Causation in a Timeless World

Abstract This paper offers a new way to evaluate counterfactual conditionals on the supposition that actually, there is no time. We then parlay this method of evaluation into a way of evaluating causal claims. Our primary aim is to preserve, at a minimum, the assertibility of certain counterfactual and causal claims once time has been excised from reality. This is an important first step in a more general reconstruction project that has two important components. First, recovering our ordinary language claims involving notions such as persistence, change and agency and, second, recovering enough observational evidence so that any timeless metaphysics is not empirically self-refuting. However, the project of investigating causation in a timeless setting has a greater relevance than its application to timeless physical theory alone. For, as we show, it can be used to model the assertibility conditions of causal claims more generally.

1. Introduction

Given recent developments in physics and metaphysics we take the unreality of time in the actual world seriously. Barbour (1999, 1994b, 1994a, Barbour and Isham 1999) and Rovelli (2004, 2007) have independently argued that a completed theory of quantum gravity will lack a one-dimensional substructure of ordered temporal instances, a substructure that provides a metric for the meaningful measure of the distance between any two time instants – what Rovelli (2004, 1995) calls ‘linearity’ and ‘metricity’ respectively. Since linearity and (arguably) metricity are necessary for the B-series—the series that orders events by the unchanging relations of earlier than, later than, and simultaneous with—Rovelli and Barbour appear to be suggesting that a completed theory of quantum gravity will make no use of a B-series ordering and thus of time.

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The idea that time does not exist has also arisen in a recent exchange between Earman (2002) and Maudlin (2002) as a possible implication of the gauge symmetry of general relativity. Similarly, in two recent papers Tallant (2008, 2010) has argued that the best way to understand the two dominant models of temporal ontology currently on offer – presentism (according to which only the present exists) and eternalism (according to which the past, present and future all exist) – is as views that endorse timelessness. Timeless theories face three challenges. First, they must be able to accommodate consciousness and the everyday experiences of conscious observers. For if they cannot, then those theories will be straightforwardly falsified by the existence of mental phenomena. Call this the consciousness challenge. Second, timeless theories must recover observational evidence. For if they cannot then those theories will be falsified by the very observational data upon which they are supposedly founded. Call this the observational challenge. Third, some account must be given of causal talk. For such talk plays a central role in our lives, underpinning (at least) moral, prudential and legal reasoning, which are central to us as agents. If, under a timeless theory, causal talk turns out to be unreasonable then that would undermine these other notions as well. Such a theory would, arguably, then be falsified by the high credence we have in the reasonableness of moral, legal and prudential phenomena. Call this the ordinary causal language challenge Until now, proponents of timeless theories have focused primarily on the first challenge: the consciousness challenge. The central aim of this paper is to meet the ordinary causal language challenge and, in so doing, provide a foundation for addressing the observational challenge. To do so we offer a schema for evaluating counterfactual conditionals and thus causal claims in a timeless setting. One way of reading this schema is as offering truth conditions for counterfactuals and for causal claims. Understood this way our account recovers causation from a timeless world. Another, weaker, reading of the schema is as offering mere assertibility conditions for

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counterfactuals and for causal claims. Understood this way our account recovers only quasicounterfactual dependence and quasi-causation in timeless worlds. We argue that, on either interpretation, our schema for evaluating counterfactuals answers the ordinary causal language challenge. It is less clear whether, on the weaker, second interpretation, our account meets the observational challenge. We argue, however, that there is an independent reason to prefer the first interpretation, which does answer the observational challenge. The independent reason we have in mind is a methodological one: to wit, that of a set of competitor semantics all of which render assertible the same set of sentences, one should prefer the semantics that is consistent with the larger range of extant scientific theories. If one accepts this methodological claim then even if one is attracted to the idea that what we offer are merely assertibility conditions and not truth conditions, one should nevertheless be motivated by this independent reason to prefer the view that these are, in fact, truth conditions. Not everyone will find themselves moved by this independent reason. But even if one accepts the weaker interpretation and concludes that our account only meets the ordinary causal language challenge that is still an important step forward. It answers one important challenge, and it leaves open that the observational challenge can be met using other resources. Moreover, the schema we offer is important for another reason, for it can be parlayed into a more general model of the assertibility of causal claims, one that can be used to model causal assertion independently of its application to timeless physical theories. We focus on a particular timeless physical theory developed by Barbour and on a particular theory of causation and causal talk, namely Lewis’s counterfactual theory. We focus on a counterfactual theory of causation because we can show how to recover counterfactual dependence from Barbour’s theory. Even if one thinks that causation proper is something other than mere counterfactual dependence, the assertibility conditions provided here remain adequate for causal talk, they just won’t provide truth conditions for such talk (this is the weak

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interpretation of our schema noted above).1 We believe that there are good reasons to focus on Barbour’s view. We therefore begin in §2 by motivating Barbour’s theory and more carefully situating, and motivating, our own project, explaining how each of the three challenges noted above arises. In §3, we briefly outline Barbour’s view before arguing that, given a Barbourian metaphysics, all causal claims are false when assessed using the Lewisian framework. In §4 we develop our new method for evaluate counterfactual conditionals and in §5 turn that account into a way to evaluate causal claims.

2. Tempus Nihil Est

The timeless physical theory developed by Barbour (1999, 1994b, 1994a) and Barbour and Isham (1999) is billed as a timeless interpretation of canonical quantum gravity. Canonical quantum gravity is obtained from the application of canonical quantization techniques to the Hamiltonian construction of general relativity. The key to Barbour’s timeless approach is an interpretation of the Wheeler-De-Witt equation – a timeless law of canonical quantum gravity – as a probability distribution over a relative configuration space, one that ties quantum probabilities to ‘time capsules’ (more on these below). Barbour’s approach is one of a class of timeless approaches to canonical quantum gravity that Anderson (2012a, p. 769) calls tempus nihil est approaches. Other timeless approaches to canonical quantum gravity have been offered by Anderson (2006, 2009), Deutsch (1997), Gell-Mann and Hartle (1993), Halliwell (1999), Halliwell and Dodd (2003), Page (1999, 2002) and Rovelli (1991a, 1991b) (see Anderson (2012, pp. 769– 772) for a full list of current tempus nihil est accounts).

