> Let's pretend that we live in a world where there are > invisible pixies > running around messing everything up. Let's say you > have 3 balls in > your left hand and 4 balls in your right hand. > Whenever you combine > the balls together, one of these invisible pixies > secretly throws in > another ball, so that whenever you count the balls, > you come up with 8 > balls. So in this pretend world, it is a known fact > that 3+4=8. Anyone > who disputes this fact in our pretend world is > considered either crazy > or stupid. > > In our real world, it is a known fact that 3+4=7. But > how can we be > sure that there are no invisible pixies running > around taking balls > away from us causing us to think that 3+4=7 when > really 3+4=8? > > My point is that mathematics is considered a > deductive science, in > which everything is absolutely certain. But how can > mathematics prove > that the above scenario cannot be true?
or worse : Banach-Tarski paradox where 1 sphere = 2 spheres.