Machine learning and information theory concepts towards an AI Mathematician
This work addresses the problem of enhancing AI's system 2 reasoning abilities for mathematicians and AI researchers, but it is incremental as it builds on existing ideas without presenting new empirical results.
The paper tackles the gap in AI's mathematical reasoning by proposing an information-theoretic approach to guide the discovery of new and interesting conjectures, rather than proving theorems, with the hypothesis that a desirable set of theorems should have a small description length and be close to many provable statements.
The current state-of-the-art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. What could be missing? Can we learn something useful about that gap from how the brains of mathematicians go about their craft? This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities -- which correspond to our intuition and habitual behaviors -- but still lacks something important regarding system 2 abilities -- which include reasoning and robust uncertainty estimation. It takes an information-theoretical posture to ask questions about what constitutes an interesting mathematical statement, which could guide future work in crafting an AI mathematician. The focus is not on proving a given theorem but on discovering new and interesting conjectures. The central hypothesis is that a desirable body of theorems better summarizes the set of all provable statements, for example by having a small description length while at the same time being close (in terms of number of derivation steps) to many provable statements.