The problem of deciding which definitions are interesting is far from being formal.
Yes, and the problem of deciding which lemmas to prove and in general what structure to try for a proof is also far from begin formal. Verifying a proposed proof is the formal bit.
Does anyone know of work that attempts to build a theorem prover by learning-from-examples? I'm imagining extracting a large corpus of theorems from back issues of mathematical journals, then applying unsupervised structure discovery techniques from machine learning to discover recurring patterns.
Perhaps a model of the "set of theorems that humans tend to produce" would be helpful in proving new theorems.
The unsupervised-structure-discovery bit does seem within the realm of current machine learning.
Any references to related work?