Mathematical Explanations
Joseph Y. Halpern
TL;DR
This work addresses how to formulate explanations for mathematical statements within the Halpern-Pearl causal framework, noting that plain mathematical facts cannot serve as explanations since they hold in all causal models. It introduces the use of impossible possible worlds to represent an agent's epistemic uncertainty about mathematical truths, enabling explanations to be judged and compared. Through concrete examples, such as Fermat's two-squares reasoning for $4373$ and the root of a polynomial $f$ at $x=2$, the paper illustrates how different causal factors can explain mathematical outcomes under this extended framework, and how partial explanations can be evaluated by prior probabilities. The approach provides a principled way to assign explanatory strength to mathematical statements and suggests a generalizable method for integrating mathematics into causal explanations with potential implications for epistemic reasoning and AI systems.
Abstract
A definition of what counts as an explanation of mathematical statement, and when one explanation is better than another, is given. Since all mathematical facts must be true in all causal models, and hence known by an agent, mathematical facts cannot be part of an explanation (under the standard notion of explanation). This problem is solved using impossible possible worlds.