When I mentioned the unsolvability of quintic to Scott in passing, it grabbed his attention, and he was visibly very curious as to how it could be possible to show that a general quintic polynomial has no solutions in terms of radicals.

Why should this be so surprising to us? I guess I think it's a bit interesting that it starts at the 5th degree rather than elsewhere, but I'm sort of used to seeing such discontinuities. Naive induction doesn't really get much of my faith anymore. Is there anything else to the problem here that I'm not seeing?