commit to directed causal relations, then these must be fixed by properties of the C-theoretic model expressible in time-direction-neutral terms, and thus left invariant under time reversal. Seen in this way, the C-theorist is committed to no causal relations being flipped by time reversal and thus to ¬CTR, contra Gold. The central problem is that the following three claims form an inconsistent triad: There are directed causal relations between events. CTR: Time reversal reverses causal relations. C-Theory: TR-twins describe the same possible world. Though each statement is independently plausible, the three jointly entail a contradiction. However, as we have seen, we may reject any one of these claims and avoid inconsistency. As should be clear, I take the second claim to be the one to reject. What is most important though is that either the second or the third claim may be rejected so as to save the first. The mutual incompatibility does not mark out the first as being the problematic claim. 4.2 Compatibilism 4.2.1 Option 2: Causal time reversal + B-theory = compatibilism Option 2 avoids incompatibility by rejecting Claim 3 of the triad (the C-theory). The B-theory holds that TR-twins describe distinct possible worlds, and this provides the logical space for there to exist directed causal relations that are flipped by time reversal without engendering a contradiction: in each B-theoretic world, the asymmetry of cause and effect is preserved. In place of the direct incompatibility of Option 1, Option 2 gives us practical and epistemological problems concerning directed causal relations in the kinds of cases in which we routinely make unambiguous causal judgements, such as in the snooker example (Figure 3). In allowing the sequences of Figure 3a–Figure 3c and Figure 3c–Figure 3a to represent distinct causal processes, Option 2: (i) leads to a problem of underdetermination, since both ‘causal’ processes are consistent with the same sets of data; and, more importantly, (ii) fails to account for why Figure 3a–Figure 3c and Figure 3c–Figure 3a are asymmetric with respect to explanation, in that only the former satisfies standard algorithms for causal discovery. These problems are unique to this approach. It is only by committing to CTR that the causal realist can entertain the possibility of processes whose causal direction is the opposite of that given by causal discovery algorithms. 4.2.2 Option 3: Non-causal time reversal + C-theory = compatibilism I have argued that Option 3 is the preferred account: by holding that TR-twins represent the same possible world (C-theory), and that cause and effect are invariant under time reversal (¬CTR), one can hold the time-reversal invariance of a theory to pose no conceptual or epistemological problems for the direction of causation. We’ve seen that Option 3 has several key benefits. First, the key sense of law-like time-asymmetry that is satisfied by irreversible or probabilistically time-asymmetric processes is captured by the C-theory. Second, combining the C-theory with ¬CTR allows for a deflationist epistemology of causal direction that (i) preserves causal direction judgements as determined by standard causal discovery algorithms, and (ii) dissolves scepticism as to whether the direction of causation matches our standard causal direction judgements. 4.2.3 Option 4: Non-causal time reversal + B-theory = compatibilism The final option, which I have not discussed up to this point, is to combine the non-causal account of time reversal with the B-theory. In principle, there are a couple of ways to do this: (i) defend a primitivist account of the direction of causation and stipulate that this should not be inverted by time reversal; or (ii) defend the same epistemology of causal direction as that of the C-theorist, but in addition, hold that TR-twins describe distinct worlds with different time-direction facts. (ii) preserves the epistemic advantages of my preferred option—Option 3—but additionally allows for two worlds to differ solely in terms of ‘earlier than’ relations. In this sense, the B-theorist can avoid the epistemological problems faced in Option 2. However, this then requires that the direction of time is wholly independent of the direction of causation. This kind of realism about the direction of time may have independent motivations and benefits that are outside the scope of this article. However, in terms of the cases we’ve considered, I take the C-theory to be the natural metaphysics of time for a non-causal interpretation of time reversal. 