Re: Dementia Redux
Edward,
The term "point" was an undefined term in Euclid's
original presentation, and it is still an undefined
term in modern geometry. Therefore, many mathematicians
have associated a very wide range of structures with
the term. In fact, anything that obeys Euclid's axioms
(or Hilbert's or anybody else's) can used as a model
of those axioms.
ED> Points, as you know are not always finite points, but
> frequently are subsets that map into points that belong
> to certain dimension (in particular a function 'add'
> can be decomposed into how many bits it can add, how many
> machine cycles does it take to execute and so on). Also
> the mathematics does not provide methodology to let you
> measure the completeness of a set. Perhaps I should,
> but will not bring Gödel in to that.
In fact, that was Descartes's most brilliant innovation,
but Gödel's theorem isn't involved:
1. Descartes associated a pair of real numbers with
each point on a plane or a triple of real numbers
with each point in space.
2. Then he proved that those pairs satisfied the axioms
of plane geometry and the triples satisfied the
axioms of solid geometry.
The result was analytic geometry. However, it is a mistake
to assume that a point in physical space "really is" a triple
of real numbers. The only thing we know is that to the best
of our ability to measure, points in space "seem to" obey
the same axioms as triples of numbers. The question of how
accurately the axioms hold is an issue to be settled by
experiment, not by mathematics.
And by the way the proof of the one-to-one correlation is
rather simple. Following is a mapping of the points in the
interior of a square of length one on each side into the
interior of a line of length one:
1. Let any point p in the square be represented by a pair
of numbers (x,y) where 0 < x < 1 and 0 < y < 1.
2. Each of the nubmers x and y can be represented by a
decimal point followed by a string of digits:
x = "." x1 x2 x3 ...
y = "." y1 y2 y3 ...
3. Let the points on the line be represented by a decimal
point followed by alternating digits taken from x and y:
"." x1 y1 x2 y2 x3 y3 ...
By generalizing this mapping to numbers of any size, you
can map an infinite 2D plane into an infinite line. And by
using triples, you can generalize it to map an infinite 3D
space into a line.
John