On 2011-01-23, Craig Feinstein <firstname.lastname@example.org> wrote: > 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.
Let's write this as 3' + 4' = 8', where the primed numerals represent numbers in pixie-world which don't behave the same way as our numbers. It only takes a moments thought to see that a' = a+1 makes them equivalent: e.g. 3' + 4' = 4 + 5 = 9 = 8'.
So it would be indistinguishable from a world where people just used different names for numbers. The arithmetic structure of natural numbers would be identical.
> 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?
Mathematics is not a science at all, as has been pointed out many times in a recent thread. Your question is founded on a false premise.