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

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

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

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?