On 01/23/2011 02:47 PM, Craig Feinstein wrote: > 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?
It all depends, really, on how you define the symbols `3', `+', `4', `7', and even `='.
One of the ways we formalize arithmetic  is via use of Peano's axioms. If we take 0 to be an arbitrary entity--for example, the state of having no balls in a set--and we add in a successor relation S(x)--for example, the state of making a new set of balls by adding another one two it--we can define numbers. Let our `3' be S(S(S(0))) and `4' be S(3) or S(S(S(S(0)))), etc. We can then define this property of addition:
a + 0 := a a + S(b) := S(a + b)
With sufficient application of this rule, we can obtain that `3' `+' `4' = `7', so we have a consistent set of definitions that can model our observed phenomena. We can't necessarily rule out other models there--what's to say that when we combine two sets of balls, they have sex and produce an extra ball which is then immediately removed by invisible ball pixies?
> 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?
We can't exactly prove that the 3 + 4 = 8 isn't true for "reasonable" definitions of those symbols, as "reasonable" can be very subjective, but it is probably the case that you can't create a notion of `+' that has the properties of addition that we're used to (most notably, x + 0 = x) while keeping the other symbols defined the same.
 I am not an expert in formalism, and I'm sure I'll be criticized for my description.
-- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth