by Scientia Salon

This is a paper by Christopher Pincock, a philosopher at Ohio State University, tackling the interesting issue of whether, and in what sense, mathematical explanations are different from causal / empirical ones. Here is the abstract:

This article focuses on a case that expert practitioners count as an explanation: a mathematical account of Plateau’s laws for soap films. I argue that this example falls into a class of explanations that I call abstract explanations. Abstract explanations involve an appeal to a more abstract entity than the state of affairs being explained. I show that the abstract entity need not be causally relevant to the explanandum for its features to be explanatorily relevant. However, it remains unclear how to unify abstract and causal explanations as instances of a single sort of thing. I conclude by examining the implications of the claim that explanations require objective dependence relations. If this claim is accepted, then there are several kinds of objective dependence relations.

And in what follows are some of the highlights from the paper, which we thought were particularly interesting while reading it. We have alerted Prof. Pincock of this, so hopefully he will chime in during the discussion.

In the philosophy of science, discussions of explanation tend to assume that most explanations are causal explanations. Disagreements arise concerning what a cause is and how the explanatorily relevant causes are to be identified. … This methodology more or less guarantees a certain kind of satisfying result: there turns out to be only one kind of explanation. However, the worry remains that this unified category of explanation is more an artifact of the tools used than an underlying unity. … The case study for this article is Plateau’s laws for soap-film surfaces and bubbles. … The power of Plateau’s laws is that they work for any such soap-film system. The frame can be any shape. In fact, there might not be a frame at all. An unconstrained soap film may form a bubble that encloses a given volume. Such a system almost trivially meets Plateau’s laws as it has only one smooth surface. A more interesting case involves more than one bubble. These systems obey Plateau’s laws as well … Taylor’s solution was to develop a new mathematical theory of surfaces that incorporated these complexities. … Prior to Taylor’s ground-breaking work in the 1970s, mathematicians were forced to resort to experiments with soap films to discern the structure of this or that surface. … With these definitions, the key result is that ‘any such configuration of surfaces must of mathematical necessity conform exactly to the three geometric principles stated at the beginning’ (Almgren and Taylor [1976], p. 86), namely, Plateau’s three laws. … We think we have an explanation when we have found a (1) classification of systems using (2) a more abstract entity that is (3) appropriately linked to the phenomenon being explained. Whenever an explanation has these three features I will say that we have an abstract explanation. … An easy way to see that causal dependence is not involved would be by emphasizing the highly mathematical character of the Taylor case. As nobody thinks that causal relations obtain in pure mathematics, we have a non-causal explanation. … Woodward is at pains to allow for explanations via causal generalizations that are not laws. These generalizations may lack the necessity or universal scope that many require laws to have, but yet they can still explain. For Woodward (writing with Hitchcock), these generalizations explain because they say how an object would change under this or that intervention … An attractive aspect of Woodward’s approach to causal explanation is that he can trace the value of explanations to the description of causal dependence relations. We can generalize Woodward’s approach by taking the essential features of abstract explanation to correspond to an abstract dependence relation. That is, we can say that a soap-film surface obeying Plateau’s laws depends on its being an instance of an almost minimal set. As with causal dependence, there can be subtle questions about exactly how we should interpret talk of abstract dependence … Strevens allows important explanatory contributions from more abstract entities like mathematical objects. But he ultimately requires these entities to represent underlying causal processes: ‘The ability of mathematics to represent relations of causal dependence—wherever it comes from—is what qualifies it as an explanatory tool’ … Jackson and Pettit distinguish between a causally efficacious property and a causally relevant property. A causally efficacious property is one whose bearer thereby gains certain causal powers, for example to bring about a certain effect in a given situation. Clearly, one sort of causal explanation would explain that effect by noting the presence of the appropri- ate causally efficacious property. However, Jackson and Pettit insist that not all causally relevant properties are causally efficacious. … Lange is clear that the value of these explanations is that they do something that causal explanations cannot do: ‘these [distinctively mathematical] explanations work not by describing the world’s network of causal relations in particular, but rather by describing the framework inhabited by any possible causal relation’ … Lange discusses the explanation for why a person cannot carry out a certain crossing over the bridges of Konigsberg. The purely mathematical part of the explanation pertains to abstract topological structure (Lange [2013], p. 489). But the rest involves ‘various contingent facts presupposed by the why question that the explanandum answers, such as that the arrangement of bridges and islands is fixed’ … According to Salmon’s classic discussion, conceptions of explanation can be divided into epistemic, modal, and ontic approaches … Few defend an epistemic conception today, primarily because the link between explanation and prediction seems too confining. Salmon himself advocates an ontic conception that ties explanations to features of the world: ‘The ontic conception sees explanations as exhibitions of the ways in which what is to be explained fits into natural patterns or regularities’ (Salmon [1998], p. 320). The more specific ontic approach that Salmon defends is of course a causal approach. … On a modal approach ‘scientific explanations do their jobs by showing that what did happen had to happen’