As a matter of fact, there is an interesting book about it: [What We Believe but Cannot Prove](http://en.wikipedia.org/wiki/What_We_Believe_But_Cannot_Prov...).

Just off the top of my head, I'd say the most obvious thing that people know that isn't scientific is to make creative leaps and discover/invent new ideas. Science can tell us a lot about how it happens, but there is no scientific formula for creating new, true, useful, or provocative ideas. When people make a creative leap, do they know it's true, in the scientific sense?

Or how about humor? Can you scientifically prove a joke to be funny? I know what I find funny, but humor is not provable, or created following the scientific method.

Demonstrably false. Going by Steve Martin's autobiography, as just one example, stand-up comics routinely submit their new material to controlled tests with sample audiences. Jokes (and performance details) that fail to elicit laughter and post-show satisfaction (i.e. as practical a demonstration of "funny" as any), are judged provably false and left out of the routine.