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From: Immortalist on 29 Apr 2010 22:02
On Apr 24, 7:25 pm, John Jones <jonescard...(a)btinternet.com> wrote:
> Immortalist wrote:
> > On Apr 24, 4:36 pm, John Jones <jonescard...(a)btinternet.com> wrote:
> >> The Peano Axioms define the way that numbers, and hence arithmetic, is
> >> "presented" to us.
>
> >> But my question is this: In such a presentation do the Peano Axioms
> >> "define" arithmetic or do they "limit" arithmetic?
>
> >> The question is important.
>
> >> 1) If the Peano Axioms "define" arithmetic then they are necessary to
> >> arithmetic and, hence, arithmetic is a branch of logic.
> >> 2) If the Peano Axioms "limit" arithmetic then the source of arithmetic
> >> isn't logical or necessary. That is, it is contingent. And the sort of
> >> contingency that that might be isn't immediately fathomable.
>
> >> But I think that there is an alternative to these two options. Any ideas
> >> what that alternative might be? That is, in what relationship does the
> >> Peano Axioms stand to arithmetic?
>
> > If Peano's set of most basic functions of ALL mathematical operations
> > is true then your asking the wrong question.
>
> If the Peano Axioms are necessarily "true" then arithmetic is either
> necessary or contingent.
> So I am asking the right question when I ask which is the case.
>

But-if arithmetic is simply used by almost everyone, for tasks ranging
from simple day-to-day counting to advanced science and business
calculations and involves the study of quantity, especially as the
result of combining numbers, then if Peano Axiom are necessarily
"true" then arithmetic is necessarily "true" and not contingently
true. You see if it we contingently true then sometimes are times when
arithmetic is false and something else is true in its place.

necessary and sufficient or necessary and contingent, fine
distinctions indeed.

> > The question should not
> > be whether Peanos Axioms are necessary or contingent for math
> > operations but instead the proper question would be; what relation
> > math has to the theory of reality. Of course math is an attempt to
> > represent our abstract notions of what reality is with a set of
> > symbols and a group of rules.
>
> Yes

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