A Tunes-esq quote...

[RE01 Rice Brian T. EM2]

this is an old post by some months, but i thought that i should comment.

> This comes from the new preface to the 20 anniversary Edition of
> Godel, Escher, Bach. It made me think of Tunes and all our continued
> discussion of reflection as a primary mechanism.
> 
>   ...one thing has to be said straight off: the Godelian strange loop
>   that arises in formal systems in mathematics (i.e., collections of
>   rules for churning out an endless series of mathematical truths solely
>   by mechanical symbol-shunting without any regard to meanings or ideas
>   hidden in the shapes being manipulated) is a loop that allows such a
>   system to "percieve itself", to talk about itself, to be "self-aware",
>   and in a sense it would not be going to far to say that by virtue of
>   having suh a loop, a formal system _acquires a self_.
> 
>      - Douglas Hofstadter
> 
while i'm not sure from this statement what exactly the "strange loop" is, i
do know what the statement identifies: a mathematical model of a theory that
extends below the level of logic.  in other words, it's not just an
inference system, it's the logical activity beneath: the activity of, say,
the processing machine involved.  otherwise, the logical structure would
confer some small meaning on the shapes generated (as is true of a Robinson
diagram or positive diagram in model theory).

the statement suggests that this loop is due to the creation of some shape
with which the processing system may identify. the presence of the
underlying system in its environment suggests self-similarity within the
closure of that system and the environment it supports.  (if we represent
the mechanical processes of these shapes via arrows, we then obtain graphs
which are both infinite and self-similar.)

of course, i suggest something more ambitious: the ability of a system to
modify itself in arbitrary ways suggests that this loop is not unique
(perhaps over time or context-shift).  so, in order to maintain this sort of
reflection, the system's environment structures (or type system, if you
prefer) must contain all of those possible selves (or models of self), or at
least maintain 'self-representation' over context-shifts and
self-modifications.

now you all have another rational for the Arrow design.