in reply to Re^7: Numification of strings
in thread Numification of strings

You went on to say

And if features of the concept of addition are not parallel, the product of the features becomes "less" Cartesian, i.e. one gets "more exceptions from the rule".

so it was clear there was still confusion. I wasn't trying to rub anything in.

and you are always right. :)

Then you missed the comma for decimals thread and the scp thread of the weekend.

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Re^9: Numification of strings
by LanX (Saint) on Aug 03, 2010 at 15:51 UTC
    so it was clear there was still confusion.

    No, no confusion, I have a clear mathematical concept!

    But I don't know how to transport mathematical concepts in a board of perl hackers, thats why I simplified to "less" cartesian and quoted less!

    Let me try:

    Universal Algebra abstracts the concept of "Cartesian product" to direct product. (A vector space is just an example of "an algebra" or variety)

    If an algebra is not a direct product but embeddable in a direct product it's a subdirect product. The dimension of an algebra is the number on non-irreducible factors.

    E.g. the algebra (e.g. a graph or a lattice) representing the state machine for the cases of + and ++ is a subdirect product of the graph representing the cases of "+" multiplied with the rule "++ := +1". Subdirect because of the missing warning.

    Sometimes exceptions will even force you to add an extra dimension, just to model this special case.

    The algebra representing perl in total is a product of the algebras of it's features, the more it differs from a direct product (i.e. the non-embeddable parts missing from sub-direct products of higher dimensions) the "less cartesian" it is. There are different metrics possible to measure the distance between algebras, but breaks in symmetry are always increasing this distance.

    So do you prefer this mathematical approach? :)

    Cheers Rolf

[reply]

      So do you prefer this mathematical approach?

      I first used the math definition, and you said you use the CS definition. Now you're using the math definition.

      Using the math approach, orthogonal is perpendicular. That's not the same as "not parallel". There are parallels between ++ and +=1 (or else you wouldn't have compared them), so talking about their orthoganility is useless.

[reply]
        > ...orthogonal is perpendicular....

        it can't be that difficult ...

        the metaphor is a space spawn by different base vectors which represent features, if ++ and + are not parallel the cartesian product has either an additional dimension or the vectors are not orthogonal anymore.

        But it's just a graphic metaphor, because you can hardly represent a computer language as a primitive n-dimensional vector space.

        Thats why you need a more abstract model from Universal Algebra (which BTW is used in CS and Math) .

        Cheers Rolf

[reply]

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