4.3 Answers Time-reversal symmetry is compatible with the existence of directed causal relations. However, realism about causal direction comes with restrictions, as shown in our inconsistent triad: either CTR or the C-theory must be rejected, as summarized in Table 2. Moreover, I have argued that the most reasonable resolution of the triad is to reject CTR: time reversal should not be understood as inverting causal relations. Table 2. Is time-reversal symmetry compatible with causation? C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ Table 2. Is time-reversal symmetry compatible with causation? C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ C-theory B-theory CTR ✗ ✓ ¬CTR ✓ ✓ Crucially, the compatibility of time-reversal symmetry and causation depends upon the interpretation of time reversal itself, and is independent of whether any particular physical theory is invariant under time reversal, contrary to standard presentations of the directionality argument. In our incompatibilist option—Option 1—the incompatibility is due to the combination of the CTR and the C-theory. For the compatibilist options, compatibility is due either to holding that time reversal does not invert causal relations (¬CTR) or to holding that TR-twins represent distinct possible worlds (B-theory). Each option is consistent with both time-reversal invariant and time-reversal non-invariant theories. 5 Outlook Causation and time-reversal invariance are not straightforwardly incompatible. Rather, the relationship between the two depends upon one’s interpretation of time reversal. I’ve shown that there are several compatibilist options available to the causal realist. Moreover, I have argued in favour of both the C-theory and ¬CTR: time reversal should be understood as a passive transformation that redescribes a single possible world, and so time reversal does not invert causal relations. This entails a suggestion about how to think of properties of instantaneous states that are acted upon by time-reversal operations: Such properties (such as velocity and momentum) are either (i) not causal, or (ii) not genuine properties of instantaneous states. That is to say, we cannot take a naïve view of velocities or momenta as telling us something about the direction of causal propagation or information flow. We can either take velocities to be non-causal in nature, so that velocities do not amount to something like causal dispositions—they just point one way or the other without contributing to the causal structure of a system—or we can take the direction of the velocity of a particle to be fixed by its position in a wider causal environment in which causal relations can be determined relative to causal discovery algorithms. This suggests a certain contextuality of such quantities—x has some velocity only relative to causal model Y.41 If we take CTR and the C-theory to both be appealing, which is reasonable, then we might think that causation is eliminated. However, this tacitly presupposes that causal facts are to be found in the microdynamics in the first place. I think this is to start off on the wrong foot. We should think that insofar as we do make causal direction judgements and wish to ascribe them to physical systems, these judgements derive from higher-level statistical observations and agential presuppositions that are themselves neutral regarding any microdynamical arrows of time or causation. Thus the time-reversal symmetry of the underlying dynamics need not require us to doubt whether there really are directed causal relations. This is welcome, since it is quite reasonable to be ambivalent about whether fundamental physics is time-reversal invariant. After all, the world of our experience accords to time-reversal non-invariant laws (for example, thermodynamics), which are underpinned by the time-reversal invariant laws of classical physics, which themselves are an approximation of quantum mechanics, which is, on many popular formulations, time-reversal non-invariant. It is desirable to avoid such worries when considering the status of causation. Footnotes 1 I assume the causal relation to hold between pairs of events for reasons of simplicity. What I say can be extended to more complicated cases, such as where an effect has multiple causes, and vice versa, and also where the relation holds between type events or possible values of variables. 2 Though these two features fall far short of a full ‘folk theory’ of causation, they suffice for the aim of this article, which is to assess whether such an account of causation is compatible with time-reversal symmetry. 3 See (Field [2003]; Ney [2009]; Frisch [2012]; Farr and Reutlinger [2013]) for versions of the directionality argument. 4 Farr and Reutlinger ([2013]) argue that Russell’s discussion of time symmetry appears to refer not to time-reversal invariance per se, but rather to the bidirectionality of the law of gravitation, in that it nomically entails the past and future trajectories of a given state—Russell ([1912], p. 15) holds that ‘the law [of gravitation] makes no difference between past and future: the future “determines” the past in exactly the same sense in which the past “determines” the future’. 5 By ‘predictive algorithm’ I have in mind the Born rule in quantum mechanics. In the case that such an algorithm is bidirectional, ‘predictive algorithm’ is a misnomer—such an algorithm would also be retrodictive. 6 Of these, only Norton explicitly endorses the claim that time-reversal invariance is incompatible with the asymmetry and time asymmetry of causation. Frisch rejects such an argument. Field and Ney both make implicit reference to both time-reversal invariance and the predictive/retrodictive symmetry of classical theories in discussing Russell’s claim. 