Is There an Intrinsic Criterion for Causal Lawlike Statements?

Abstract

A scientific mathematical law is causal if and only if it is a process law that contains a time derivative. This is the intrinsic criterion for causal laws we propose. A process is a space-time line along which some properties are conserved or vary. A process law contains a time variable, but only process laws that contain a time derivative are causal laws. An effect is identified with what corresponds to a time derivative of some property or magnitude in a process law, whereas the other terms correspond to the cause(s). According to our criterion, causes are simultaneous with their effects and causality has no temporal direction. Several examples from natural and social disciplines support the applicability of our criterion to all scientific laws. Various objections to our proposal are presented and refuted. The merits our intrinsic theory of causality vis-à-vis the Salmon–Dowe conserved quantity theory are discussed.

Acknowledgements

A previous version of this paper was presented at the annual Philosophy of Science Conference that took place at the Inter-University Centre in Dubrovnik in April 2012. We wish to thank Philippe Chatelain, Augusto Ponce, Howard Sankey, John Sipe, and especially Theo Kuipers and two anonymous referees of this journal for their very useful comments on previous drafts of this paper.

Notes

We make the customary distinction between a lawlike statement and law, the latter being what makes the former true. But, in the paper, we will often use the latter for the former.

This idea was already proposed by Philipp Frank in 1932 in the context of physics (Frank 1997). But our account is more developed and also generalized to other scientific disciplines. We also disagree with Frank on a number of points. For example, unlike him, we take causes and effects to be simultaneous.

The heat flux is not the variation of a property of the system with time, although it (of course) has the same unit as the time derivative of internal energy [Watt = Joule/second], which is the reason for the use of a dotted q to represent it.

Schrödinger's law is deterministic in the following sense. Its integration leads to solutions which allow the calculation of the value of the state function Ψ(t) for any instant of time from the value of the state function at any other instant. Thus, according to the standard interpretation, it is possible to calculate the precise evolution of probabilities of measurement results in time. Since Schrödinger's equation determines the evolution of probabilities in time we could say that it is a deterministic—causal—law which deals with distributions of probabilities. It is not a probabilistic law because the connection between the causes and the effect (the variation in time of the probability distribution) is deterministic. A probabilistic law states a probabilistic link between some events, such as the law that states the probability of decay of an atom of plutonium within some time interval. In our view, such a law is not causal. In order to explain the decay of a particular atom at a particular instant of time, we should be able to describe the internal dynamics of the atom. Needless to say, such an explanation is as of today unavailable and there might be none at all. Probabilistic laws do deliver predictions, but not explanations. On the other hand, a law that links a variation of the value of a probability in time to some causal factors is a causal law. A so-called ‘master equation’ that describes the evolution in time of the probability for a system to be in some discrete state, such as in a Markovian process, can be considered to be a causal law.

We wish to stress that the principle of least action is not mathematically equivalent to the Euler–Lagrange equation (which is causal), but the latter is a consequence of the former. Satisfaction of the Euler–Lagrange equation is equivalent to the existence of a critical or stationary point, which need not be an extremum (minimum or maximum) (Abraham and Marsden 1982, 248). An extremum is necessarily a critical or stationary point, but not conversely. Thus, the satisfaction of the Euler–Lagrange equation is not a sufficient condition for the function of the Lagrangian to have an extremum and, consequently, for the general satisfaction of the principle of least action. We wish to thank our colleague in mathematics, Augusto Ponce, for having clarified this issue to us.

Dowe grounds the (not temporal) direction or asymmetry of a causal process on open conjunctive forks (Dowe 2008, 192ff).