Non-Omniscience, Probabilistic Inference, and Metamathematics

We suggest a tractable algorithm for assigning probabilities to sentences of firstorder logic and updating those probabilities on the basis of observations. The core technical difficulty is relaxing the constraints of logical consistency in a way that is appropriate for bounded reasoners, without sacrificing the ability to make useful logical inferences or update correctly on evidence. Using this framework, we discuss formalizations of some issues in the epistemology of mathematics. We show how mathematical theories can be understood as latent structure constraining physical observations, and consequently how realistic observations can provide evidence about abstract mathematical facts. We also discuss the relevance of these ideas to general intelligence.