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alfredo_at_ncs.es (Alfredo Novoa) wrote in message news:<40cdb9dd.19202972_at_news.wanadoo.es>...
> On Thu, 10 Jun 2004 22:32:19 GMT, "Bill H"
> <wphaskett_at_THISISMUNGEDatt.net> wrote:
>
> >> An axiom is a proposition regarded as self-evidently true without
> >> proof.
> >>
> >> http://mathworld.wolfram.com/Axiom.html
> >
> >I think this definition is too rigid.
>
> No, it is correct.
Yep, it sure is a good statement of the concept.
> >An axiom can easily be thought of as both a self-evident truth (so what's
> >self-evident?)
>
> Absolutely trivial and self contained. You don't need to operate with
> the statement to see that it is true.
>
> For instance here is the fitst of Euclid's postulates:
>
> "A straight line segment can be drawn joining any two points."
>
> This is contained in the line definition. Nothing new.
Axioms are based within a system of thought. For example, Euclid was thinking about planar geometry. Is it possible that if your straight line bent by space could not connect two points in that space? Ah, then you might think, "Well then, it's not a straight line anymore." But from who's perspective? I'm thinking about Einsteinian physics, or even touching on n-dimensional concepts. Axioms just set down rules (and the rules don't have to make 'sense' in the real world) for a logical system. They are not 'true' inherently to the real world. They simply are a base for logical deduction. Although looking back at your post, we may be thinking the same thing.
> >or an assumption to use to base a further analysis. Newton's
> >3 laws of motion are generally referred to as axioms that are used as
> >assumptions (or postulates) for further theoretical analysis.
Referring to this earlier post, I'd say: Newton's laws are not postulates (axioms). They are theorems in physics based upon his original hypotheses. These physical theorems, as far as I know, are different than mathematical theorems, where the former are elucidations about the physical world we perceive, the latter are conclusions derived from the original axioms with certain rules applied to those axioms. Newton's laws, in other words, make bad examples in this discussion about axioms.
>
> It is a very bad use of the terms. Postulates are not assumptions,
> postulates are axioms: truths.
Well said, but, truths in the real world, or within the system? Because I can build any logical system with a set of axioms. They will always be true (if they don't contradict each other) because that's where I started. I made them true, like an act of God. I said, "This is how it is; where do we go from here." IMO, I think that is what the Wolfram definition is stating rather clearly.
>
> Newtos's 3 laws of motion are not evident, self consistent nor true.
Correct.
Todd
"Nothing is True" -- not a Zen koan, but very paradoxically
self-referential
Received on Mon Jun 14 2004 - 16:05:14 CDT