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From: visser@ph.tn.tudelft.nl (Boudewijn W. Ch. Visser)
Subject: Re: rereRe: The end of god
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References: <36vt2m$g6m@scapa.cs.ualberta.ca> <371epj$8gn@engnews2.Eng.Sun.COM> <1994Oct12.133530.10573@cc.ic.ac.uk> <visser.781973314@galaxy.ph.tn.tudelft.nl> <CxpMF7.E6J@twisto.eng.hou.compaq.com>
Date: Sun, 16 Oct 1994 10:01:44 GMT

ops@houp14.im.Hou.Compaq.com (IM Ops) writes:

>In article <visser.781973314@galaxy.ph.tn.tudelft.nl>, visser@ph.tn.tudelft.nl (Boudewijn W. Ch. Visser) writes:
[some comments about Goedel & incompleteness deleted]

>|> >>However, there are *many* formal systems that *are* provably complete,
>|> >>for example, Euclidean geometry is provably complete, propositional logic
>|> >		^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>|> 
>|> >Are you sure? Is not his 4th axiom that paralell lines never converge (or
>|> >something), which (in real space) is not true and (in theoretical space)
>|> >cannot be proven?
>|> >flame me if I'm wrong...
>|> >-- 
>|> >Piers the Hamster, p.p.williams@ch.ic.ac.uk
>|> 
>|> >"...and I've got this terrible pain in all the diodes down my side..."
>|> 
>|> About the paralell axiom,it is an axiom because it can't be proved !
>|> (if it could be proved from the other (not from an equivalent) axioms,it
>|> wouldn't be necessary.
>|> It is true that the 'real' world is not euclidean.However,the parallel
>|> axiom makes euclidean geometry to be euclidean.If you use some other
>|> axiom about parallel lines ,you end up with non-euclidean geometry.
>|> Consider geometry on a sphere,for instance.It is definitly non euclidean.
>|> No need to flame,or maybe only about 'axiom ... cannot be proven ?'

>I must disagree with you here regarding the status of axioms. Axioms are
>not arbitrarily chosen nor can they be altered for "use". They represent
>fundamental attributes which are self-evident and irrevocable.

>Euclidean geometry merely notices that given certain conditions, a planar
>field, a line and a point not on the line, there will be only one line
>parrallel to the line intersecting the point. It does not specify the
>result in a "saddle" field or a spherical field.

>We need this sort of information, the field, about space in a local
>area to determine the answer to 52nd theorum. This makes in not an
>axiom but a synthetic concept based on the spatial field - which is
>axiomatic (it's existence, that is).

I don't quite see the point you are making.It is true that I was  
not very accurate.Let me try to make things clearer.
Euclids postulates are two sets of five :
the five 'common notions' ,such as `Things which are equal to the same
thing are also equal to one another','If equals are added to equals,the sums
are equal' etc.
and five postulates:'it is possible to draw a straight line from any point
to any other point','a finite straight line can be extended continiously in
a straight line',[..] and the infamous fifth:
if a straight line falling on two straight lines makes the interior angles
on the same side less than two right angles,the two straight lines ,if
produced indefinitly,meet on the side on which are the angles less than
two right angles.

It is no suprise that people found the fifth ugly,and tried to prove it out
of the other 4 (+common notions).
Unfortunatly,this can't be done !
Around the 18th century mathematicians tried to prove this postulate by
showing that is one substited it by something contradictory to it,the 
resulting geometry would be inconsistent.
However,they found that one could build a complete consistent geometry based
on a postulate contradictory to Euclid's fifth.
Indeed axioms can't be arbitrarly chosen,but it is possible to build
consistent geometrys out of axioms contradictory to the ones Euclid gave.
Euclids geometry is an execllent approximation of our world;On very large
scale however,the universe is non-euclidian.

Perhaps we should end this thread,as it seems no longer to fit in the newsgroups
it appears in ?

Boudewijn


-- 
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|Boudewijn Visser       |E-mail:visser@ph.tn.tudelft.nl |finger for | 
|Dep. of Applied Physics,Delft University of Technology |PGP-key    | 
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