Determinism localised

The argument from nomic bidirectionality hinges on the determinism of physics. However, determinism is not a self-explanatory term and there are various ways of understanding it. In what follows, I will argue in favour of a local definition of determinism. Building up on that, in the next section I will show that causation is compatible with local determinism.

Here we find a first definition that I call global determinism:

Global determinism: Let W be the set of all possible worlds that satisfy the laws of physics. Then world W 2 W is deterministic iff, for every W and all times t, if W (t) = W0_{(t), then W (t}0_{) = W}0_(t0_{). (see Earman 1986, 13)}

However, while this definition tells us when a world is deterministic, the argument from nomic bidirectionality establishes a conflict between causation and a physical system. Hence, what we actually need to know is what it means for a physical system to be deterministic. I wish to put forward the following definition:

Local determinism: Let T be a theory. Let MT be the set of all dynamically

possible models that satisfy T , and M 2 MT. Let M(t) ⇢ M be the model at

a time t, and s(t) ✓ M(t) be the state of a system. Then M is deterministic iff, for every s and all times t, there exists within T a unique bijective map T

such that T : s(t)! s0(t0).

What characterises a model is notoriously hard to define. However, for present purposes we might understand it simply as a set of states that solve the laws of T. Accordingly, I take the term ‘physical system’ used earlier to be synonymous with model. The term ‘state’ is ambiguous as it can refer either to an element of a model, say the state of a single particle, or a subset, say the state of multiple particles. The map T is determined by the dynamical equations of T, such that if a state

interacts with other states of the model, then this interaction must be included in the dynamical equations.

Global determinism follows from local determinism, if s is a state of the whole world. Thus local determinism can recover the more common global determinism as a special case. Sometimes determinism is also defined as theory determinism: “We say that a theory is deterministic if and only if: any two of its models that agree at a time t on the state of their objects, also agree at all times future to t.” (Butterfield 1998, 37) Again, local determinism recovers theory determinism, iff all M 2 MT are

deterministic. However, as I will explain now, there are good reasons to prefer local determinism over the other formulations.

First, if the definition of determinism should be applicable to theories at all, then we might actually have no other choice than abandoning global determinism. Physical theories in general do not describe worlds, but systems. These systems can in principle be as large as what we call a world, but they do not have to be; for the theory this is irrelevant. Hence, the term ‘world’ is not well defined by a theory. Furthermore, global determinism is practically inapplicable. Given that we have no theory of everything and given our limited capacities, we cannot give a description of the whole universe, and whether this will ever be possible is an open question. Hence, global determinism is void in the context of today’s theories. With Wüthrich (2011, 368) we might see this simply as a consequence of naturalism, that is, a focus

Second, if for the sake of the argument we apply global determinism to a theory, then presumably the theory would describe nomic dependencies between states of the whole universe. As a consequence, a valid formulation of the argument from nomic bidirectionality must assume that there are causal relations between states of the whole universe. Whether there are these global causal relations is a contentious matter, but what is clear is that most theories of causation are applicable to smaller objects, and intuitively most would agree that there are causal relations between smaller objects if there are causal relations at all. Or in other words, global causal relations, if they exist, are arguably not fundamental, but build up from local causal relations. Insisting on global determinism would rule out any approach to answer the question on the local level by definition. Local determinism, on the other hand, is more liberal, since a model can in principle be the whole universe, but not always is.

Third, physical theories such as Newtonian physics or quantum mechanics have models describing deterministic as well as models describing indeterministic processes (see Earman 2007; Norton 2008; Wüthrich 2011). As a consequence, one might call these theories indeterministic as a whole, however, a balanced approach seems more feasible. If a theory is deterministic or not relative to a model, then determinism of a theory should be defined relative to a class of models. This leaves no other choice than to abandon global determinism as well as theory determinism.

Contrary to that, Hoefer (2010, sec. 2.2) argues in favour of global determinism, for otherwise there could be influences from outside the local system that would lead to a spurious breakdown of determinism. Thus, it could be the case that M(t) = M0_(t),

but M(t0_{)6= M}0_(t0_{), because there was an influence on M from outside the system}

described in the model. Of course, this is not what is meant by indeterminism. However, as Butterfield (1998, 36-37) explains, this problem is easily avoided if a model is regarded as representing an isolated system that has no interaction with other systems, as it is commonly done in physics. It might be objected that real systems are never completely closed. However, if being closed is not taken as a condition for the systems in question, then, following Hoefer, we usually would not have deterministic systems, which in turn would make the directionality argument invalid and thus brighten the prospects of squaring causation and physics rather than dampening them.