Can you name me a few theorems that were discovered without first observing some special cases? In other words, by brute logic: Starting from the known and logically deducing the unknown?
EDIT: As an example, we all know that the quadratic reciprocity law was not discovered in this way.
Jones' index rigidity formula comes to mind. In fact, there was no way to "observe" it in the special cases. It was widely believed that the index could take all values between $1$ and $\infty$- so the theorem came as a surprise. (The theorem states that the index takes values in a discrete series between $1$ and $4$).