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We suspect that the method we use here for recovering counterfactuals can also be used to recover a semantics for

a more ‘meaty’ account of causation, such as the process theory defended by Dowe (1992, 1999) and Salmon (1994). We describe how this might be done in footnotes 5 and 8.

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Timeless approaches have been developed in response to the following ‘problem of time’ in quantum gravity. Our two most well-confirmed empirical theories – quantum mechanics and general relativity – are inconsistent with one another. Theories of quantum gravity seek to reconcile these two basic theories to produce a quantum account of gravity. One of the central ways in which quantum mechanics and general relativity are at odds concerns their differing notions of time; quantum mechanics makes use of an absolute time variable; relativistic mechanics does not. To handle this inconsistency a canonical quantization of general relativity to produce a viable theory of quantum gravity strips out the time variable entirely (Anderson 2012a, pp. 757–758). The problem then is what to say about this situation. Should time be recovered post quantization or not? Given the loss of the time variable in the quantization of canonical quantum gravity we are faced with a choice; we can either try to recover time from the theory or we can accept what the quantization appears to be telling us, and deny the existence of time. Tempus nihil est approaches take the second option. Both options remain open. However, tempus nihil est strategies allow us to avoid a range of problems associated with recovering time post quantization (the problems are rather technical and need not concern us here, see Anderson (2012a, pp. 766–768)). There is also a sense in which it is the most straightforward approach of the two. After all, the mathematical results yielded by the quantization of canonical quantum gravity carried out to solve the problem of time appear to be telling us that time is not there. Taking this result at face value and eliminating the time variable is, perhaps, the path of least resistance. To be sure, canonical quantum gravity is not the only game in town. One of its chief rivals, string theory, reconciles general relativity and quantum theory by positing up to 11 extra dimensions. Quite apart from the trade-off in simplicity between this theory and tempus nihil est versions of canonical quantum gravity (which remove dimensions, rather than adding them), it is not obvious that string theory is successful in avoiding the problem of time (see Anderson (2012b, p. 19) for discussion). When the theory is quantized we may still lose the time variable. It 5

may turn out, then, that we are forced to consider a timeless approach to quantum gravity when quantizing string theory as well. This is all to say that we have good reason to take tempus nihil est theories seriously; they constitute some of our best solutions to a serious problem in quantum gravity. But why focus on Barbour’s theory in particular? To answer this question, we must briefly consider three closely related arguments against timeless theories each of which correspond to one of the three challenges identified in §1. According to the first, observation requires conscious observers. But in a timeless world there could be no conscious observers since consciousness supervenes on processes in time. This is a particularly devastating objection since, on the assumption that we know that actually there are consciousness beings, we can show that timeless theories are straightforwardly false. The second, related, argument is that since observation is not instantaneous it presupposes the existence of conscious observers and events located at different times such that the former can causally interact with the latter. But if there is no time then there are no such conscious observers and events that can causally interact. A fortiori there cannot be observational evidence for timeless theories (this problem is articulated by Healey (2002, pp. 310–312)). The third argument is related to our causal talk. On the not unreasonable assumption that time is necessary for making sense of causation, the loss of time threatens to undercut the coherence of causal notions. This, in turn, jeopardises a range of other phenomena for which causation appears necessary, the most serious of these being the reasonableness of moral, prudential and legal reasoning, all of which appear to be built on the back of causation. A central challenge for all timeless theories is to show that such talk is in good order even though time has been excised from reality. Of course, we do not suggest that making sense of causal talk is sufficient for completing the more general project of recovering notions that appear to require causation; but we do think it renders that project tractable.

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The first argument is what, in §1, we called the consciousness challenge, the second is the observational challenge and the third is the ordinary causal language challenge. Of these only the first is currently recognised by proponents of tempus nihil est theories. One of their main goals is to recover the appearance as of time through an extensive reconstruction project whereby our experiences are recovered intact using the metaphysical resources afforded by the theory. The consciousness challenge is why Barbour’s theory is so important. Of the tempus nihil est theories offered to date, theories in the neighbourhood of Barbour’s have perhaps made the most significant progress toward developing a plausible approach to that challenge. This is via the development of what Anderson (2012a, 769–772) calls a ‘histories’ approach to consciousness, whereby timeless worlds are imbued with a rich representational structure that can provide a basis for experience (these are the aforementioned time capsules; again, more on this below). So insofar as there is reason to take tempus nihil est approaches to canonical quantum gravity seriously, which we believe there is, we have reason to take Barbour’s view seriously; it is a leading contender in this area. Moreover, even if Barbour’s tempus nihil est theory is unsuccessful it is a useful model for investigating such theories more generally. That’s because other tempus nihil est theories on the market are similar to Barbour’s theory with regard to their metaphysical core. Accordingly, lessons learned for the relationship between causation and time for Barbour’s view are likely to be lessons learned for such timeless theories more generally. We assume, then, that the consciousness problem can, in principle at least, be solved as proponents of Barbourian tempus nihil est theories contend, by appealing to something like the complex representational structure of a timeless world (see Ismael (2002) for discussion). It is not so clear, however, how the histories approach can solve either the observational challenge or the ordinary causal language challenge. That’s because, as we will show in the next section, causation, at least of the counterfactual kind, is difficult to square with Barbour’s theory.

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3.

