Suppose now that assumptions (1)–(4) hold for a theory T with differential equation E but (5) does not. Then it follows from formulation (A) of the received view and from definition (D) that T is not deterministic. Is there an account of laws of nature which, conjoined with formulation (B) of the received view, renders T deterministic according to (D)?
The set of physically possible worlds of formulation (A) is the union of the set of solutions of E that are deterministic and the set of solutions of E that are not deterministic. In order to achieve determinism the set of physically possible worlds of formulation (B) must be narrower: it may contain all deterministic solutions, but it can not contain more than one solution from any of the ‘non-determinism bouquets’ (from any of the sets of solutions of E that agree in some of their states). We are thus seeking an account of law of nature which crowns E as the representation of the law of possible worlds that are represented by deterministic solutions, but which crowns E as the representation of the law of at most one of the possible worlds that are represented by solutions in a non-determinism bouquet.
What properties may tell apart deterministic and not deterministic solutions? The following difference is clear: for deterministic solutions E is maximally informative in the sense that being supplemented by a state it furnishes all non-nomic facts. E however is not maximally informative in this sense for a not deterministic solution, and in fact may fare pretty badly in narrowing the set of non-nomic facts that may obtain. This difference in informativeness of E for deterministic and not deterministic solutions signals that some variant of the Best Systems account of laws may do the job for us.
An account of laws that exploits the difference in informativeness of E between deterministic and not deterministic solutions may help ensuring determinism of the theory. Suppose that s is a not deterministic solution of E: albeit s satisfies E, E is not necessarily a law of the world represented by s, because there may be an alternative proposition (not necessarily a differential equation) \(L'\) that our account of laws crowns as the law of the world represented by s on the grounds that \(L'\) is more informative of s than E. If such an alternative \(L'\) exists, then s is not physically possible according to the (B) formulation of physical possibility, since the (B) formulation requires a physically possible world to have the same laws as the theory, but E is not a law of s. To generalize, if for any not deterministic solution there existed an alternative proposition that is crowned as the law of said not deterministic solution, then determinism of the theory would be restored. If the account of laws is Best Systems type then the alternative proposition \(L'\) would need to provide a better balance of informativeness and simplicity than does the law represented by E. To sum up, if for any not deterministic solution such an alternative \(L'\) existed, then the Best Systems view, together with assumptions (D), (1)–(4), \(\lnot\)(5) and formulation (B) would entail that the theory T is deterministic, despite T not being deterministic under assumptions (D), (1)–(4), \(\lnot\)(5) and formulation (A).
This problem defines a research project whose success would entail the non-failure of determinism. On the mathematical-physical side of the project we would inquire whether for every not deterministic solution of a physically relevant differential equation we could find another simple proposition that is more informative about this solution than the original differential equation. On the philosophical side of the project we would seek to independently motivate the notions of simplicity and informativeness that characterize said proposition, and on the basis of these notions we would seek the variant of the Best Systems view which crowns this and such propositions as laws of the corresponding not deterministic solutions. We now take a look at the plausibility of success of this research project to save determinism.
Could Our Research Project Plausibly Succeed?
In abstracto the prospects of our research project of saving determinism seem bleak. For a simple example consider John Norton’s Dome as analyzed by Norton [32, 33] and Malament [27]. Imagine a ball resting on the top of a carefully designed Dome-shaped surface. The ball can move frictionlessly, but it is restricted to move on the surface, and is only influenced by a homogeneous gravitational field. Our physical theory T is classical mechanics with Newton’s laws, in particular the second law: \(F = ma\). In order to find how the ball moves we need to solve the initial value problem, where the force F of Newton’s law is determined by the shape of the Dome and by the gravitational field, and where the initial values are the initial position and momentum of the ball. If this were a commonplace problem in classical mechanics we would get a unique solution telling us how the ball moves. However the shape of the Dome is trickily designed so that our initial value problem yields many different solutions: the ball spontaneously starts to roll from the top of the Dome, but classical mechanics can’t tell us when this starting moment happens. In other words, the Dome shows that under the assumptions (1)–(4) property (5) fails in classical mechanics. Given definition (D) and formulation (A) the solutions of the Dome are not deterministic, and hence we get the somewhat counterintuitive conclusion that determinism fails in classical mechanics. Thus for the research project to succeed we ought to find, for each not deterministic solution of the Dome, a proposition that provides a better balance of simplicity and informativeness about the solution than \(F=ma\).
Substituting the force determined by the shape of the Dome and by the gravitational field, Newton’s law in the Dome world becomes
$$\begin{aligned} x^{(2)} = \sqrt{x}, \end{aligned}$$
(1)
where x is the distance of the ball from the top of the Dome on its surface and the \(^{(2)}\) superscript denotes the second derivative with respect to the time parameter. (As the reader can readily check, the family of “Dome solutions”,
$$\begin{aligned} x(t) = \left\{ \begin{array}{ll} 0 &{} ~if~ 0 \le t \le \tau \\ \frac{1}{144} t^4 &{} ~if~ \tau < t, \end{array} \right. \end{aligned}$$
parametrized with the time \(\tau\) when the ball spontaneously starts to roll from the top of the Dome, all solve equation 1 for the same initial values \(x(0)=0\), \(x^{(1)}=0\). A particular choice of \(\tau\) yields a particular Dome solution.)
Albeit equation (1) contains some “ugly” mathematical notation it may strike us as “simple”. Can we really hope to find a proposition which better balances simplicity and informativeness for a particular not deterministic Dome solution than the equation (1) itself?
