Seminal Papers in Complexity

Posted by glen ep ropella on
URL: http://friam.383.s1.nabble.com/Seminal-Papers-in-Complexity-tp524047p524103.html

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Michael Agar wrote:
> As described in past posts, that's exactly what I'm trying to figure  
> out--formal math definition doesn't help, metaphorical use too vague.  
> Whatever the solution is, it's likely to be propositional/schematic  
> rather than numeric and involve observer perspective/background  
> knowledge. I'll write more to the list when I think I'm onto a solution.

Formal math definitions do help.  You just can't be myopic about it and
restrict yourself to arithmetic.  Open it up to higher math.

It seems you want to generalize linearity to apply to _other_
composition functions.  The typical definition of linearity applies only
to addition, i.e. f(x+y) != f(x) + f(y).  If you abstract up just a bit,
linearity means "on the same line", which is a way of saying "in the
same space" where the space is 1 dimensional.  It's simply a closure
under addition.

But, there's no reason you couldn't define the same _type_ of thing with
other composition operators.  All you need to do to have an unambiguous
definition of what you mean by "linearity" is to a) define the
composition operator you're talking about and b) define the closure of
that operator.  Of course there are plenty of such constructs already,
they just aren't referred to with the word "linearity".

For example, there's a thing called an "affine plane":  a set, P, of
points, together with a set, L, of subsets of P called lines.  The
points and lines must satisfy the following incidence axioms.

i) any two distinct points lie on exactly one line.
ii) for each line l and point x not on l, there exists a unique line m
containing x and not meeting l.
iii) there exist 3 points not lying on a line.

Basically, this is a topological construct wherein lines don't cross or
jump out of the plane.  It's a kind of closure in that sense.  A
composition where a line _did_ cross another or jump out of the plane
would be non-affine-planar (similar to non-linear).

- --
glen e. p. ropella, 971-219-3846, http://tempusdictum.com
A day an hour of virtuous liberty is worth a whole eternity of bondage.
- -- Cato

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