Nick -
So, ummm . . . in a carefully done axiomatization of Euclidean
geometry, the terms "point", "line", "plane" (among others . . .) are
left explicitly *undefined* . . . See, for example, Hilbert's
axiomatization as described here:
http://www.math.umbc.edu/~campbell/Math306Spr02/Axioms/
Hilbert.html
There are very good reasons for leaving terms such as these
explicitly undefined -- this allows a multiplicity of models for a
given axiomatic system . . .
A book I like on a variety of these issues is "Introduction to
Model Theory and Metamathematics" by Abraham Robinson (North Holland
Press, 1965) (warning: this is a real mathematics book, probably not
for the faint of heart . . . :-)
tom
On Jul 14, 2008, at 8:28 PM, Nicholas Thompson wrote:
No! You have gone a bridge to far, unless you are willing to
rewrite the role of definitions in axiom systems.
In a system in which a definition is, "a point is a position in
space lacking dimension"
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
