An answer: Counter-examples are important in Maths.

Anecdote: Oliver Sacks wrote about autistic twins [1] who could not speak or communicate well to people but who could perform prodigious feats of calculation, including factorising large pimes. Sacks sat with them with a table of prime numbers, and swapped some really large ones after which they took him seriously. Sacks took their reaction to his large primes as proof of their arithmetical ability.

The mathematician John Conway (the game of life chap) was quoted on an online forum decades later as saying that Sacks should have suggested some large non-prime numbers as a test of the twins. I'm searching for a reference!

[1] http://math-frolic.blogspot.co.uk/2013/02/prime-savants-fasc...