From: wilson@ctron.com (David Wilson)
Newsgroups: sci.math
Subject: Re: Fundamental Theorems?
Date: 17 Aug 92 18:16:35 GMT
Reply-To: wilson@ctron.com (David Wilson)
Organization: Cabletron Systems Inc

Sorry for the cascade, but

	Take all that follows with a grain of smiley.

In article <19305@nntp_server.ems.cdc.com>, mstemper@ems.cdc.com writes

>In article <4711@balrog.ctron.com>, wilson@web.ctron.com writes:
>|>     Fundamental Theorem of Geometry:
>|>
>|> 	The Independence of the Parallel Postulate:
>|> 
>|> 	The parallel postulate is independent of the axioms of absolute
>|> 	geometry.
>|> 
>|> 	Try to beat this one for historical importance as well as its 
>|> 	bearing on mathematical thought.
>
>Important, yes. Accepted, yes. Theorem, no.
>How would you prove this? Reductio ad absurdum? :->

	To prove this, take a suitable model of the hyperbolic plane
	within the Cartesian plane, say the Cayley-Klein incidence
	plane with suitable interpretation of line, point, etc., and
	proceed to show that this model fulfills the axioms of absolute
	geometry but not the Parallel Postulate.

	What?  This merely shows that the independence of the parallel
	postulate from absolute geometry is only as good as the
	consistency of the real numbers?  Forgive me, as I am one of
	the old school who believes that the real numbers may even be
	more consistent than past policies of the Roman Catholic Church
	regarding the subject.  (At any rate, I lose little sleep worrying
	what the real numbers are going to do before morning comes.)

	What?  We didn't show that the Parallel Postulate is independent
	using Euclid's Axioms?   It seems to me that as the Fundamental
	Theorem of Calculus relies strictly on the Axioms of Calculus,
	it is only proper that the Fundamental Theorem of Geometry
	should be founded exclusively on the Axioms of Geometry.  I'm a
	little rusty, you'll have to refresh me on those...

>|>     Fundamental Theorem of Mathematics:
>|> 
>|> 	The principle of mathematical induction:
>|> 
>|> 	Let P(x) be a proposition about natural number x.  If
>|> 	    1.  P(0)
>|> 	    2.  For all x, P(x) ==> P(x+1)
>|> 	Then P(x) is true for all numbers.
>|> 
>|> 	This principle pervades mathematics.  To understand, appreciate,
>|> 	and use mathematical induction is the entry ticket to mathematics.
>|> 
>|> 	(Bob Silverman points out that mathematical induction is an
>|> 	axiom.  I believe that this makes it no less a theorem).
>
> I thought that axioms were what you used to prove theorems?

    If one regards a theory as simply a collection of statements closed
    over some logic, then there is really no distinction between axiom
    and theorem.  Any axiom set and logic will do as long as we agree
    on all of the conclusions.

    In short, what you must prove depends on what you assume.  One man's
    axiom is another man's theorem.  In my book, an axiom is a statement
    that someone at some time didn't feel compelled to justify.

    Let us first agree on the Axioms of Mathematics before we argue the
    Fundamental Theorem of Mathematics.  Until then, my vote for
    mathematical induction stands.


David W. Wilson (wilson@ctron.com)

Disclaimer: "Truth is just truth...You can't have opinions about truth."
- Peter Schikele, introduction to P.D.Q. Bach's oratorio "The Seasonings."
Newsgroups: sci.math