My general answer is a lot in pure mathematics, but mathematics in full generality is incredibly hard, so we haven’t even mapped the mathematical territory very well yet.
In scenarios where you actually have to interface with our specific physical/mathematical universe, I believe the answer is 0 times, or close to it, and Einstein cut the search space from thousands to millions of initially valid theories to 4 theories, but he didn’t get to the correct theory 0-shot, and he did have to rely on experiment a little bit:
My general answer is a lot in pure mathematics, but mathematics in full generality is incredibly hard, so we haven’t even mapped the mathematical territory very well yet.
In scenarios where you actually have to interface with our specific physical/mathematical universe, I believe the answer is 0 times, or close to it, and Einstein cut the search space from thousands to millions of initially valid theories to 4 theories, but he didn’t get to the correct theory 0-shot, and he did have to rely on experiment a little bit:
https://www.lesswrong.com/posts/GSBCw94DsxLgDat6r/interpreting-yudkowsky-on-deep-vs-shallow-knowledge#6HPjxMvTnP9JeibXZ