Barbour Worlds Behaving Badly

The backdrop to Barbour’s theory is the notion of a relative configuration space. A configuration space,

, for a single particle in a 3-space (within which a vector has coordinates

x, y and z) is (roughly) the space of all possible coordinates that the particle might take. The configuration of n particles in a Euclidean 3-space is

; that is, a space with 3n dimensions. A

point in the space represents the configuration of n particles in a 3-space (i.e. a three-dimensionalor 3- geometry). The entire universe at a time can be assigned a particular configuration in the configuration space space

, one that specifies the coordinates of all existing particles. The entire

can then be thought of as all possible configurations of the particles in the universe in

a Euclidean 3-space. By taking

and factoring out the six frame variables specifying the centre of mass co-

ordinates and the orientation of the system, Barbour extracts a relative configuration space Points in

are configurations in a Riemannian 3-space.

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lacks an absolute frame description

of the universe in terms of location or orientation. Barbour calls this relative configuration space: Platonia. As Butterfield (2002, p. 303) writes: [A point in Platonia] is a specification of all the inter-particle distances (and so of all the angles) at some instant, without regard to (a) where the system as a whole is in absolute space, nor to (b) how it is oriented, nor to (c) its handedness.” Barbour makes three central claims about Platonia. First, the actual world is a single point in Platonia. Because points in Platonia are points in a Riemannian 3-space, any such point is purely spatial. In short, Barbour’s claim is that our world lacks a temporal dimension. It is, as it were, a single instant; though ‘instant’ is itself a too temporally loaded term since it suggests temporal connections to other times. Better to say then that the world is a relative configuration of all particles in the universe in three dimensions. Second, Platonia charts the space of possibility; it is the space of all possible ways for the universe to be. Third, some points in Platonia possess

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‘traces’. Consider a particular point, p, in Platonia. Point p contains what seem to be records of a past: fossils, tree rings, written documents, apparent memories and so on. The sum total of traces at a point is a time-capsule (twice-mentioned above). Time-capsules constitute the representational structure of a point in Platonia and, importantly, represent other points in the relative configuration space

. This representational structure is quite rich, in two

senses: (i) one time capsule can bear traces of another time capsule at another point without the reverse being true and (ii) one time capsule can have representations of another time capsule which, itself, represents another time capsule and so on. This ‘nested’ representational structure gives the appearance of a history. Crucially, however, this is mere appearance. Points in Platonia are not temporally connected to other points and so no point has a genuine past.2

3.1 Counterfactual Dependence

Given a standard Lewisian semantics for counterfactuals Barbour worlds are ones in which no claim of the form P □→ Q will come out true. Here is why. According to the standard Lewisian semantics for counterfactuals:

2

Barbour’s view can be thought of as the combination of presentism with modal realism (Butterfield (2002)). As

Butterfield notes, care is warranted in treating Barbour’s view as a version of presentist modal realism, for two reasons. First, under presentism it is typically thought that the past existed and the future will exist, and thus that which ‘instant’ is present changes as time passes (c.f., Tallant (2010) and Fiocco (2007)). The points in Barbour’s configuration space are static: there is no sense in which the instant that exists will change to a different instant. Each instant exhausts the ontology of its own little universe; all that exists, existed and will ever exist from the perspective of an instant is a single, three-dimensional configuration. Second, under Lewis’ full-blown modal realism, each world is a four-dimensional space-time. However, under Barbour’s view, each world is not a complete space-time as under Lewis’ view; each world is merely a point in a configuration of particles in Riemannian 3-space.

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CC1

‘P □→ Q’ is true iff there is some possible world in which P and Q that is closer to the actual world than any possible world in which P and ~Q.

And where, for Lewis, similarity between worlds is determined via the following four-fold recipe:

(1) It is of the first importance to avoid big, widespread, diverse violations of law. (2) It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails. (3) It is of the third importance to avoid even small, localized, simple violations of law. (4) It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly (Lewis 1979, p. 472).

The metric does a lot of the work in CC1. When applying the metric, Lewis assumes the relevant laws to be deterministic and henceforth so shall we.3 A violation of the laws is a miracle. Miracles are defined intra-worldly, as follows. For any two worlds w1 and w0:

A miracle at w1, relative to w0, is a violation at w1 of the laws of w0, which are at best the almost-laws of w1. The laws of w1 itself, if such there be, do not enter into it. (Lewis 1979, p. 469)

Suppose, then, that we aim to evaluate an instance of the following counterfactual schema:

CF

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If x had not occurred, then y would not have occurred.

As we discuss later on, this assumption can be given up.

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According to Lewis, we do so as follows. First, we match worlds based on laws and past history leading up to the time, t, at which x occurs. Those worlds that have the same history until x occurs and similar laws are the closest worlds. Of these worlds, we focus on those in which x does not occur at t and look to see what happens in the short period after t. If there is one world in which y does not occur that is closer to the actual world than any world in which x does not occur at t and y occurs anyway, then the counterfactual is true, and not otherwise. Return now to Platonia. There are two problems with applying Lewis’s account of counterfactual dependence in a Barbour world (i.e. a point in Platonia). 4 First, in order to evaluate a counterfactual we must be able to apply Lewis’s closeness metric to Platonia, ordering the points therein. For Lewis, the closest worlds are the worlds with similar laws and past history to the actual world. But those cannot be the closest worlds in Platonia. Points in Platonia do not have a past history. So they cannot be matched in this way. Moreover, by ‘laws’ Lewis has in mind dynamical laws; laws that can be used to extrapolate the dynamics of a system over time.

