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

[Andreas Johansson <andreasj@gmail.com>]

On 6/24/06, Phil Bigelow <bigelowp@juno.com> wrote:
> 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.

An "observation", in the context of the scientific method, is some
kind of record of the external world. A test is the comparison of this
observation to a theoretical prediction, not a test in itself. This is
why I said physical tests aren't observations.

Calling thought experiments "observations" seems to me obfuscatory.

>  The axioms, themselves, are based on earlier real-world
> observations.

Not necessarily.

> >  > >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").

No, there must not. You can conjure up any self-consistent set of
axioms and explore the resultant theorems, and you're doing
mathematics.

What axiomatic system we use depends on what we want to do.

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

It doesn't exist, for a start.

--
Andreas Johansson

Why can't you be a non-conformist just like everybody else?