in reply to MOPT01  assumptions and spaces
Excellent post, I would like to see more of this
type of analysis here.
I'll try to keep in a positive spirit and offer a few
suggestions.
There are seven basic kinds of information spaces
Lest anyone get bored with having only seven information
spaces, there are plenty more to explore. For example,
there is Quaternions and Rotation Sequences:
A Primer with Applications to Orbits, Aerospace and Virtual Reality
which describes a four dimensional complex space.
I enjoyed reading Flatterland for an
entertaining tour of many geometric spaces.
Or is there some sort of proof that all spaces can
be categorized into one of seven types of information
spaces?
Technically, computers can't handle irrational spaces at all.
Languages like Macsyma
handle irrational numbers nicely. The MACSYMA program:
integrate(sin(x)*exp(x^2),x)
returns
1/4 2 %I x + 1 2 %I x  1
%E SQRT(%PI) (ERF()  ERF())
2 2

4
which seems like a reasonable way to handle the irrationals.
Update:
I forgot about numeric mode in MACSYMA, it does both
numeric and algebraic. As far as algebraic versus
numeric, it's all just LISP as far as I know, which
does both relatively efficiently.
integrate(sin(x)*exp(x^2),x,0,1)
1/4 2 %I + 1 2 %I  1 1/4
+ 1
%E SQRT(%PI) (ERF()  ERF()) %E SQRT(%PI)
+ ERF()
2 2
+ 2
  
+
4 2
End of update
It would be useful also to name a few modules that work in
these different spaces, especially the builtin modules
that fill the gaps where there are no appropriate
keywords. For example, date operations are available in POSIX
support the interval space, and complex space is
supported in Math::Complex.
Thanks for the post! It has inspired me to go try out
Math::Calc::Units.
It should work perfectly the first time!  toma
Re2: MOPT01  assumptions and spaces by mstone (Deacon) on Dec 16, 2002 at 22:00 UTC 
Or is there some sort of proof that all spaces can be categorized into one of seven types of information spaces?
Hardly.. we haven't even touched things like modular spaces, yet.
Don't put too much weight on anything I say as a complete and accurate statement of mathematical truth. I'm chopping things up like mad, to keep from overloading everyday humans with the kinds of details necessary to be mathematical rigorous. At the same time, I'm trying not to stray too far from what a real mathemetician would be willing to accept in casual conversation.
Languages like Macsyma handle irrational numbers nicely.
Hmm.. interesting.
Personally, I'd call that an algebraic package rather than a numerical package per se, but that's really not a point I'd want to argue in any detail. You're right that the formula in question does represent a whole set of irrationals, which just goes to show that you can change the ground rules completely by changing your basic assumptions. ;)
 [reply] 

Don't put too much weight on anything I say
as a complete and accurate statement of
mathematical truth.
I think it would be better to at least attempt to make sure
that your statements are correct,
even if they finesse rigor or completeness.
If there are statements that are just flat wrong, it would
be better to fix them somehow.
Otherwise, the meditation is in danger of being only
psychobabble, and unworthy of being corrected by
those here who know better.
It should work perfectly the first time!  toma
 [reply] 
