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

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

On Sun, 25 Jun 2006 01:25:16 +0200 Andreas Johansson <andreasj@gmail.com>
> 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.

>From Webster's Dictionary:
Axiom:  a statement that needs no proof because its truth is obvious;

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.

-- Andreas Johansson

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