Jaakko Kuorikoski (Associate professor, New Social Research) alustaa:
According to a widely accepted view, explanation is about tracking objective dependency. Many have recently proposed that the general idea is also applicable to various non-causal forms of explanations (e.g., Huneman 2010; Pincock 2016; Reutlinger forthcoming; Saatsi and Pexton 2013), including explanations that many take to be essentially mathematical. In this paper I argue that if we accept the general idea of objective dependence as the basis of explanation, there cannot be mathematical explanations. What appear to be mathematical explanations are either highly abstract mechanistic explanations or reconceptualizations of the explanandum phenomenon in which mathematics as such does not have an explanatory role.
The main problem lies in providing truth conditions for the relevant counterfactuals for mathematical or abstract dependence. This is problematic whether one entertains a realist metaphysics of abstract or mathematical entities or not. On the one hand, realist metaphysics is more congenial to the idea of true ontic dependence between abstract entities, but the epistemology and metaphysics of such dependencies are even murkier than the corresponding problems related to the mere existence of abstract objects. On the other hand, if one is non-realist with respect to abstract objects, then one faces the challenge of explaining how abstract dependencies meet the requirement of specifically ontic dependence.
I argue that in apparent pure cases of mathematical explanation, what is perceived as an explanatory advance with respect to phenomena is better seen as an increase in formal understanding, i.e., an increase in the understanding of our systems of reasoning and representation. Formal understanding can be given a similar, broadly inferentialist, analysis as explanatory understanding in terms of answers to what-if questions, but the crucial difference is that it is about our tools of reasoning, not the world itself.
Jani Hakkarainen firstname.lastname@example.org