SUO: *Date 23 Apr 2002 -- Quotient Structures
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John,
I would like to extract the concept of a quotient structure
from our discussion of lattices of theories, as I think that
this is a very important way to think about the relationship
between formal systems of symbols and sentences and so on and
the sorts of objective realities that we all really care about
the most when you really get down to it.
The general idea is this: The equivalence classes that we form
among the elements in one domain corresond to the elements that
exist in another domain.
In our present situation, we have been considering equivalence classes
of syntactic entities and their correspondence to mathematical objects.
For instance, the syntactic entities could be finite axiom sets and the
mathematical objects could be nodes in the objective lattice of theories.
Thinking about things this way helps me to work toward a real implementation,
because it sorts out the things that will be parsed into data structures and
directly manipulated by algorithms from the other sorts of things that exist
outside of the computational, or even the concrete conceptual representations.
Here is the kind of picture that I usually have of the situation:
| Object Domain Syntactic Domain
|
| o-----------o
| o~~~~~~~~~~~~~~~~~~/| s s s ... |\
| / \ / o-----------o \
| / \ / \
| / \ o-----------o \
| o~~~~~~~\~~~~~~~~~~~| s s s ... | \
| \ \ o-----------o \
| \ \ \ o-----------o
| \ o~~~~~~~~~~\~~~~~~~~| s s s ... |
| \ / \ o-----------o
| \ / \ /
| \ / \ o-----------o /
| o~~~~~~~~~~~~~~~~~~\| s s s ... |/
| o-----------o
|
| Figure 1. Objects Inducing A Sign Partition
They are shutting down our power today so I have to run.
Jon Awbrey
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