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*To*: dinosaur@usc.edu*Subject*: Re: WAS-- Re: Hanson 2006, Mortimer, Baeker response*From*: Phil Bigelow <bigelowp@juno.com>*Date*: Sat, 24 Jun 2006 17:42:31 +0000 (pd)*Reply-to*: bigelowp@juno.com*Sender*: owner-DINOSAUR@usc.edu

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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