Outlines of a theory of structural explanations

Abstract

This paper argues that in some explanations mathematics are playing an explanatory rather than a representational role, and that this feature unifies many types of non-causal or non-mechanistic explanations that some philosophers of science have been recently exploring under various names (mathematical, topological, etc.). After showing how mathematics can play either a representational or an explanatory role by considering two alternative explanations of a same biological pattern—“Bergmann’s rule”—I offer an example of an explanation where the bulk of the explanatory job is done by a mathematical theorem, and where mechanisms involved in the target systems are not explanatorily relevant. Then I account for the way mathematical properties may function in an explanatory way within an explanation by arguing that some mathematical propositions involving variables non directly referring to the target system features constitute constraints to which a whole class of systems should comply, provided they are describable by a mathematical object concerned by those propositions. According to such “constraint account”, those mathematical facts are directly entailing the explanandum (often a limit regime, a robustness property or a steady state), as a consequence of such constraints. I call those explanations “structural”, because here properties of mathematical structures are accounting for the explanandum; various kinds of mathematical structures (algebraic, graph-theoretical, etc.) thereby define various types of structural explanations.