On 24 Jan., 02:20, Joshua Cranmer <Pidgeo...@verizon.invalid> wrote: > 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.
It may happen that such "sufficient application" leads one to believe after some time of pen-and-pencil work that 1275174 * 8127 = 10363239098 is true, yet it is false (isn't it?) Errare humanum est - what is the smallest number result where we are absolutely sure to do the calculations right?
> -- > Beware of bugs in the above code; I have only proved it correct, not > tried it. -- Donald E. Knuth