Determinism and indeterminism

4. The notion of state

To a philosopher, our definition of determinism looks very formal. And indeed, it is closely related to purely mathematical questions. For a physical theory is often presented as a set of equations, so-called ‘differential equations’, governing how physical magnitudes (for example, numerically measurable quantities like distance, energy and so on) change with time, given their values at an initial time. Our definition then corresponds to such a set of equations having a unique solution for future times, given the values at the initial time; and whether a set of differential equations has a unique solution (for given initial values) is a purely mathematical property of the set.

But we should beware of identifying determinism with this purely mathematical property: there are conceptual, indeed metaphysical, matters behind the mathematics. The reason lies in the notion of state. We have taken states to be simply the properties ascribed by a theory to objects of a certain kind (the theory’s ‘systems’); and so as varying from one theory to another. But there are two general features which the notion of state needs to have if our definition of determinism is to be sure of capturing the intuitive idea. These features are vague, and cannot be formalized: but without them, there is a threat that our definition will be intuitively too weak.

First, states need to be intrinsic properties. It is notoriously hard to say exactly what is meant by ‘intrinsic’, but the idea is to rule out properties which might code information about how the future just happens to go, and thus support a spurious determinism. Thus, to take an everyday example, ‘Fred was mortally wounded at noon’ implies that Fred later dies. But the property ascribed at noon is clearly extrinsic: it ‘looks ahead’ to the future. And so this implication does not show that there is any genuine determinism about the processes that led to Fred’s later death.

Second, states need to be ‘maximal’, that is they need to be the logically strongest consistent properties the theory can express (compatibly with their being intrinsic). For in an intuitively indeterministic theory, there might well be some properties (typically, logically weak ones) such that models agreeing on these properties at one time implies their agreeing on them at all later times.

Do physical theories’ notions of state have these two features? The question is vague because there is no agreed analysis of the ideas of an intrinsic, or a maximal, property. Perhaps ‘maximal’ can be readily enough analysed in terms of logical strength, as just hinted. But it is notoriously hard to analyse ‘intrinsic’. But, by and large, the answer to this question is surely Yes. Physics texts typically define, or gloss, ‘state’ and similar words as a system’s maximal (or ‘complete’) set of intrinsic (or ‘possessed’) properties; and in philosophy, the most commonly cited examples of intrinsic properties are the magnitudes figuring in the states of familiar physical theories, such as mass or electric charge. So it seems there is no widespread problem of spurious satisfactions of determinism.

But although there is not a problem, the need for these features brings out the main point: determinism is not a formal feature of a set of equations. Indeed, there are many examples of a set of differential equations which can be interpreted as a deterministic theory, or as an indeterministic theory, depending on the notion of state used to interpret the equations.

The idea of states as intrinsic also brings out two other points, one philosophical and one technical. The philosophical point concerns the ideas at the end of §2 about laws of nature, and the whole universe (as against a given theory) being deterministic. One of course expects that making sense of the idea of laws of nature will involve the theory of properties. But we now see a more specific point: that making sense of the universe being deterministic will involve the general analysis of ‘intrinsic’.

The technical point concerns theories that treat all the states up to the given time, taken together, as contributing to determining the future states. (There are a few such theories. It does not matter here whether all these earlier states taken together do determine all the future states: as, one might say, whether there is determinism of the future by the whole past.) At first sight, it looks as if such theories add to the usual intrinsic notion of state, a highly extrinsic notion – for which the state at a given time encodes some of the information in all earlier intrinsic states. What is going on?

In fact, in all such theories (so far as this author knows) the extrinsic notion is a technical convenience, rather than a new notion of state. The theory refers to the arbitrarily distant past (typically in some time-integral from minus infinity to the given time) just as a mathematically tractable way of stating information about the state at the given time, information that contributes to the future development of the system. (Wanting to state this information of course reflects the idea of states as maximal.) For example: in statistical physics, some such time-integrals define correlations in the present state; and in theories that study systems interacting with their environment, the past states of the system yield useful information about the present influence of the environment (which is otherwise not represented in the formalism).

Note that this explanation accords with a familiar tenet about causation: that past states influence the future, but only via their influence on the current state – there is thus no ‘action at a temporal distance’. This tenet is widely held by philosophers; and to the extent that one can talk about causation in physical theories, it is upheld in physics. This is especially true of relativity theory; for relativity both unifies space and time, and upholds the principle of contact-action (see Relativity theory, philosophical significance of §3; Spacetime). The tenet is also closely related to a very common property (being Markovian) of probabilistic theories, both in physics and beyond. Indeed, according to some probabilistic theories of causation, the tenet is equivalent to the theory being Markovian.

Theories that refer to past states are relevant to our final topic: the fact (mentioned in §2) that there are some uncontroversial variations on our definition of determinism. So far we have for simplicity assumed that there is a single intuitive idea of determinism: the idea of the present state determining future states. But as we have just seen, there is an analogous idea: that all the states up to the present, taken together, determine future states. It just so happens that (using an intrinsic, maximal notion of state!) this idea is not obeyed in known physical theories: they have no ‘action at a temporal distance’. But that is no reason to deny to the idea the name ‘determinism’; or, more clearly, ‘determinism of the future by the past’ (rather than by the present).

This point is reinforced by other analogous ideas, ideas which are obeyed in known physical theories. Thus in general relativity, and in quantum field theory, diverse technical reasons make it much easier to define a state on an interval of time (called a ‘sandwich’ of spacetime!) than at an instant of time (a ‘slice’ of spacetime). There is no hint here of action at a temporal distance: the interval can be arbitrarily short – it is just that for technical reasons it must have some duration. But such states prompt rather different definitions of determinism, requiring (roughly speaking) that for any interval, no matter how short, states to the future of that interval are determined by the state on it. And these definitions are often satisfied.

One can instead strengthen the definition of determinism, requiring the state at the given time to determine not only future states, but also past states. Many important physical theories have a property called ‘time-symmetry’ or ‘time-reversal invariance’, which implies that they satisfy this stronger definition, if they satisfy our first one. A famous example is Newtonian mechanics. Indeed, it may well be that Laplace had in mind this point (rather than just the intelligence having a memory), when he said in the quotation above ‘as the past’ (see Mechanics, classical; Thermodynamics §§4–5).