choo@cs.yale.edu (young-il choo) (08/02/90)
In article <1990Aug1.152945.12826@ux1.cso.uiuc.edu> morrison@thucydides.cs.uiuc.edu (Vance Morrison) writes:
I also had this idea, (of representing real numbers as functions that
'generate' them). The apeal is of course that finally you have something
that is TRUELY a real number (in the mathamatical sence), not just an
approximation.
...[discussion on the difficulty of "=" for reals as functions] ...
Now this it not to say the method does not have merit, if fact, if anything
I believe it casts doubt on wheter the real number system 'EXISTS' in
any pratical sence. (After all, the only 'real' numbers we EVER deal
with are the ones that we manipulate in some way. These happen to be
EXACTLY those which can be repesented by programs).
I can't explore these issues as much as I would like at present, but
hopefully in the future I will have the opertunity. I would like to
see what properties this new algebra has and how it relates to the
standard real numbers. I would certainly love to hear from anyone
who may have insights in this matter.
Vance
You may want to look into work done on constructive reals. A very good
reference (though requiring some mathematical background) is "Constructive
Analysis" by Bishop and Bridges (Springer-Verlag: 1980, ISBN 0-387-15066-8).
Since testing the equality of two reals is not decidable (in general), there
are weaker (decidable) notions that can be used instead.
As computer technology improves, I can forwee having special "constructive
real" processors, just has we now have math co-processors (which actually only
do finite precision floating point arithmetic).
-- Young-il