7 In classical Hamiltonian mechanics, a state is given by the three-dimensional position and momentum values of the particles. Here, the momenta are vectorial properties—they are the product of velocity and mass. As such, everything I say about the direction of velocities can be translated to talk about the direction of momenta should the reader wish. This distinction makes no difference to the points made about time reversal and causation. 8 For instance, the standard set of time-reversal transformations in electrodynamics inverts the magnetic field, in quantum mechanics inverts spin, and so on. See (Sachs [1987]) for a detailed account of time-reversal operators across physics. 9 The idea of an instantaneous state being time directed has itself been taken to be conceptually problematic. Albert ([2000], p. 18) rhetorically asks ‘what can it possibly mean for a single instantaneous physical situation to be happening “backward”?’. Callender ([2000], Footnote 4) objects that ‘it just does not make sense to time-reverse a truly instantaneous state of a system’. 10 For instance, Arntzenius and Greaves ([2009], p. 563) note that ‘any theory, including ones that are (intuitively!) not time reversal invariant, can be made to come out “time reversal invariant” if we place no constraints on what counts as the “time reversal operation” on instantaneous states’. 11 Insofar as time-reversal operations may be applied to in-principle unobservable processes (such as the quantum-mechanical evolution of a system between measurements), the idea of a time-reversed state ‘appearing’ a certain way or being ‘viewed’ backwards in time is a heuristic metaphor. 12 For instance, from a recent article in Nature Physics: ‘Under time reversal […], states should become effects and vice versa’ (Oreshkov and Cerf [2015], p. 3). 13 In cases in which a model is its own time reverse—for example, a stationary particle—the same model describes both ‘forwards’ and ‘backwards’ versions of the relevant process. 14 The C-theory of time is presented and defended in (Farr [unpublished]). The claim that TR-twins are equivalent descriptions of a single state of affairs is entailed by the C-theory but not exhaustive of it. For the aims of the present article, this claim is the relevant feature of the C-theory. 15 Although McTaggart ([1908], [1927]) consistently refers to the C-series as ‘non-temporal’, this is due to precisely the same reasoning for which he takes the B-series to be non-temporal, that is, that neither series contains ‘real’ (A-series) change—in neither series is there a division between past, present, and future that changes. Farr ([unpublished]) argues that a C-theory of time is defensible once we relax the assumption that time requires A-series change. 16 Max Black ([1959]) similarly distinguishes the ‘order’ and ‘arrangement’ of a series of events, claiming that only the former is observable and hence fundamental. 17 See (Reichenbach [1956], Chapters 2–6). 18 In particular, any quantities that differ between TR-twins (such as instantaneous velocity, spin, and so on, as discussed above) can be considered descriptive artefacts that equally correspond to a single time-direction-independent (C-theoretic) state of affairs. 19 This is a non-standard way of presenting the commitments of a B-theory of time. This is due to the fact that the B-series is standardly presented in negative terms—in that it does not commit to the A-series’ properties of ‘pastness’, ‘presentness’, and ‘futurity’, nor an objective passage of time. However, the B-series is characterized by the inclusion of ‘earlier than’ relations that are not present in the C-series. Farr ([unpublished]) argues that the standard focus on the negative and not positive aspects of the B-series is due to the historical prominence of the debate over temporal passage, which separates the A-series from the B- and C-series. The separate issue of the directionality of time, which separates the B- and C-series, has occupied far less literature. 20 Earman’s criticism here is specifically aimed at the passive interpretation of time reversal defended by Black ([1962]), but also applies to his other targets, Reichenbach and Gold. Black claims that it would follow from time-reversal invariance of fundamental physics that ‘earlier than’ is a three-place relation (such that x is earlier than y only relative to some third term z—for example, an observer, some process, and so on). Earman rightly notes that Black’s conclusion is actually a consequence of the passive interpretation of time reversal, and follows regardless of the time-reversal invariance. 21 Indeed, Earman ([1974], p. 24) points to this distinction: […] the Reichenbach–Gold position [that is, the C-theory] cannot be based solely on time reversal invariance, but must rely on specialized assumptions about the nature of time reversal invariance. These assumptions have never been explicitly stated, much less justified. I should add that these specialized assumptions concern time reversal and not invariance under time reversal per se. 22 Though I use irreversibility as an illustrative example, the following line of reasoning equally well applies to probabilistic time asymmetries that are weaker than strict irreversibility. 