4.3 The compatibility of causation and physics
4.3.4 Reconciling causation and physics
As Eagle (2007, p. 158) proposes, the problem of bidirectional nomic dependence could easily be solved by relying on the temporal asymmetry of causal processes, for which I argued in section above 4.3.1. According to Eagle, it could be maintained that causation is a symmetric dependence relation and that the asymmetry of causation is provided by the temporal asymmetry. Given that physical processes in the world are temporally asymmetric, causation would then obviously be compatible with the bi-directional nomic dependence of physics, since causation itself would be a bidirectional dependence relation. Hence, causes would as much determine their
effects, as effects would determine their causes. Only, because of the temporal asymmetry of processes in the world, one is contingently realised before the other in time, which makes it appear as if causes determine their effects, but not vice versa. Even though I believe this is a promising line of thought, I do not wish to follow it here. Historically it seems more accurate to understand causation itself as being asymmetric in two ways, temporally and causally (cf. my discussion in section 3.3.5), and following Eagle’s idea is in danger of changing the concept, rather than illuminating it (cf. my discussion of Ney 2009 in section 4.2.3). Instead, in what follows I wish to argue that, presupposing the factual temporal asymmetry of physics, despite the nomic bidirectionality of physical laws, processes can show nomic directionality that is suitable for causation.
It is Eagle again, who pointed towards a possible reply to Russell’s argument that he, however, did not work out. According to Eagle (2007, p. 158), a system being deterministic does not preclude the possibility of asymmetric determination relations:
Even in a deterministic system, it is possible that both (i) all trajectories which feature this event type c as part of some global state s at t have some further type of event e as a feature of a state s� at t�, and (ii) not every global state that features e at some time lies on a trajectory which involves some past state that features c. So the occurrence of c determines the occurrence of e in a way that e doesn’t determine the occurrence of c. Focusing attention on events of a purely local character, rather than the entire state of the system, might very well give us an asymmetry of determination between particular events.
Thus, Eagle’s idea is that in a system S, which is smaller than the whole universe, it is compatible with determinism that every state S1(t1) evolves deterministically
into a later state S2(t2), but not every state S2(t2) has evolved from an earlier
state S1(t1). Unfortunately, Eagle merely highlights the mere possibility of such a
structure, and does not explicate further, how it could be realised. This will be my task in what follows.
Already Earman has presented a similar possibility and concluded that: “causal directionality is not incompatible with determinism” (Earman 1976, 18). Accordingly, there could be an intervention from God, or more realistically, an force from outside the system, that, like the Lorentz-Dirac equation changes the state of the system before it is turned on. This intervention, however, would typically violate the laws of T, and is in conflict with the condition applied here that the deterministic system is closed. Why is this a problem? The reasoning below will argue that for a given theory T determinism and causation can be reconciled. Changing that theory is in danger of being ad hoc. The closed system condition, on the other hand, is needed to prevent spurious breakdowns of determinism, and it seems reasonable to demand of a causal interpretation that it can include closed systems. Furthermore, Earman’s argument relies on the existence of such a force, for which there currently is no empirical evidence.
As a first step in concretising Eagle’s idea, I wish to make use of work on asymmetric causal structures, which has long been popular in economics, but gained less attention
in philosophy. In particular Simon & Rescher (1966) have worked out a precise notion of causal structures that can serve as the blueprint to spell out Eagle’s idea, and show how bidirectional nomic dependence can be compatible with causation. According to Simon & Rescher (1966, p. 324) a “causal relation is not a relation between values of variables, but a function of one variable (the cause) on to another (the effect).” The basis of a causal structure is a set of one or more functions that has to be self-contained, that is, it has to contain as many variables as functions. A larger self-contained structure can show a kind of asymmetry in that some functions can be solved without first solving others, but not vice versa. This will become clearer with an example, even though I can hardly pay justice to all the subtleties of Simon & Rescher’s idea.33
Suppose that two billiard balls collide elastically at time t0. Their momenta
before the collision are fixed to ¯p1(t < t0) and ¯p2(t < t0). This determines the
momenta after the collision, p1(t > t0) and p2(t > t0). The bar on the former
two momenta indicates that they are exogenously fixed to certain values, while the latter two momenta are variables. By the conservation of momenta, we have
¯
p1(t < t0) + ¯p2(t < t0) = p1(t > t0) + p2(t > t0). Following the notation in Simon &
Rescher, this set of four variables can be put into four functions, together forming a self-contained structure.
