Computation of Probabilities in Causal Models of History

The aim of this paper is to investigate the ascription of probabilities in a causal model of an episode in the history of science. The aim of such a quantitative approach is to allow the implementation of the causal model in a computer, to run simulations. As an example, we look at the beginning of the science of magnetism, “explaining” — in a probabilistic way, in terms of a single causal model — why the field advanced in China but not in Europe (the difference is due to different prior probabilities of certain cultural manifestations). Given the number of years between the occurrences of two causally connected advances X and Y, one proposes a criterion for stipulating the value pY/X of the conditional probability of an advance Y occurring, given X. Next, one must assume a specific form for the cumulative probability function pY/X(t), which we take to be the time integral of an exponential distribution function, as is done in physics of radioactive decay. Rules for calculating the cumulative functions for more than two events are mentioned, involving composition, disjunction and conjunction of causes. We also consider the problems involved in supposing that the appearance of events in time follows an exponential distribution, which are a consequence of the fact that a composition of causes does not follow an exponential distribution, but a “hypoexponential” one. We suggest that a gamma distribution function might more adequately represent the appearance of advances. Why did the history of a scientific field follow a certain path and not another? The answer for this kind of question usually involves the identification of a set of historical causes. Causal relations, however, may be quite complex, and one way of expressing this complexity is by means of probabilistic causal models. The present paper is part of a project for investigating how the history of science may be expressed in terms of such models. The problem to be addressed is how to compute probabilities for an overall process, given the probabilities for intermediate steps, and how to estimate the probabilities of such intermediate steps. Principia, 10(2) (2006), pp. 1–124. Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil. 110 Osvaldo Pessoa Jr. 1. Causal Model for an Episode in the History of Science In a previous work (Pessoa 2005), an overview was presented of the approach to the history of science based on causal models.1 This project started out as an exploration of a method for postulating counterfactual histories of science (Pessoa 2001), and led to the development of a theory of science based on general units of knowledge, which may be called “advances” (or “achievements”, or “contributions”). Advances are passed on from scientist to scientist, and may be seen as “causing” the appearance of other advances. This results in networks which may be analyzed in terms of probabilistic causal models, which are readily encodable in computer language.2 Consider the following representation for the steps leading to the development of a rudimentary form of the magnetic compass (Fig. 1). This is a modification of the model presented in Pessoa (2005), but in which no calculations of overall probabilities were given. In a single diagram, consisting of advances connected by causal relations, one attempts to account for two independent factual histories of the early science of magnetism, those occurring in China and Europe. According to this reconstruction, the difference between the two histories was due mainly to the strong presence of divination techniques in China (Needham 1962). Although such cultural manifestations associated with the lodestone (magnetic ore) were also present (to a lesser extent) in Europe, for instance in Samotracia,3 we have simplified the situation by considering that the prior probabilities for the divination techniques B and E in Europe were zero, while in China they were 1. The path leading to the first magnetic compass, the lodestone spoon compass (F), started from the discovery and exploration of the “lodestone effect” (A) (the mutual attraction of magnetic ore and the attraction of iron to magnetic ore) which occurred both in China and in Europe. However, in China there was a divination technique done with a greased iron needle floating in water (B), which led to a variation involving a floating lodestone needle (C). With this practical arrangement, the discovery that the lodestone needle aligns along the NorthSouth direction (D) was highly probable, and in fact occurred in China around the beginning of the Christian Era, but not in the West. After this discovery, the development of a rudimentary compass (F) was a small step. Causal connections are represented as probabilistic relations, the values of which are a rough estimate of the probability for the occurrence of an effect in a typical reference time interval, in this case taken to be Tref = 400 years. A Principia, 10(2) (2006), pp. 109–124. Computation of Probabilities in Causal Models of History of Science 111 Directive Loadstone property of effect loadstone Floating Loadstone loadstone spoon needle compass Divination by Diviner’s floating needle board A pEur(A) = 1 pChi(A) = 1 HHHH HHj J J J JJ Qs p(D/A,¬C) = .2