23 The experimental violation of the combination of charge and parity symmetry (CP symmetry) in particle is well documented. For a discussion of CP violation in K0 meson decay, see (Sachs [1987], Chapters 8–9); for B0 mesons, see (Abe et al. [2001]). 24 This point is made at length by Price ([1996b]), particularly Chapters 2 and 7. 25 With respect to GRW, the key content of the irreversibility of collapses is that the set of collapses is co-oriented with respect to time such that in each GRW model there is, relative to a choice of time direction, either: (i) only collapses; or (ii) only ‘anti-collapses’. 26 Even supposing there is a privileged direction of time along which processes ‘really’ occur, we evidently do not need knowledge of this to collectively prefer to say that entropy ‘increases’ rather than ‘decreases’. 27 In other words, in principle, we might have preferred to describe processes in our world from future-to-past rather than from past-to-future without getting anything ‘wrong’. 28 Note, however, that on the B-theory, relative time reversal (change of the C-series) can, in principle, be carried out in two different ways: first, by holding the laboratory’s time orientation fixed and time reversing the experimental system; second, by holding the experimental system’s time orientation fixed and time reversing the laboratory. 29 Whether or not these logically possible worlds are deemed physically possible depends upon whether the relevant physical theory is time-reversal invariant. 30 It is also tenable for the B-theorist to adopt ¬CTR, as is discussed in Section 4.2. 31 The statistical dependence here is merely implicit. Given that the example depicts only a single (though abstract) run of the process, there is merely an apparent coincidence in that the initial conditions are highly improbable—they appear fine-tuned to entail coordinated behaviour. On multiple runs of this exact scenario, the statistics produced would provide a straightforward violation of the CMC, which holds that causally independent variables (relative to their causal pasts) are statistically independent (cf. Hausman and Woodward [1999]). 32 This point can be quite easily restated in terms of Lewis’s ([1979]) counterfactual theory of causation: according to Lewis's possible world semantics, the coincidences in the processes in Figure 3c–Figure 3a entail that in such a case, the past ‘overdetermines’ the future, and thus there is counterfactual dependence of earlier events upon later events but not vice versa. 33 For example, common-cause principles and screening-off conditions (cf. Arntzenius [2010]). 34 cf. (Pearl [2000]; Spirtes [2001]; Woodward [2003]). 35 Stipulations about agency and control play a key constitutive role in causal modelling. In general, multiple causal models will be compatible with the statistical data concerning relationships between variables of a system, and designating certain variables as ‘exogenous’ (that is, ‘free’ variables that are not effects of other variables in the system) narrows down the set of viable causal models for the system. 36 See (Farr [2016]) for a discussion of this issue in the context of the debate between John Norton ([2009]) and Mathias Frisch ([2009], [2014]) about causal reasoning in time symmetric systems. 37 The C-theorist could instead commit to a symmetric notion of ‘causal betweenness’, which provides an ordering that is invariant across TR-twins. This route appears to be taken by (Reichenbach [1956], p. 191). 38 See (Price and Weslake [2010]) for a critique of hyperrealist accounts of the direction of causation. 39 The patterns of statistical (in)dependence with which causal discovery algorithms are concerned are themselves neutral with respect to the direction of time. For instance, retrocausality—whereby a pair of cause-and-effect events are such that the cause event is later relative to clock time (for example, of a laboratory) than the effect event—is conceptually possible relative to such algorithms. 40 As Price and Weslake ([2010]) argue, hyperrealism about causation requires a denial of physicalism. 41 Price ([1996a]) suggests this kind of case as a problem for reducing causal direction to the fork asymmetry of causal models in microphysics: the direction of a causal process will ultimately be determined by which variable one includes in one’s model. This is suggestive of an arbitrariness of causal direction as determined by causal models. In particular, an open issue for this approach is what to make of ‘post-selected’ causal models, whereby the data are chosen in such a way to reveal patterns of correlations that suggest the opposite causal direction to that which we take to hold in the world. Whether we can reject the significance of such apparently causal relations on grounds of being artificial or unnatural is interesting; but this is an issue for another article. Acknowledgements This work was funded by the Templeton World Charity Foundation (TWCF 0064/AB38). Thanks to Jeremy Butterfield, Fabio Costa, Natalja Deng, Sam Fletcher, Huw Price, Sally Shrapnel, Magdalena Zych, and two anonymous referees for helpful comments. I dedicate the paper to the memory of my snooker buddy and dad, Peter Robert Farr. References Abe K. , Abe R. , Adachi I. , Ahn B. 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