Jesse Mazer wrote:
>
>
> > Date: Tue, 9 Jun 2009 12:54:16 -0700
> > From: meekerdb.domain.name.hidden
> > To: everything-list.domain.name.hidden
> > Subject: Re: The seven step-Mathematical preliminaries
> >
>
> > You don't justify definitions. How would you justify Peano's axioms
> as being
> > the "right" ones? You are just confirming my point that you are
> begging the
> > question by assuming there is a set called "the natural numbers"
> that exists
> > independently of it's definition and it satisfies Peano's axioms.
>
> What do you mean by "exists" in this context? What would it mean to
> have a well-defined, non-contradictory definition of some mathematical
> objects, and yet for those mathematical objects not to "exist"?
A good question. But if one talks about some mathematical object, like
the natural numbers, having properties that are unprovable from their
defining set of axioms then it seems that one has assumed some kind of
existence apart from the particular definition. Everybody believes
arithmetic, per Peano's axioms, is consistent, but we know that can't be
proved from Peano's axioms. So it seems we are assigning (or betting
on, as Bruno might say) more existence than is implied by the definition.
When Quentin insists that Peano's axioms are the right ones for the
natural numbers, he is either just making a statement about language
conventions, or he has an idea of the natural numbers that is
independent of the axioms and is saying the axioms pick out the right
set of natural numbers.
Brent
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Received on Tue Jun 09 2009 - 15:22:10 PDT