The Prospects for a Monist Theory of Non-causal Explanation in Science and Mathematics

Abstract

We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation (CTE) for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics.

Mathematical Explanations in Evolutionary Biology or Naturalism? A Challenge for the Statisticalist

“…In order to reach his goal, Walsh moves from Woodward's interventionist account of explanation, which is usually understood to be able to provide an adequate account of causal explanations, and then he tries to extend this conception to cover noncausal explanations. This strategy has been adopted in recent years by several philosophers who aimed at providing an adequate account of non-causal explanations in different domains (see Ariew et al, 2015, for non-causal explanations in population genetics; Reutlinger, 2016, for non-causal explanations in metaphysics; Baron et al, 2017;2019, for mathematical explanations;Reutlinger et al, 2020, for non-causal explanations in science and pure mathematics). All of those philosophers elaborate on Woodward's counterfactual theory of explanation, according to which explanation "is a matter of exhibiting systematic patterns of counterfactuals dependence" (Woodward, 2003, p. 191).…”

Counter Countermathematical Explanations

“…BCR's (2017, p.7) proposal for evaluating countermathematicals keeps classical logic fixed when varying mathematical facts. After all, the ordinary cases of mathematical explanations, including the instances discussed by BCR (2017BCR ( , 2020, Reutlinger (2016), Reutlinger et al (2020), andWoodward (2018), are generated from the mathematical facts of classical logic. I will discuss what happens to their account if we move to a domain of mathematics based on a contradiction-tolerant logic in the next section.…”

“…If it could be done, we would be on the road to acquire a general theory of explanation in science and mathematics. Generality, some argue, is a virtue that ideally a theory of explanation should satisfy (Nickel, 2010;Reutlinger et al, 2020). Moreover, the success of a counterfactual theory of mathematical explanation would have resounding impacts on the debates about metaphysical explanation, grounding, logical explanation, artificial intelligence explanations, and non-causal explanations more generally (Schaffer 2016;Wilson 2018a,b;Maurin 2019;Baron 2019;Kasirzadeh and Smart 2021).…”

“…Recently, there have been several attempts to liberate counterfactual accounts of explanations from their causal trappings (see for instance, Reutlinger, 2016;Baron et al, 2017;Woodward, 2018;Baron et al, 2020;Reutlinger et al, 2020). Among these, the most elaborate and systematic endeavor to extend the counterfactual theory of causal to mathematical explanations is due to Ripley (2017, 2020), which I will abbreviate henceforth by BCR.…”

“…It should be mentioned that BCR's account, as the most elaborate, is central to the current debate about the viability of a counterfactual account of mathematical explanations. Some of the proponents of countermathematical explanations, such as Reutlinger et al (2020), simply presuppose the validity of BCR's account. Others, such as Woodward (2018), presumably require arguments very much along the lines of BCR to defend their accounts of countermathematical explanations -at least if a mathematical impossibility is supposed to figure in the antecedent.…”