4

As pointed out to us by an anonymous referee, when applying Lewis’s account to Barbour’s picture and thus to

points in Platonia, we are actually moving away from Lewis’s core account. This is so for two reasons. First, because the space of possible worlds to which Lewis appeals is bigger than Platonia. Lewis’s space of worlds includes all possible worlds; Platonia only includes physically possible worlds. Second, Lewis’s space of worlds includes worlds with real past histories (i.e. worlds with a four-dimensional space-time manifold). Platonia, by contrast, includes only three-dimensional configurations – there are no worlds with real past histories. In the current context, however, these differences between Platonia and the space of worlds to which Lewis appeals do not matter, since the same result can be obtained using the unrestricted space of Lewisian worlds just as easily, so long as the actual world is a point in Platonia. Which is to say that even on the unrestricted picture, Lewis’s framework as applied to the actual world treated as a three-dimensional configuration will render counterfactuals trivially false.

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But there are no such laws at a point in Platonia. So, again, the worlds cannot be matched in this way.5 The only way to determine similarity between points in Platonia is (i) via similarity with regard to relative configurations and (ii) via any non-dynamical laws that govern such configurations, such as there are. Even if we can order points in Platonia in this way, however, there is a much deeper problem to contend with. For CF to be true there must be some point p in Platonia at which x does not occur and y does not occur within the short period after p, a point that is closer to actuality than any point p* in Platonia at which x does not occur and y occurs anyway in the short period after p*. The trouble is that while there are configurations in the relative configuration space that bear witness to x failing to occur, there are no points at all at which x fails to occur and y occurs / does not occur in the short period following the failure, just because there are no short periods at all for points. Points lack the temporal width to allow both for x’s non-occurrence and for any ‘downstream’ consequence. So there is no world in which x fails to occur and y does not occur that is closer to actuality than any world in which x does not occur and y occurs anyway, just because there are no relative configurations that bear witness both to x’s non-occurrence and to the non-occurrence of y. CF is trivially false for any value of x and y.

3.2 Causation

We come now, at last, to causation in a Barbour world. According to a standard Lewisian theory of causation, the semantics for a causal claim is as follows:

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Note that the problem is not that there are no dynamical laws, the problem is that there are no dynamical laws. I.e.,

the problem is not that there are no dynamical generalisations; the problem is that these are not laws. We discuss this issue in more detail in §4.

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CAUS1

‘x causes y’ is true iff there is a causal chain leading from x to y consisting of events a, b, c ... such that c depends counterfactually on b, b depends counterfactually on a and so on. (Lewis 1973, p. 563)

According to CAUS1, ‘x causes y’ is true only if there are relations of counterfactual dependence between events, a, b, c, etc. that form a causal chain between x and y. We have just seen that, according to a standard semantics for counterfactual conditionals, ‘b counterfactually depends on a’ will be false in a Barbour world. Thus ‘x causes y’ will be trivially false in such a world if we evaluate statements of that form according to CAUS1.6 Barbour worlds are therefore not just timeless, they appear to be lacking in causation as well.

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We have focused on counterfactual theories of causation, but a similar result can be obtained for process

theories. According to the process theory of causation developed by Dowe (1992, 1999) and Salmon (1994), ‘x causes y’ is analysed as follows:

CAUSPROC

‘x causes y’ is true iff (i) x’s worldline in spacetime intersects with y’s and (ii) at the point of intersection x transfers a conserved quantity to y.

Here conserved quantities are things like energy, velocity, and momentum. A causal process is thus the worldline of an object that possesses a conserved quantity. All causal claims analysed via CAUS1PROC are trivially false in a Barbour world. Barbour worlds are relative configurations of particles in a 3-space. According to the process theory, however, causation requires the existence of four-dimensional space-time. Since processes are temporal or, at the very least, require the existence of multiple intra-world instants that are appropriately connected to one another, no such processes can occur in a Barbour world and so no causal claim so understood can be true.

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4. Evaluating Counterfactuals

Because causal claims are trivially false in Barbour worlds the ordinary causal language challenge appears pressing; this is the challenge, it will be recalled, of showing that causal talk is in good order in a timeless setting. There are two ways to address this challenge. First, one might provide truth conditions for causal claims that vindicate causal talk in a world without time, and then argue that these are, in fact, the correct truth conditions for such talk. That would be a strong vindication of ordinary causal language. Second, one might provide assertibility conditions for causal claims that render causal talk assertible in a timeless setting. One might then argue that these are, in fact, the correct assertibility conditions for causal talk. This would be a weaker vindication of ordinary causal language, but it would, arguably, be enough to solve the ordinary causal language challenge; if causal claims are assertible then, even if such claims are false, that would show that such talk is coherent and acceptable, at least in some sense. This last point is similar to a thought defended by, for instance, mereological nihilists when they attempt to make sense of ordinary talk about macro-sized composite objects. Nihilists hold that there are no such objects, but that some ordinary claims about them, though false, are assertible, and other ordinary claims are not even assertible. The difference between the claims that are assertible and those that are not is an important difference; it tracks something real (arrangements of simples) and it tracks something that matters to us. Whether it is assertible that ‘there is a tiger in the room’ but strictly false because there is just a set of simples arranged tigerwise, as opposed to it being not assertible (and false) because there is not even a set of simples arranged tiger-wise, makes a big difference to one’s survival prospects. What one should care about, in this case, is not whether ‘there is a tiger in the room’ is true or false, but rather, whether it is assertible or not. Our way of understanding causal talk in terms of assertibility conditions is like this. There is something interesting that is being tracked by such talk, which makes it assertible to make claims 14

