Re: WAS-- Re: Hanson 2006, Mortimer, Baeker response

[Phil Bigelow <bigelowp@juno.com>]

On Sun, 25 Jun 2006 02:00:40 +0200 "Andreas Johansson"
<andreasj@gmail.com> writes:
> On 6/24/06, Phil Bigelow <bigelowp@juno.com> wrote:
> >
> >
> > On Sun, 25 Jun 2006 01:25:16 +0200 Andreas Johansson 
> <andreasj@gmail.com>
> > writes:
> > > On 6/23/06, Phil Bigelow <bigelowp@juno.com> wrote:
> >
> > > The test of a scientific theory is if it agreess with 
> observation.
> > > The
> > > test of theorem is if it follows logically from the axioms.
> >
> >
> > While there is a difference, it appears to be rather minor.
> 
> It appears quite major to me. In science, you start with 
> observations
> and see what theory you can concoct to describe them, in maths you
> start with axioms and see what theorems follow. Induction vs
> deduction.
> 
> > One is
> > physical, the other is mental.  Is a mental "test" itself an 
> observation?
> 
> I'm not sure what you're getting at here, but I'd say neither 
> physical
> nor mental tests are observations.


In order to test something, even mentally, you make an "observation" (in
this case a virtual observation, to borrow from computer animation
rhetoric).   "Thought experiments" are a form of mental "observation".
The assumptions that the mind uses in these thought experiments are based
on axioms.  The axioms, themselves, are based on earlier real-world
observations.



>  > >From Webster's Dictionary:
> > Axiom:  a statement that needs no proof because its truth is 
> obvious;
> > self-evident.
> >
> > So, how do mathematicians canonize axioms?  In other words, what 
> is the
> > process involved in determining that a mathematical concept is 
> "obvious"
> > or "self-evident"?
> >
> > If there *is* such a process, then I'll wager it probably involves 
> some
> > form of testing.  Which is not that different than the testing of
> > scientific hypotheses and theories.
> 
> That's not how the word "axiom" is used in mathematics.  You don't
> determine that something is an axiom - you declare it to be so by
> fiat.


>From Websters Dictionary:
Fiat: An order or issued by authority; sanction.

If we exclude those fiats issued by madmen, then there must be a widely
accepted logical rationale that backs up a mathematical fiat (a "declared
axiom").

What, exactly, is this mental process?  Why should it be considered
different from a part of the scientific method?

<pb>
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