[FRIAM] not enough of Robert Rosen

Tue Jan 15 11:18:16 EST 2008

Sorry for the delay.  I had several neglected conversations I had to
catch up on.... other communities were blissfully free of my annoying
presence... can't let that happen.

> - what does 'fragile to ambiguity' mean ?

Ambiguity is a pretty large word.  But for this purpose, it means an
unanswerable question to which one needs an answer.  The typical logical
problem would be a situation where neither "true" nor "false" is the
right answer; instead the answer lies somewhere in between.  (This
generalizes to the vernacular sense of the word "ambiguity", but that's
not necessary for now.)

When a system comes to such a problem, it has to "handle" it.  There are
various ways of handling it, of course.  It might just choose randomly.
 It might be endowed with some fuzzy decision making method.  It might
have some heuristics available.  Etc.

If, however, it can't handle it.  I.e. if the system _must_ have an
answer and it can't find a way to coerce the answer into "false" or
"true", that means it's fragile to that particular ambiguous situation.

If there exists even a single ambiguous situation to which a system is
fragile, then it is fragile to ambiguity.

> - what would a 'holarchy of formal systems' look like ? Is't a  
> holarchy a structure where influence is not only top-down but also  
> bottom-up ? And how could any such bi-directionality ever exist in  
> some kind of nesting of formal systems ?

Yes, [de]composition are both complete in a holarchy.  (We do have to
allow that a holarchy might be a platonic ideal.... but let's just
assume they exist in reality for now.)

As we practice math (and programming) now, it seems like we have a
holarchy already.  Whenever we come to an axiom (A_1) or an irreducible
theorem in our current systems, we (humans) can hop outside of that
formal system, build a finer-grained formal system wherein we can derive
A_1 from smaller building blocks.  There doesn't seem to be a limit to
our (human) ability to do that.  Likewise, when we reach an undecidable
proposition (P_1) in any particular formal system (F_1), we can hop
outside F_1 and either modify it by adding P_1 to the axioms to create
F_1' or build a different formal system F_2 wherein we can derive P_1 or
we could build a different formal system F_3 that holds P_1 as an axiom.

Now, when we do such things, A_1 and P_1 have different _meanings_ (by
definition) because different formal systems have different semantic
grounding.  But, somehow, as humans, we can do it and, in doing it,
satisfy ourselves of the reasonableness of A_1 or P_1.  We may not be
able to say A_1 or P_1 are sound (true in reality).  But they are valid
according to their respective formal systems and if those formal systems
seem reasonable, then we can just accept it and get on with our work.

The fact that we, as humans, can engage in such system-hopping, at will,
and that there doesn't seem to be a limit on when/where/how-often we can
do it, leads one to believe that we have access to a formal systems
holarchy.  (If you're a platonist, the holarchy already exists and we
humans just explore it.  If you're a constructivist, then we're
constructing the holarchy as we go along.)

I hope that helps.  Sorry if it doesn't answer your question directly.

- --
glen e. p. ropella, 971-219-3846, http://tempusdictum.com
Never ascribe to malice that which is adequately explained by
incompetence.  -- attributed to Napolean Bonaparte

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