such as ‘global warming is caused by humans’ as opposed to claims of the form ‘cheese rolling causes global warming’. Both claims are false if ours is a timeless world; but on our account only the former is assertible and that matters. Indeed, what should matter to us, when evaluating causal claims, is not their truth of falsity but their assertibility (or lack thereof), since that is primarily what guides action. Our first step toward meeting the ordinary causal language challenge, then, is to recover truth/assertibility conditions for counterfactual conditionals. Our strategy is to take the standard Lewisian semantics and for each of the ‘moving parts’ in that semantics locate an analogue in Barbour’s picture capable of playing that role. As touched on above, there are, then, two ways to interpret the schema that we offer. It can be interpreted in a strong sense, as yielding truth and assertibility conditions for counterfactuals, or in a weak sense, as yielding assertibility conditions only. This will become important in the following section. For now and for the sake of clarity, we lay out both interpretations. As we saw, the central difficulty with evaluating counterfactuals in a Barbour world is that the worlds against which the counterfactuals are evaluated – namely other points in Platonia – are too impoverished; they do not have the temporal width needed to make sense of counterfactual dependence between (what we take to be) distinct events. Counterfactuals therefore cannot be evaluated against worlds on this picture. So we need something else to play this role. Our suggestion is to make use of a further piece of structure within Barbour’s picture: paths through Platonia. A path through Platonia is a curve through configuration space and can be modelled as a totally ordered subset of points in the relative configuration space

(see Figure 1).

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PLATONIA

P1 T P2 TP3 TP4 TP5 TP6 TP7 T P8

Figure 1

A totally ordered subset of points {P1, P2, P3, P4, P5, P6, P7, P8} in Platonia.

Paths through Platonia can be generated via a process of ‘best matching’. The key to best matching is similarity. Call the set of all points in Platonia, S. To perform best matching, we find a relation that partially orders S, a relation that compares points in some physical respect.7 A path is some subset of S that is totally ordered by such a relation. For instance, we might take a relation that partially orders S in terms of entropy. Note that in doing so, we are careful to take only one point from any equal-entropy equivalence class so as to avoid duplication. The entropybased paths are then those subsets of S that are totally ordered via their relative similarity in entropy (see Figure 2).8 Such a path might, for example, be a subset S1 of S where the partial 7

Where ‘’ is a partial order over a set S iff for any points x, y and z such that x  S, y  S and z  S: (1) x  x. (2) If x  y and y  x then x = y. (3) If x  y and y  z then x  z.

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Where ‘T’ is a total order over a set S iff for any x, y and z such that x  S, y  S and z  S: (1) If x T y and y T x then x = y. (2) If x T y and y T z then x T z. (3) x T y or y T x.

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order of all points in S in terms of entropy totally orders all of the points in S1 in terms of increasing entropy, starting from the lowest entropy point until the point with the highest entropy is reached. PLATONIA

S1

Figure 2

P1 T P2 T P3 T P4 T P5 T P6 T P7 T P8

A totally ordered subset of points in Platonia. Each of the Pn represents a point.

Paths through Platonia look very much like four-dimensional Lewisian possible worlds. If paths were induced by, say, temporal connections between the points, then it would be tempting to say that a path produced by a process of best matching just is such a world. But paths through Platonia are not induced by anything other than similarity. Accordingly, as Barbour suggests, though definable, these paths have no deep physical or metaphysical significance. In short, Platonia permits of pathing, but these paths are not reflective of any real structure in the space, i.e. any physical connectedness between points either temporal, spatial or, indeed, lawful. Importantly, because there are many ways in which points can be compared there are many potential paths through Platonia (see Figure 3).

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PLATONIA

Figure 3

Paths through a point in Platonia; each path is a subset that is totally ordered by some similarity relation that induces a partial order over points in Platonia. Each path has the central point as a member.

To make clear that paths through Platonia are not four-dimensional worlds but merely mimic such things, call paths through Platonia quasi-worlds. Quasi-worlds can be plugged into a Lewisian semantics for counterfactuals to yield the following assertibility conditions:

CC2

‘P □→ Q’ is assertible iff there is some quasi-world in which P and Q that is closer to the actual quasi-world than any quasi-world in which P and ~Q.

In what follows we consider two views. According to the first, these assertibility conditions are also truth conditions. According to the second, they are not. Thus according to the first view the following is a correct analysis:

CC2(a)

‘P □→ Q’ is true iff there is some quasi-world in which P and Q that is closer to the actual quasi-world than any quasi-world in which P and ~Q.

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According to the alternative, CC2 fails to provide truth conditions for counterfactuals. Rather, the correct truth conditions are the standard Lewisian ones identified in §3.1. In that case one holds the following:

CC2(b)

(i) ‘P □→ Q’ is true iff there is some possible world in which P and Q that is closer to the actual world than any possible world in which P and ~Q and (ii) ‘P □→ Q’ is assertible iff there is some quasi-world in which P and Q that is closer to the actual quasi-world than any quasi-world in which P and ~Q.

According to the first interpretation of CC2 some counterfactual conditionals are both true and assertible in Barbour worlds. According to the second interpretation no counterfactual conditional is true in a Barbour world, but some are assertible. Henceforth we will say that if ‘P □→ Q’ is assertible, but not true, then Q quasi-counterfactually depends on P. We return to the distinction between quasi-counterfactual dependence and counterfactual dependence shortly. For now we will focus on explicating the right hand side of the biconditional found in CC2 and its two sub-kinds. To render CC2 viable we must do two things. First, we must explain what it means for a quasi-world to be ‘actual’. Second, we must specify a closeness metric over quasi-worlds that can play the same role as Lewis’s closeness metric over worlds. Addressing the second point will help to address the first so we’ll start there. Lewis’s closeness metric makes heavy use of deterministic laws of nature. We need something very much like a deterministic law of nature that holds at a quasi-world and that can be plugged into CC2. While the points in quasi-worlds do not bear genuine nomic connections to one another, for some quasi-worlds it is possible to extrapolate a set of deterministic ‘quasi-laws’. 19

