Re: Euclids postulates and non-Euclidean geometry

In article <1141779781.332801.72780@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
RCA <rcanand@xxxxxxxxx> wrote:
Thanks all for your great responses. My key takeaway from all your
responses are:

My interest is logical and not historical - most of the literature
seems to imply that the fifth postulate is more questionable in an
absolute and obvious sense, which is what confused me. This discussion
has clarified that it is not obvious.

I found an interesting relevant discussion about this issue at
http://www.cut-the-knot.org/triangle/pythpar/Geometries.shtml

1. There are several nonscientific (historical - Euclid didnt introduce
it for a while, aesthetic - it "looks" uglier) reasons why it turned
out that the fifth postulate was debated more than the others.

Careful: Euclid "introduced" it at the same as all the other
postulates. It just takes him a while before he starts ->using<- it.

2. The negation of the fifth postulate led to practically useful
geometries and thus came to the forefront.

That was more accidental than anything else. But since people had
spent a fair amount of time during antiquity and the Middle Ages
attempting to ->prove<- the fifth postulate, that was the thin end of
the wedge and it was how people started coming up with examples where
the "obvious" wasn't so obvious after all. This would eventually lead
to the Hilbert school and take on logic, that the axioms of geometry
should be as valid if they speak of points, lines, and planes, as if
they speak of beers, tables, and beerhalls.

I also failed to mention Desargues, whose projective geometry also led
the way towards the discoveries of Bolyai et al, in that he showed
that it was possible to consider the postulates and make them say
something else if one 'translated' some of the key undefined terms
like "point" and "line" into some other meaning than the usual.

3. There is active work in logic that investigates negating the other
postulates as well, based on (Bolyai et al's) further abstraction of
denying the common meanings of line and the plane.

There was, certainly. Hilbert clarified Euclid's geometry by
introducing many more axioms (Euclid has a lot of lacunae based on an
incomplete formalization: how do you know the two circles in
Proposition I[?] intersect at a point, for example?), and discussed to
some degree how the negation of certain axioms would lead to
spherical, hyperbolic, and projective geometries, to mention a
few. Whether or not this is an "active" area of study I am in no
position to tell, but I suspect that the ground for the particular
case of (Hilbert's axiomatization of) Euclid's geometry is pretty
well-trodden by now.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... First Postulate: To draw a line from any point to any point. ... Axioms imply the existence of only countably ... so RF geometry is no longer a model. ... FG's claim that no circles ...
    (sci.math)
  • Re: Making money from Java
    ... >> Propositions and the whole of these constitute the Geometry. ... >> system' but to state 'since Euclid's system is used to derive facts about ... >time, Euclid was only aware of this one geometry (because as we now know, ... the parallel line postulate says (or is ...
    (comp.lang.cobol)
  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... up to Euclid and Hilbert: ... First Postulate: To draw a line from any point to any point. ... Axioms imply the existence of only countably ... so RF geometry is no longer a model. ...
    (sci.math)
  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... up to Euclid and Hilbert: ... First Postulate: To draw a line from any point to any point. ... Axioms imply the existence of only countably ... so RF geometry is no longer a model. ...
    (sci.math)
  • Re: Making money from Java
    ... >>> With all due respect, Mr Wong, the Euclid I studied made ... >>> concluded that the sum of the angles of a triangle is equal to the sum ... > Propositions and the whole of these constitute the Geometry. ... the parallel line postulate says (or is ...
    (comp.lang.cobol)