1. f1[¯p1(t < t0)] = 0
2. f2[¯p2(t < t0)] = 0
3. f3[¯p1(t < t0), ¯p2(t < t0), p1(t > t0)] = 0
4. f4[¯p1(t < t0), ¯p2(t < t0), p2(t > t0)] = 0
The first two equations signify that the values of ¯p1(t < t0)and ¯p2(t < t0)are pre-set
to certain values, say, by the experimenter. The latter two equations signify that the values p1(t > t0) and p2(t > t0) can be calculated from the momenta before
the collision. This system is directed in that equation f1 and f2 can be solved
before solving the other two equations, simply by putting in the values that the momenta are fixed to before the collision. The momenta after the collision can only be calculated if we already know the momenta before the collisions. Thus, f3 and
f4 can only be solved, if f1 and f2 are already solved. In the notation of Simon &
Rescher, this directionality can be captured by two diagrams:
33I have chosen Simon & Rescher as the reference work here, because they address the topic of
directionality most explicitly. A much more detailed account of defining causal relations in their spirit can for example be found in Pearl (2000). It might be noted that Pearl discusses Simon & Rescher’s approach in chapter 7.2.5 and finds that “[t]he asymmetry that characterizes causal relationships in no way conflicts with the symmetry of physical equations.” (Pearl 2000, p. 228) I wish to point out, that I am not trying to reduce causation to structural equations (see a critique of such an approach in Hall 2007 and a reply in Hitchcock 2009).
f1 & % f2 ( f3 f4 , ¯ p1 & % ¯ p2 ( p1 p2
We can say that ¯p1 and ¯p2 are exogenous variables, while p1 and p2 are dependent
variables. Given this directionality, Simon & Rescher feel confident enough to say the exogenous variables are causes, while the dependent variables are effects. Hence, in the above example one can conclude that the momenta ¯p1 and ¯p2 jointly cause p1
and p2.
To clarify where the advantages of Simon & Rescher’s account lie, it is worth taking a quick look on another attempt to formalise causal relations. According to Frisch (2010, sec. 3), the causal relation can be represented by defining an asymmetric, transitive and non-circular relation C between two states S of a system. “C is interpreted as the causal relation: S(t2) bears C to S(t1)exactly if S(t1)is a
cause of S(t2). If two states do not stand in relation C then they are not causally
related. The result is a class of what I want to call potential causal models of a theory.” (Frisch 2010, p. 79) I wish to point out one problem with Frisch’s account that shows the superiority of Simon & Rescher. As the latter observe, a function f (x) = y usually possesses an inverse f 1(y) = x. If now this function is in some way supposed to describe a causal relation, x causes y, then it seems plausible that the inverse function shows the inverse causal relation, y causes x. Now, while this is unproblematic for functions, the directionality of causation should prohibit the causal relation from having an inverse. This problem is recognised by Simon & Rescher and they solve it with the notion of self-contained systems; the asymmetric structure a self-contained system exhibits cannot be inverted in that way. The same, however, is not true for Frisch’s causal relation C. Prima facie there is no reason why C should not have an inverse C 1 that is also a causal relation; even the asymmetry
of C does not preclude this (which is just to repeat the point that asymmetry is not sufficient for directionality). There then seems to be no grounds to decide which relation, C or C 1, is the real causal relation. Now, it might be possible for Frisch
to add the further stipulation that C does not possess an inverse or that C, but not C 1 is a causal relation. However, in the present context this would not bring
us any further, since this move would not provide any understanding about where this directionality comes from. Contrary to that, the directionality follows naturally from Simon & Rescher’s account.
It is clear that simply stipulating a causal relation, as Frisch does, cannot serve as an argument for the existence of causal relations, and, to be fair, it is not meant as such by Frisch. Needless to say, the same is true for Simon & Rescher; that they chose to call their asymmetric relation causal does not mean that it actually is. Thus, a further argument has to be made to underpin Simon & Rescher’s view that they are indeed describing causal relations. The point I wish to make is that this argument comes from Woodward’s interventionism.
An obvious objection against the billiard example is that even though the way I described the collision forms a self-contained system, this is not sufficient to establish that the process is a causal one. The problem is that in this example the choice of ¯p1 and ¯p2 as exogenous and p1 and p2 as dependent variables is arbitrary. For
what reason should the former have any priority over the latter? I believe this objection can be answered with the help of the interventionist theory by Woodward, which provides a structure that is that of a self-contained system with directional dependence relations. Recalling the discussion from section 3.3.4, it is part of the definition of an intervention that it has to be exogenous to a causal process, that is, if there is an alleged causal process in which X causes Y, then an intervention I has to come from outside of this process and break any previous influences on X, such that the variable X that causes Y is entirely set by I (cf. Woodward 2003, p. 94). Back to Simon & Rescher’s formalism, this is what provides the priority of some variables over others; only if we already know the value of the intervention we can calculate the effect, but from the effect we cannot calculate whether there was an exogenous intervention or whether it has been caused solely by an endogenous cause. Following the notation of Simon & Rescher (1966, sec. 6), interventions, Ii,
can be made explicit in the formalism, which, in the billiard case gives the following structure: I1 ! ¯p1 & % I2 ! ¯p2 ( p1 p2
What we have so far is the result that even a system of functional relations can be directed, and that this directionality can be understood as causal. It is clear, however, that this explication is not a reduction of (CD) to non-causal directionality. However, such a reduction was neither aimed at, nor will it be necessary for the overall argument. Prima facie, (CD) can be compared with physics even if it is a genuinely causal concept. Analogously, for example, one can ask whether there are laws in physics, even if the term ‘law’ is not fully reduced to non-nomological concepts. What is necessary, however, is some understanding of (CD) that we can work with, that is more than the statement ‘causes are causes’, and the above analysis provides such an understanding.