Though not laws strictly speaking, deterministic quasi-laws for a quasi-world h1 defined by total ordering T1 are such that (i) for any point p, the quasi-laws can be applied to p as if p were a time in a four-dimensional world to yield a total ordering T2 of a sub-set of points in Platonia of which p is a member and (ii) total order T1 = total order T2. Deterministic quasi-laws are extracted from a quasi-world in the same way that deterministic laws are extracted from Lewisian possible worlds. For instance, if the deterministic laws supervene on the Humean mosaic within a Lewisian possible world then so too do the quasi-laws supervene on the Humean mosaic defined by a path. There are three reasons to think that quasi-laws are not genuine laws of nature. First, it is part of the concept of a law that there is a fact of the matter about what they are, in the following sense: one and only one set of laws governs the actual world. There is, however, no fact of the matter of this kind about what the quasi-laws are with respect to the actual world, which is a point in Platonia. There are many sets of quasi-laws (because there are many paths) and no sense in which only one set of such quasi-laws governs the actual world. Second, genuine laws tell us something about the nature of the actual world; they are high-level structural constraints over the goings-on in reality. Quasi-laws tell us nothing about the actual world (again: a point in Platonia); they do not constrain actuality in any way. Arguably, the only thing that constrains actuality in Platonia is the timeless Wheeler-De-Witt equation from canonical quantum gravity and its associated probability distribution over time-capsules. The third reason for thinking that quasilaws are not laws is that laws operate within a world. Quasi-laws do not. Quasi-laws operate across worlds in so far as they ‘operate’ at all, stringing together distinct, instantaneous possible worlds into a trajectory through the configuration space. In order to put the quasi-laws to work in evaluating counterfactuals we require the notion of a quasi-miracle. A miracle, as we saw, is a violation of the deterministic laws and can be defined intra-worldly. A quasi-miracle, then, is a violation of the deterministic quasi-laws and can be defined over quasi-worlds. Thus, a quasi-miracle at a quasi-world h1, relative to another quasi20

world h0, is a violation at h1 of the quasi-laws of h0, which are at best the almost-quasi-laws of h1. Lewis defines the size of a miracle in terms of the spatiotemporal region across which there is a violation; the larger the region, the larger the miracle. Since there are no such regions in quasiworlds this won’t do for our purposes. Instead, we use regions in relative configuration space within a quasi-world. A large region is a large subset of the points in the total order for that quasi-world. A small region is a small subset. A quasi-world h0 in which there is a large quasimiracle relative to a distinct quasi-world h1, then, is one in which there is a large difference between the set of points corresponding to h0 and the set of points corresponding to h1; a small quasi-miracle is one in which there is a small difference between the sets of points; where a difference in the sets corresponds to a difference in the elements of those sets. Using quasi-laws and the notion of a quasi-miracle we can modify Lewis’s metric for similarity across worlds as follows:

(1) It is of the first importance to avoid big, widespread, diverse violations of quasi-law. (2) It is of the second importance to maximize the region of configuration space throughout which perfect match of particular fact prevails. (3) It is of the third importance to avoid even small, localized, simple violations of law. (4) It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.

We require one final piece of the puzzle: the notion of an actual quasi-world. This brings us to the first point noted above regarding the viability of CC2. An actual quasi-world is required to anchor the closeness metric thereby rendering the analysis tractable. Because the actual world is a single point p in Platonia, an actual quasi-world will be some path through p. Because Lewis assumes that the actual laws are deterministic we shall assume the same for the quasi-laws of the actual quasi-world. A candidate to be the actual quasi-world, then, is any totally ordered sub-set 21

of points in Platonia with p as a member and that possesses deterministic quasi-laws. This narrows the field somewhat (see Figure 4). PLATONIA PLATONIA

Figure 4

Narrowing the field: red paths possess deterministic quasi-laws.

Because there are many different versions of the deterministic quasi-laws there are still many different paths of this kind through p. So we need some way to narrow in on a particular path. We do this as follows. As discussed above, points in Platonia (at least those that are candidates to be actual) are imbued with a rich representational structure. This structure includes (apparent) memories, as well as the full range of evidential features including (what appear to be) records of the past, historical evidence and empirical data more generally about (what appears to be) a fourdimensional universe; evidence that has led agents like us to generate an account of (what we take to be) laws of nature governing such a universe. It is this rich representational structure that we can use to fix on a path as the actual quasiworld. Of the paths through configuration space that pass through p and possess quasi-laws, the actual quasi-world is the path through p such that (i) the points in the order prior to p agree as far as possible with whatever evidence exists at p regarding what (appears to be) the past history of a four-dimensional world and (ii) the quasi-laws defined by that path correspond as far as possible to whatever laws can be extrapolated from the empirical evidence at p by conscious agents at that world. If all goes well, we wind up with a single path (see Figure 5).

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PLATONIA

Figure 5

The actual quasi-world: red paths possess deterministic quasi-laws; the blue path accords with the representational structure of the central point.

If things do not go well, we may still end up with multiple paths. We will return to that possibility in a moment. For now, let us grant that there is a unique path corresponding to the actual quasi-world defined by the evidence. We now have enough structure to evaluate a candidate counterfactual.9 Consider:

(CF1) 9

If Suzy had not thrown the rock, then the bottle would not have broken.

Once we have come this far, we can also develop a process-like semantics using quasi-worlds to produce a proxy

for process causation. Because an actual quasi-world mimics an actual spacetime there will be process-like phenomena within actual quasi-worlds that can underpin a quasi-process semantics. The basic idea would be to use quasi-worlds to recover quasi-world lines through those worlds. We could then reformulate CAUSPROC from footnote 4 to give us assertibility conditions for causation as follows:

CC3

‘x causes y’ is assertible iff (i) x’s quasi-world line in a quasi-world intersects with y’s quasi-world line and (ii) at the point of intersection x quasi-transfers a conserved quantity to y.

Where ‘quasi-transference’ is a matter of (i) x at a point p1 in Platonia in quasi-world h1 having a conserved quantity C and (ii) at a second point in Platonia p2 in quasi world h1 such that p2 is posterior to p1 in the total order for h1, x lacks the conserved quantity, and y possesses that quantity.

