Re: definition of parallel lines

["Frohne, Vickie" <VFrohne@BEN.EDU>]

Actually, the "Parallel postulate" IS a formal mathematical definition. It
was Euclid's genius to recognize that it could not be deduced from the other
four postulates.  See
http://mathworld.wolfram.com/Non-EuclideanGeometry.html  for a brief summary
of the topic. Geometry on a sphere is described in
http://mathworld.wolfram.com/SphericalGeometry.html

It is necessary to recognize that the word "definition" means something
different to mathematicians than it means to the editors of Webster's. The
art of mathematics is to take a small set of ground rules (called axioms,
postulates or definitions), then use logical deduction to build more complex
ideas (theorems and lemmas).  The chain of reasoning leading to a theorem is
called a "proof." Proofs may refer to previous theorems, but every chain of
reasoning must lead all the way back to the postulates. Eventually, the
collection of theorems becomes a large logical structure.  One such
structure is called the "Euclidean Geometry," which is based on only five
postulates.  Changing one of the postulates and building another logical
structure results in a different structure or "geometry."  There are several
distinct non-Euclidean geometries, depending on what set of postulates one
chooses to build from.  In some sense, math is like playing with kid's
building blocks. The types of structures you can build depends on the types
of blocks (postulates, Legos, Tinkertoys) that you start with. Theorems that
are true when based on one set of postulates are often not true if the
postulates are changed. Which is why it's always important to state
assumptions (postulates) very clearly.

Most people on this listserv probably know all of the above already, but
it's a story worth telling to the students. If I had known this "big
picture" sooner, high school geometry would have made a lot more sense, and
I would have been more aware of underlying assumptions in introductory
physics.

Vickie

-----Original Message-----
From: John S. Denker [mailto:jsd@MONMOUTH.COM]
Sent: Thursday, September 05, 2002 5:26 PM
To: PHYS-L@lists.nau.edu
Subject: Re: definition of parallel lines


"Frohne, Vickie" wrote:
> In
> plane geometry, ... there exists only one line
> through a specified point that
> is parallel to a given line.

True, but not a definition.

> ... surface of a globe, ... There are NO parallel lines
> in this space. Every pair of great circles will intersect.

Evidently taking non-intersection to be the definition
of parallel.

That is one possible definition, but not the only possible
definition.  According to my dictionary, possibilities include
 a) non-intersection
 b) everywhere equidistant
 c) locally equidistant
 d) common perpendiculars

... all of which are equivalent in flat space.

Note (c) and (d) hinge on _local_ properties and hence are
favored by physicists.  These also agree with vernacular
usage, and are best in tune with the question that started
this thread.

Note that the well-known "parallels" of lattitude are
equidistant (but not straight) "lines".
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