Any theory about improvement raises the question: how is the knowledge of how to make that improvement created? Was it already present at the outset?… - David Deutsch

Experience is indeed essential to science, but its role is different from that supposed by empiricism. It is not the source from which theories are derived. Its main use is to choose between theories that have already been guessed. That is what ‘learning from experience’ is.

As I understand it, the claim is that the less you use Homeopathy, the better it works. Sounds plausible to me.

Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The object of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be, mathematical explanation.