
Re: Is it possible that 3+4=8?
Posted:
Jan 23, 2011 8:20 PM


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 [1] is via use of Peano's axioms. If we take 0 to be an arbitrary entityfor example, the state of having no balls in a setand we add in a successor relation S(x)for example, the state of making a new set of balls by adding another one two itwe 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 therewhat'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.
[1] 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

