Baoyu Dai
University of Wisconsin
Mancosu (2008) defines intra-mathematical explanations (IME) as the use of mathematical facts to explain other mathematical facts. For example, Cantor’s theorem can explain the mathematical fact that there is no largest infinity. Recently, Baron et al. (2020) proposed a counterfactual approach to account for IMEs, which consists of two parts. The first part involves constructing a counterfactual structure for the explanandum and explanans, while the second part focuses on evaluating the counterfactual constructed in the first part. This paper will critically discuss the counterfactual account and then present my own attempt to study the explanatoriness of IMEs.
I argue that the counterfactual approach faces two problems. Firstly, Baron’s evaluation strategy of “altering the mathematical fact in the antecedent” leads to contradictions. Secondly, the counterfactual account analyzes IMEs from an ontic perspective, which, in my opinion, should be avoided because ontic explanations are not genuinely explanatory when compared to epistemic explanations.
After critiquing the counterfactual account, I propose to investigate IMEs from an epistemic perspective. I argue that the explanatoriness of IMEs is sensitive to different agents, and I propose that an IME is explanatory only if the agent accessing the explanation possesses sufficient background knowledge, and the domain of explanation aligns with the agent’s interest domain.
Furthermore, I argue that mathematical practices inform us that different IMEs possess varying degrees of explanatory power. Explanatoriness comes in degrees. To support this point, I introduce three explanatory virtues: simplicity, basicness, and salience. Simplicity concerns the number of premises, basicness relates to the amount of preliminary knowledge required, and salience refers to the ease with which agents understand a given explanation. An IME that exhibits these three virtues will be considered to possess a higher degree of explanatoriness.
Chair: Gabriel Levc
Time: September 8th, 14:00-14:30
Location: SR 1.006
SOPhiA 2026 