Determinism and indeterminism

3. Defining determinism

In §2 we said that a theory is deterministic if and only if for any two systems of the kind described by the theory: if they are in exactly the same state as one another at a given time, then according to the theory, they will at all future times be in the same state as one another. But this formulation is still rough.

The main problem is that, whatever theory one considers, its systems are continually interacting with their environment: as physicists put it, no system is ‘completely isolated’. For example, each system feels the gravitational pull of other objects. These interactions make determinism, as just formulated, an impossibly tall order. For, first, it will be very rare for two systems to be in exactly the same state at a given time. And even if they are, it will be virtually impossible that their subsequent interactions with their environments match so exactly that they are also in the same state as one another at all future times. But surely, determinism should not be so formulated that it will fail because of the vagaries of interactions with other systems: whether it fails or holds should be a matter internal to the theory considered.

The remedy is clear enough. To set aside such interactions, we need to formulate determinism in terms of completely isolated systems. But we cannot just think in terms of two systems in an otherwise empty universe. For in general the theory will take these two systems to interact with each other; so that again determinism can fail in a spurious way. That is to say, even if we suppose that at a given time the two systems are in the same state, at some future time they may well not be: their interaction, as described by the theory, might lead to their states differing. (The problem is of course aggravated if we think of the systems as also interacting in other ways, not described by the theory.)

To avoid this kind of spurious failure of determinism, we need to think of the theory as describing single completely isolated systems, each one alone in its universe. Let us say that a sequence of states for such a single system, that conforms to the laws of the theory, is a model of the theory. So a model contains a system of the theory’s kind, undergoing a history allowed by the theory: the model is a ‘toy universe’ or ‘toy possible world’, according to the theory. (This use of ‘model’ is common in general philosophy of science. In particular, it is often useful to consider a scientific theory as the class of its models in this sense, rather than in the traditional manner of logicians – as a set of sentences closed under deduction; see Models; Theories, scientific.) Using this notion of model, we can give a better definition of determinism, which avoids the problem of interactions. 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.

(This definition returns us to the first of the two questions at the end of §2: namely, should we apply the idea of determinism to theories of the whole universe, for example cosmologies? We now see that our strategy for avoiding spurious violations of determinism, due to interactions between systems, commits us to answering Yes to this question. For by taking a model of any theory to describe a single completely isolated system, alone in its universe, we are in a sense treating any theory as a cosmology. But since each of a theory’s possible universes contains just one system of the kind treated by the theory, it is typically a humble, even a dull, cosmology!)

This definition is still a bit vague: precisely how should we understand a single time t in two models, and two models ‘agreeing’ on their states at t? The answers to these questions lie in the idea of isomorphism of models, or parts of models; in the usual sense used by logicians. (There is no need for a ‘meta-time’ outside the two models, in terms of which their time series can be compared: thank goodness, since that would be very questionable!). Thus we can speak of an ‘instantaneous slice’ of one model (that is, the part describing the system at a single time) being isomorphic to an instantaneous slice of another model. And similarly, we can speak of isomorphism of ‘final segments’ of two models: that is, isomorphism of parts of two models, each part describing the system at all times future to some time within the model. Determinism is then a matter of isomorphic instantaneous slices implying that the corresponding final segments are isomorphic (where ‘corresponding’ means ‘starting at the time of the instantaneous slice’). That is: we say that a theory is deterministic if, and only if: for any two of its models, if they have instantaneous slices that are isomorphic, then the corresponding final segments are also isomorphic.