Conceptual (and Hence Mathematical) Explanation, Conceptual Grounding and Proof

Abstract

This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent research in metaphysics on the pair metaphysical grounding-metaphysical explanation, and especially with the literature in philosophy of science on the pair causality-causal explanation.

Mathematical Explanations: An Analysis Via Formal Proofs and Conceptual Complexity

“…seeFine (2012). In this paper we rather think of the notion of conceptual ground, which has been receiving an increasing attention recently, e.g.,Betti (2010),Poggiolesi and Genco (2023),Smithson (2020).3 In this respect our approach is similar to the one ofWilhelm (2021). However, two important differences prevent us from a stricter comparison.…”

“…Mathematical explanations are a special kind of scientific explanation and as such they arguably share the same key components: ground(s) and conclusion, as analogue of cause(s) and effect, but also some type of generalizations. Poggiolesi and Genco (2023) have discussed at length the need for generalizations in mathematical explanations, noting that generalizations are in a mathematical context different from laws; they rather amount to theorems or definitions which connect the ground and the conclusion of the mathematical explanation in question. In the explanatory proof of the Quadrangle Theorem, for example, the generalization at issue corresponds to Theorem (i), which states that any quadrangle can be seen as the sum of two triangles.…”

“…Some exceptions areBetti (2010),Mancosu (1999),Pincock (2015a),Poggiolesi and Genco (2023).2 In this paper, we use as synonymous the words "ground" and "reason." However, we do not take grounding to be a metaphysical relation as it is commonly assumed to be in the contemporary literature, e.g.…”

Scientific explanation as a guide to ground

“…Moreover, even if we were to accept that some mathematical explanations are grounding explanations, we think that the kind of grounding at work in the mathematical case is conceptual or representational rather than worldly. Namely, it deals with connections of semantic priority between propositions or truths due to the (mathematical) concepts involved in them (Poggiolesi & Genco, 2023;Smithson, 2020), rather than facts having individuals, properties, and relations as constituents. For example, a historically significant case of grounding in mathematics is Bolzano's theory.…”

From thin objects to thin concepts?

“…There are two features of the relation of conceptual grounding which have been highlighted recently and which are important for our purposes. On the one hand, several scholars (see, e.g., Poggiolesi & Genco, 2021) have been underlining the key role that mathematics plays for conceptual grounding. In other words, paradigmatic and highly interesting cases of conceptual grounding are examples coming from the mathematical world, namely a world where truths are connected in virtue of the mathematical concepts they contain.…”