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According to CC2, CF1 is assertible at the actual world iff there is some quasi-world in which Suzy throws the rock (i.e. there is a point, or set of points, in the total order for that quasi-world which bears witness to this fact) and the bottle breaks that is closer to the actual quasi-world than any quasi-world in which Suzy throws the rock and the bottle does not break. In short, if it would take more of a quasi-miracle to get to a quasi-world in which Suzy throws the rock and the bottle does not break than to get to a quasi-world in which Suzy throws the rock and the bottle does break, then the counterfactual is assertible and not otherwise. If the counterfactual is vindicated by CC2 and is thus assertible then, according to CC2(a), the counterfactual is also true. According to CC2(b), the counterfactual is merely assertible because, when evaluated using the standard Lewis semantics for counterfactuals, it is false. The beauty of evaluating counterfactuals via either version of CC2 is that any difficulties arising for the evaluation are very likely to be difficulties for the application of Lewis’s semantics to ordinary counterfactuals. For instance, we have used deterministic quasi-laws throughout to evaluate counterfactuals. It might be thought, however, that the restriction to determinism is ad hoc, and that we should cast our net wider by judging closeness using indeterministic quasi-laws. That, however, is just as much an issue for Lewis’s original counterfactual theory of causation. Lewis’s analysis is, in the first instance, explicitly restricted to the case of determinism. He goes on, however, to generalise the analysis to the indeterministic case (see Lewis (1986, pp. 58–65)). Any such generalisation will carry over to the quasi-world analysis of counterfactuals deployed above. There is one problem, however, that might appear unique to CC2. In order to apply the analysis we must anchor the closeness metric to the actual quasi-world. The actual quasi-world is a total ordering of a subset of points in Platonia that possesses the actual world, p, as a member. The total ordering is special in that points prior to p in the ordering agree with whatever evidence exists at p regarding the past history of a four-dimensional world and the quasi-laws defined by 24

the total order correspond as far as possible to the laws suggested by the evidence at p. The trouble is that, as noted, for all we have said, these two conditions may not determine a unique quasi-world. There may be more than one quasi-world that coheres with p in this way. Moreover, there is no obvious way to break ties since there is no fact of the matter about which quasi-world is the actual quasi-world over and above the evidence contained within p. A similar problem does not arise for Lewis’s analysis. For Lewis, there is a fact of the matter about which world the actual world is, one that might ultimately come apart from any evidence we have to hand, evidence that leads us to form beliefs about which world is actual. As a consequence, when the evidence underdetermines which world is actual, the metaphysical facts about which world really is actual can be used to determine a unique world for the purposes of anchoring counterfactuals. CC1 and CC2 therefore come apart in the following way. For Lewis, the epistemic question: “given the evidence, which world should we believe is actual?” comes apart from the metaphysical question: “which world is actual?” While there may be no unique answer to the first question, there is always a unique answer to the second. For the quasi-world approach, however, the epistemic question: “given the evidence, which quasi-world should we believe is actual?” does not come apart from the metaphysical question: “which quasi-world is actual?” The answer to the first question determines the answer to the second. So if there is no unique answer to the first question, then the second question will also lack a unique answer. Should we be concerned? Well, while it is true that there are metaphysical facts about which world is actual that can be used to anchor Lewis’s analysis, we do not know what those facts are because we are not privy to all of the fundamental truths about the nature of the actual world (as far as we know). So in so far as that analysis is actually put to work by Lewis and others to recover our semantic intuitions about which counterfactuals are true and which are false, we do so by anchoring that analysis to whatever world we believe to be actual, based on what we know about the laws and the history of the world from the available evidence. In short, any tangible 25

results yielded by the application of Lewis’s CC1 are based on whatever answer to the epistemic question we have, rather than the one unique answer to the metaphysical question, of which we are ignorant. Any such results can therefore be recovered from CC2. That’s because the quasi-world believed to be actual lines up with the Lewisian possible world believed to be actual, since the method for working out the answer to the epistemic question in both cases is based on the same evidence and so should yield exactly the same results. Thus our everyday judgements about the truth, or assertibility, of counterfactuals are likely to agree for both CC1 and CC2.

5. Causal Claims

This brings us back to causation. Recall Lewis’s counterfactual analysis of causal claims:

CAUS1

‘x causes y’ is true iff there is a causal chain leading from x to y consisting of events a, b, c ... such that c depends counterfactually on b, b depends counterfactually on a and so on.

Combining CAUS1 with CC2(a) will yield some true claims of the form ‘x causes y’ in a Barbour world since according to CC2(a) there is a b that counterfactually depends on a, and so forth. Combining CAUS1 with CC2(b), however, will not yield any true claims of the form ‘x causes y’ since, according to CC2(b), some counterfactuals are assertible in a timeless world, but none are true. Thus we need an alternative analysis of causation for those attracted to CC2(b). Two suggestions present themselves:

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CAUS2

‘x causes y’ is true iff there is a quasi-causal chain leading from x to y consisting of events a, b, c ... such that c depends quasi-counterfactually on b, b depends quasi-counterfactually on a and so on.

CAUS3

‘x causes y’ is assertible iff there is a quasi-causal chain leading from x to y consisting of events a, b, c ... such that c depends quasi-counterfactually on b, b depends quasi-counterfactually on a and so on.

