  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.