The next and final step now is to show that this directionality is compatible with determinism. To answer this question, I wish to build up on local determinism in order to formalise Eagle’s idea as follows:
Asymmetric determination: Let T be a deterministic theory with models MT.
Let si(t1)✓ Mi and si(t2)✓ Mi, t1 < t2, be two states of the model Mi2 MT.
Both states stand in an asymmetric determination relation iff, there exists a model Mj 2 MT with a state sj(t)✓ Mj such that si(t) = sj(t)for all t t2
Thus, there are two models that are isomorphic after t2, but not before t1. Why is
this asymmetric determination? Because given any state s(t t1), using the laws
of T, we can find a unique bijective map such that T : s(t t1)! s(t t2), but
there is no unique bijective map for the converse, that is, given a state s(t t2).
The reason for this is that a state at t t2 can have evolved from either of the
non-identical states si(t t1)or sj(t t1). In conclusion, asymmetric determination
is consistent with local determinism, given that each model M is deterministic and the asymmetry is a relation involving two different models. According to the earlier discussion, this amount to asymmetric nomic dependence, since from any si(t t1)
and sj(t t1) the later state s(t t2) follows, but not vice versa.
The crucial step now is to ask in which sense these asymmetric dependence relations can be interpreted as asymmetric causal relations? Earlier, I have argued that causal directionality can be analysed into asymmetric dependence relations within complete structures. Furthermore, I explicated these structures in terms of an interventionist account of causation. The asymmetry we found consists in that the effect depends on exogenous intervention, but not the converse. With a local definition of determinism, we are now in a position to write down this asymmetry of interventions more formally, to see whether it is indeed consistent with local determinism.
Asymmetric causal structure: Let I(t1) ⇢ M1 be an intervention on a state
s1(t1) such that according to the laws of T at a later time t2 we find the
state s1(t2) ⇢ M1, t1 < t2. Let s2(t2) ⇢ M2 be a state that is isomorphic
to s1(t2), and M1(t > t2) is isomorphic to M2(t > t2), but M1(t < t1) is not
isomorphic to M2(t < t1). If M1, M22 MT, then T has models that establish
an asymmetric causal structure.
In other words, the asymmetry consists in that both M1(t < t1) and M2(t < t1)
determine the occurrence of s2(t2), but the latter does not determine the occurrence
of I(t1) or s2(t1), given that both are compatible with the laws of T and dynamical
evolutions involving s2(t2). Or to say it in the terms of Simon & Rescher s2(t2) is a
function of I(t1) or s2(t1), but not vice versa.
Perhaps the most obvious objection to the account here presented is that it works in both temporal directions and therefore fails to circle out one direction of nomic dependence over the other. Prima facie, the intervention can lie either in the past or the future of its effect, and there might even be symmetric cases with two interventions. So far, neither possibility can be excluded. However, at this point I wish to make use of the previously established result that processes in the world have a factual direction in time. This rules out interventions that lie in the future of their effect, since a past event cannot be manipulated by a future intervention when time has the corresponding direction. It would be question begging to rule out future to past interventions by merely appealing to the causal fact that we cannot intervene on the past. This is not what I propose. Rather, what is doing the work in ruling out future to past interventions is the temporal directionality of the physical facts, which was established independently of any causal concept.
Let me evaluate this point in a little more detail. If, for the sake of the argument, we assume that causal relations in both temporal directions are possible, then, according to Woodward’s condition (NC), the following two counterfactuals could be true: 1. ‘If the intervention I(t1) was carried out, then the value of X(t2) would
change.’ 2. ‘If the intervention I(t2) was carried out, then the value of X(t1) would
change.’ However, if contrary to that we assume that time is directed from t1
to t2, then whatever is done at t2 will not change anything at t1 and the second
counterfactual would be false. This is not the result of an asymmetry of interventions, but an asymmetry of time.
Two clarifications: First, it is not the fact that one event comes before the other that makes the earlier the cause and the later the effect. Rather, it is the fact that interventions, that is, manipulating one variable by manipulating another, only work towards the future that establishes the causal order. Second, the causal directionality defended here is not the simple fact that ‘we can only manipulate the future’, but comes from an asymmetry between possible models. The temporal directionality of interventions is not used to establish the directionality of causation,34 but merely to
pick out one causal direction over the other.
Another objection might be that in the world only one possible model can be realised. Hence, while one might agree that there is an asymmetry between possible models, one might ask where this asymmetry is gone when only one model is actual. However, as Woodward (2003) has argued, there is a close link between interventions and counterfactuals. For a process to be causal, an intervention does not actually has to happen, it only has to be counterfactually possible.35 Accordingly, for an
asymmetric causal structure, it is not necessary that two models are realised. It suffices that an alternative model involving an intervention is possible according to the laws of T . What I must presuppose, however, is that all models that are considered have the same temporal direction.
I wish to finish this section by addressing a last worry. What we set out for was to analyse the asymmetry of causation and to ask whether this is compatible with physics. What we ended up with is an asymmetry between different models, different