From my limited mathematical experience, it seems as though lots more tough-to-prove conjectures end up being true than being false, e.g. Fermat's last theorem, Poincaré conjecture, etc. (for the full list, see wikipedia). Anyone have any thoughts on this?
For me, this raises interesting questions about how we think of mathematics. Mathematicians are able to come up with amazing conjectures that end up being true, though at the time of inception there was no way to prove them. It was more about intuition than any empirical method of proof.
Sorry if this is really confusing; l'm happy to clarify anything.