Well, simplicity is in the eye of the beholder. Contrast (1) with the following differential equation:
$$\begin{aligned} x^{(k)} = 0, \end{aligned}$$
(2)
where k is a yet undetermined number. A charitable reader is going to agree that, at least at a first blush, it is not unreasonable to hold that equation (2) is “simpler” than equation (1). There may be issues with the number k not being “simple” enough, but (2) is homogenous and only makes reference to differentiation, while (1) is inhomogeneous and involves, besides differentiation, the square root of the variable as well.
The solutions of equation (2) are polynomials up to the k-th degree. Polynomials can be used to approximate finite trajectories and thus they can approximately describe the path the ball rolling down on the Dome takes in any particular Dome solution. Mathematically speaking for any finite stretch of a solution s of equation (1) we can find a value for k so that a solution \(s'\) of equation (2) stays within our desired level of approximation to s. By appropriately choosing k number of initial values we can uniquely determine this \(s'\) solution. Thus, assuming our Best System only systematizes a finite lifespan of the Dome universe, equation (2) may count as a proposition of a deductive system which approximates truth, is informative and is simple.
As equation (2) is merely approximately true while equation (1) is exactly true of a Dome solution, an account of laws that crowns (2) as a law for such a solution would need to allow approximate truth to be sufficient for lawhood. The Best Approximately True Systems is such an account. If we accept that equation (2) is simpler than equation (1), it becomes a question of balance whether (2) or (1) should count as a law in a Dome solution according to BATS. As the loss in truth due to the approximation could be made arbitrarily small, if a gain in simplicity and informativeness has an effect on the balance of informativeness, simplicity and approximation to truth, then the overall balance may be improved by a suitable choice of small approximation. In case it turns out that equation (2) provides a better balance than equation (1), we get the conclusion that in formulation (B) of physical possibility a not deterministic Dome solution is not physically possible according to classical mechanics, since its laws are not those of Newton’s. The laws of deterministic solutions of \(F=ma\) would however be still those of Newton’s, since for deterministic solutions we would loose informativeness by moving to an approximation like (2). Thus deterministic solutions are physically possible according to classical mechanics, while not deterministic solutions are not.
This conclusion does not depend on the assumption of having only one instance of the Dome-particle system being present in the world. Taking x to be a 3N-dimensional vector equation (2) applies to a system of N point particles and hence it has solutions approximating any finite lifespans of worlds that contain, say, N number of Dome-particle systems, without losing any simplicity or informativeness. Nevertheless the argument has many shortcomings, notably the assumption that our Best System only systematizes a finite lifespan of the possible world. The gain in simplicity by equation (2) might turn out to be too small to favor one system over the other, and it may even be illusory. The distance measured between trajectories by the supremum norm might not be physically relevant. A further nagging point is that equation (2) requires k number values to determine the solution instead of equation (1)’s 3 (two plus the time when the ball starts to spontaneously roll), their only advantage being that all k of them are initial values. This brings up the question whether informativeness or simplicity of laws should depend upon their ability to get combined with additional accessible informative and simple propositions in order to produce further informative and simple propositions about the world (for an elaboration on this point, see [21]).
In general, instead of attempting to find some abstract scheme to approximate not deterministic solutions of arbitrary differential equations, it seems worthwhile to investigate the relationship of various concrete differential equations in physics. Among the partial differential equations the typical sources of failure of uniqueness are the so-called parabolic and elliptic equations, such as the classical heat equation or the Laplace equation. There is a general sense in which such equations can be approximated by quasilinear first order hyperbolic equations whose initial value problems have a unique solution. Robert Geroch, one of the main authorities on partial differential equations in physics opines that
A case could be made that, at least on a fundamental level, all the “partial differential equations of physics” are hyperbolic – that, e.g. elliptic and parabolic systems arise in all cases as mere approximations of hyperbolic systems. Thus, Poisson’s equation for the electric potential is just a facet of a hyperbolic system, Maxwell’s equations. ([19], pp. 2, 3)
Geroch then proceeds to show that a general symmetrization procedure is available for quasilinear first order hyperbolic systems; for symmetric systems general theorems on existence and uniqueness of solutions are available.
If conversely a not deterministic solution s of the parabolic or elliptic equation E of a physical theory T can be approximated by a deterministic solution \(s'\) of a hyperbolic equation \(E'\) of another physical theory \(T'\) then physics, beyond helping with the problem of rendering the theory T deterministic, may also help to alleviate the philosophical problem of vagueness plaguing the key concepts—informativeness, simplicity, and approximation—of the Best Approximately True Systems account of laws. As \(E'\) represents a law of another physical theory, it is likely going to pass both as simple and informative. Having unique initial value problems, \(E'\) is also likely going to be more informative than E. Moreover, since the main motivation for upholding simplicity and informativeness as defining characteristics of laws is that these seem to be true of the fundamental equations that appear in our physical theories, we may even bypass the problem of having to define what we mean by informativeness and simplicity altogether. The sense in which s and \(s'\) approximate each other is made explicit by the mathematical claim of the approximative relationship itself, and this sense of approximation can also provide the relevant notion of approximation to truth required by the Best Approximately True Systems account of laws.
This analysis is cursory, but it adds yet an additional reason for why thorough investigation of the approximative relationships of different physical theories should further be pursued. The prospects of our research project to save determinism could only be adequately judged after such detailed investigation is brought to fruition.