R Hansen says: >For example, if I asked you if two equally sized circles can intersect at more than two points (forgetting the case where they are one and the same) you would, beyond picturing the event in your mind, have to develop some circle theory to reply with some certainty.
I'm certain two equal radius circles can coincide. Voilà!
Its a nice example though. You could "prove" it as Euclid did, or use Descartes great device. Its only your claim, though, that that constitutes the only path to "some certainty".
Regardless, I find this debate always confusing two separate things.  What is the mathematical standard of proof? (Or, certainty)  How do people think while solving mathematical problems?