hagman
Posts:
1,923
Registered:
1/29/05
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Re: Is it possible that 3+4=8?
Posted:
Jan 24, 2011 2:50 PM
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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 [1] 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
What a perfect quote matching this ;)
hagman
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