CAUS2 and CAUS3 only disagree about whether, when the conditions on the right hand side of the biconditional are met, the claim ‘x causes y’ is true, or merely assertible. We have, then, three possible analyses of ordinary causal talk: CAUS1, CAUS2 and CAUS3. Both CAUS1 and CAUS2 recover causation; they vindicate some ordinary causal claims as being true. Accepting either view would straightforwardly solve the ordinary causal language challenge as stated in §2. Moreover, if ordinary causal claims are true then there are good prospects for showing that we are justified in believing in a timeless theory on the basis of observation. After all, observational claims will come out as true (or false) according to this semantics. So it would solve the observational challenge as well. Compared to CAUS2, CAUS3 gives us something weaker – it renders causal claims assertible. If one thinks that these assertibility conditions are also truth conditions then one also accepts CAUS2. We, however, are interested in those who accept CAUS3 but think that the assertibility conditions are not also truth conditions. Where this is the case we will say that we recover quasicausation. That is, we will say that ‘x quasi-causes y’ is true iff ‘x causes y’ is false, and ‘x causes y’ is assertible. If CC2 is accepted as the correct account of the assertibility conditions of counterfactuals then CAUS3 would be sufficient to meet the ordinary causal language challenge, since it would render causal claims assertible in a timeless setting. Does CC2 represent the correct assertibility 27

conditions for causal claims? We believe that it does, at least assuming a counterfactual theory of causation. To see this, set aside the issues surrounding timelessness and consider the relationship between the assertibility of causal claims and evidence more generally. Intuitively, a causal claim is assertible just when there is substantial evidence for that claim. This means that causal claims can legitimately be asserted, even when they are false. That’s because there may well be substantial but misleading evidence in favour of a causal claim. Because the quasi-world approach to assertibility just is a precisification of the relationship between evidence and causal claims, it is well-placed to vindicate this intuition. Indeed, what is striking about the configuration space that Barbour offers is that it can be used to provide a broader ‘epistemic model’ of assertion for causal claims that has implications beyond its relevance for timelessness. The configuration space, the paths through that space and the appropriate narrowing provided by the evidence at any given point in the space can all be used to model the assertibility conditions for claims about causation independently of any timeless physical theory. In other words, if we want to determine when a causal claim is – or is not assertible – for an agent at a time, given the available evidence, a relative configuration space of the kind that Barbour uses to interpret canonical quantum gravity will provide a model for doing so. We have used quasi-causation to recover the assertibility of causal claims in a timeless world. But it could equally be used to map the assertibility of causal claims in any world that has evidence, whether it is timeless or not. That is why CC2 constitutes the correct assertibility conditions for causal claims and that is also why CAUS3 provides a solution to the ordinary causal language challenge. One might wonder, though, whether recovering only quasi notions will go any way towards answering the observational challenge. While ordinary claims of the form ‘yesterday I did an experiment which had the following results’ may be assertible, they are not true. Since it is their truth that matters when we are attempting to justify belief in a timeless theory via appeal to

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observational evidence, recovering quasi-causation is not enough to make any headway with the observational challenge. Even if that is where matters stand, progress has still been made. Answering the ordinary causal language challenge is important, and it may be possible for defenders of a timeless physical theory to provide other resources to answer the observational challenge. Still, we are tempted to make one last attempt at answering the observational challenge by suggesting that there is a reason to prefer CAUS1 or CAUS2 to CAUS3. We see no point appealing to apparent features of our concept of counterfactual dependence or causation, or to intuitions about the right semantics for causal claims to adjudicate this dispute. Those who are drawn to CAUS3 over the other two options simply have somewhat different intuitions than their disputants about these matters. What we can say, however, is the following: all three accounts render assertible the same sentences. When we focus narrowly on causal discourse, then, pragmatically speaking these three different ways of evaluating causal claims are of a kind. Where they differ is that while CAUS1 and CAUS2 render causal claims true, CAUS3 renders causal claims false, and hence on the face of it makes it difficult, and perhaps impossible, to answer the observational challenge. While this might lead one to be suspicious of timeless physical theories as self-undermining, it seems odd that a difference regarding whether one takes a set of conditions to provide truth, or merely assertibility, conditions for certain claims could potentially make the difference between a live scientific theory being self-undermining, and it failing to be self-undermining. Semantics should not, we think, have this kind of clout. Thus we think there is a plausible methodological principle in the offing: if there are competitor semantics for a discourse, such that all of the competitor semantics render assertible the same set of sentences, then one should choose the semantics that is consistent with the largest range of live scientific theories. Then either (a) CAUS3 is consistent with timeless physical theory because there is, given its correctness, a way to meet the observational challenge or (b) CAUS3 is not consistent with timeless physical theory because 29

there is, given its correctness, no way to meet the observational challenge. If (a) is true then both challenges can be answered whichever of the three options for evaluating causal claims one chooses. If (b) is true then one has independent reason, via our methodological claim, to prefer CAUS1 or CAUS2 to CAUS3 whatever one’s other semantic intuitions. In either case, both the ordinary causal language challenge and the observational challenge will be answered.

6. Conclusion

We have presented a new way to evaluate counterfactual conditional claims and causal claims with a view to solving one or both of the ordinary causal language challenge and the observational challenge for a particular timeless approach to canonical quantum gravity. We have argued that, at worst, our account recovers mere quasi-causation and thus classes (some) ordinary causal claims as assertible, though false. This addresses the ordinary causal language challenge but not the observational challenge. We have, however, suggested that there are reasons to prefer one of the candidate ways of evaluating counterfactuals that addresses both of the challenges. At best, then, both challenges are met. At worst, only one is met and additional resources must be marshalled to meet the other. Moreover, our strategy for evaluating counterfactuals is independently useful, for it serves as a model for understanding the assertibility conditions of causal claims outside of a timeless setting. This is important as it sheds light on the relationship between truth and assertion for causal talk. When fed back into the discussion of timelessness, this suggests a more general result for tempus nihil est approaches to canonical quantum gravity. For any such approach, if that approach allows for the existence of rich evidential structures capable of grounding quasi-causation, there will be a way to recover the assertibility of everyday causal claims. 30

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