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in reply to MOPT-01 - 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)

[download]

returns 
 
                1/4                2 %I x + 1        2 %I x - 1
              %E    SQRT(%PI) (ERF(----------) - ERF(----------))
                                       2                 2
              ---------------------------------------------------
                                       4

[download]

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)

[download]

       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

[download]

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

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Re2: MOPT-01 - 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]

In Section Meditations