Apriori Knowledge in an Era of Computational Opacity: The Role of AI in Mathematical Discovery

Abstract

Can we acquire apriori knowledge of mathematical facts from the outputs of computer programs? People like Burge have argued (correctly in our opinion) that, for example, Appel and Haken acquired apriori knowledge of the Four Color Theorem from their computer program insofar as their program simply automated human forms of mathematical reasoning. However, unlike such programs, we argue that the opacity of modern LLMs and DNNs creates obstacles in obtaining apriori mathematical knowledge from them in similar ways. We claim though that if a proof-checker automating human forms of proof-checking is attached to such machines, then we can obtain apriori mathematical knowledge from them after all, even though the original machines are entirely opaque to us and the proofs they output may not, themselves, be human-surveyable.