  How did humanity coax mathematics from the aether? We explore the Platonic
view that mathematics can be discovered from its axioms - a game of conjecture
and proof. We describe Minimo (Mathematics from Intrinsic Motivation): an agent
that jointly learns to pose challenging problems for itself (conjecturing) and
solve them (theorem proving). Given a mathematical domain axiomatized in
dependent type theory, we first combine methods for constrained decoding and
type-directed synthesis to sample valid conjectures from a language model. Our
method guarantees well-formed conjectures by construction, even as we start
with a randomly initialized model. We use the same model to represent a policy
and value function for guiding proof search. Our agent targets generating hard
but provable conjectures - a moving target, since its own theorem proving
ability also improves as it trains. We propose novel methods for hindsight
relabeling on proof search trees to significantly improve the agent's sample
efficiency in both tasks. Experiments on 3 axiomatic domains (propositional
logic, arithmetic and group theory) demonstrate that our agent can bootstrap
from only the axioms, self-improving in generating true and challenging
conjectures and in finding proofs.
