Determinism and indeterminism

2. Controversy

But this consensus has been badly misleading. First of all, formulations of determinism in terms of causation or predictability are unsatisfactory. And once we use a correct formulation, it turns out that much of classical physics, even much Newtonian physics, is indeterministic; and that parts of relativity theory are indeterministic (owing to singularities). Furthermore, the alleged indeterminism of quantum theory is very controversial – for it enters only, if at all, in quantum theory’s account of measurement processes, an account which remains the most controversial part of the theory.

Formulations of determinism in terms of causation or predictability are unsatisfactory, precisely because of philosophers’ interest in assessing determinism by considering physical theories. That interest means that determinism should be formulated in terms that are clearly related to such theories. But ‘event’, ‘causation’ and ‘prediction’ are vague and controversial notions, and are not used (at least not univocally) in most physical theories. Prediction has the further defect of being an epistemological notion – hence Laplace’s appeal to an ‘intelligence’; while the intuitive idea of determinism concerns the ontology or ‘world-picture’ of a given theory (see Causation; Events).

Fortunately, the intuitive idea of determinism can be formulated quite precisely, without these notions. The key idea is that determinism is a property of a theory. Imagine a theory that ascribes properties to objects of a certain kind, and claims that the sequence through time of any such object’s properties satisfies certain regularities. In physics, such objects are usually called ‘systems’; the properties are called ‘states’; and the regularities are called ‘the laws of the theory’. Then we say that the theory is deterministic if and only if for any two such systems: if they are in exactly the same state as one another at a given time, then according to the theory (for example, its laws about the evolution of states over time), they will at all future times be in the same state as one another. (Montague 1974, pioneered this kind of formulation.)

We can make determinism even more precise in the context of specific physical theories, by using their notions of system, state and law (regularity). But the classification of theories as deterministic or indeterministic is not completely automatic. For the notions of system, and so on, are often not precise in a physical theory as usually formulated. So various different formulations of determinism are often in principle legitimate for a given theory; and there is room for judgment about which formulation is interpretatively best.

However, it is well-established that the main conclusions are as reported above. The philosopher who has done most to classify physical theories in this way is Earman (1986), who upholds these main conclusions. We shall just briefly support these conclusions with two points that his book does not cover. (The details of the above formulation of determinism will not be needed for these points.)

First, much Newtonian physics is indeterministic. Indeed, indeterminism is lurking in the paradigm case discussed by Laplace: point-masses influenced only by their mutual gravitational attraction, as described by Newton’s law of gravitation.

But surely the motion of each point-mass is determined by thus forces on it, in this setting the gravitational force (together with its initial position and velocity)? (See Mechanics, classical §2.) Indeed it is, locally. That is: given the initial positions, velocities and forces, the motion of each point-mass is determined, for some interval of time extending into the future. But it might be a very short interval. (Technically, the equations of the theory have a unique solution for some interval of time, perhaps a very short one.) Furthermore, as time goes on, the interval of time for which there is such a solution might get shorter, shrinking to zero, in such a way that after some period of time, the solution does not exist any more. In effect, determinism might hold locally in time, and yet break down globally.

One way this might happen is by collisions: in general, Newtonian mechanics is silent about what would happen after two or more point-masses collide. But more interestingly, it seems that it might also happen without collisions. Thus it was conjectured in 1897 that one might somehow arrange for one of the point-masses to accelerate in a given spatial direction, ever more rapidly and at so great a rate, during a period of time, in such a way that it does not exist in space at the end of the period! By that time, it has disappeared to ‘spatial infinity’. (The source for the energy needed by the acceleration is the infinite potential well-associated with Newton’s inverse-square law of gravitation.) That this can indeed happen with just Newtonian gravity was finally proved true by Xia in 1992 (using a system of five point-masses). So now we know that, even setting aside collisions, Laplace’s vision of Newtonian determinism is only valid for local intervals of time.

Second, quantum theory can be interpreted as being deterministic. De Broglie and Bohm showed that such an interpretation of elementary quantum theory is possible, despite the alleged proofs that it was impossible (given in the 1930s by some of the discoverers of quantum theory). The basic idea is that a quantum system consists of both a wave and a particle. The wave evolves deterministically over time according to the fundamental equation of quantum theory (the Schrödinger equation) and it determines the particle’s motion, which therefore also moves deterministically, given the wave (hence this interpretation is also called the pilot wave interpretation). This contrasts with the orthodox interpretation. Roughly speaking, the orthodox interpretation accepts only the wave, and accommodates particle-like phenomena by having the wave evolve indeterministically (violating the Schrödinger equation) during processes of measurement (see Quantum mechanics, interpretation of §3; Quantum measurement problem). In recent years, the de Broglie–Bohm approach has been greatly developed so as to yield a deterministic interpretation of more and more of advanced quantum theory, including quantum field theory (see Cushing 1994). Suffice it to say, a deterministic interpretation of quantum theory is entirely coherent.

There remain two other controversial matters; which return us to general metaphysics and philosophy of science. First, should we apply the idea of determinism, as we have formulated it, to theories of the whole universe, that is, cosmologies? If so, then the ‘systems’ in question will be universes or ‘possible worlds’, that is, total possible courses of history. So one will in general not require that the systems whose states one compares must lie in the same possible world (see Possible worlds).

Second, should we accept the idea, for a given kind of system, of a complete theory, a theory that in some sense describes the whole truth about the systems? Some philosophers hold that this idea is incoherent: at least if it is filled out as allowing that such a theory is never formulated by humans; or at least if it allows that humans might be in principle incapable of formulating such a theory (see Scientific realism and antirealism §1). But if we accept some version of this idea, then we can reasonably talk of the systems, or perhaps the kind of system, being deterministic: namely, if and only if the systems’ final theory is deterministic.

So the cautious answer to these questions is No. To answer Yes is to be bold: (some would say, incoherent). In particular, if we answer Yes to both questions, we are in effect accepting that it is meaningful to talk of the whole universe being deterministic. For we are accepting the idea of a complete theory of a given possible world (a total possible course of history). So we can reasonably call this theory the ‘theory of the world’, and its general propositions ‘the laws of nature’ (see Laws, natural §1). (Again, humans are unlikely to have much idea of this theory or its laws.) And then we can say that the given world is deterministic if and only if its theory is. That is, the given world is deterministic if and only if any two worlds, obeying this theory, that have exactly the same state (in the sense of this theory) as one another at a given time also have exactly the same state at all later times.

The rest of this entry is restricted to discussing determinism for given physical theories. It thereby answers No to the second question; and, cosmological theories apart, it also answers No to the first question. But before embarking on this cautious strategy, we should briefly note that, historically, the bold (perhaps incoherent) idea of the entire world being deterministic, irrespective of any theory, has been very important; it has been the focus of countless philosophers’ discussions of determinism (both for and against it).