Re: SUO: Theory Query
Concerning the differences in theories
mathematical and physical:
[Patrick Cassidy wrote]
>>The way I am using "theory" is in
>>the sense of an ontological theory, which is
>>a "theory of the real world" -- one which
>>addresses the question of which formally
>>expressed concepts and relations
>>best represent the objects and processes of the
>>real world for the purpose of automated reasoning.
>>
>
. . .
>
>>This definition specifically excludes mathematical
>>theories which do not purport to represent objects
>>or events on the real world.
>>
John F. Sowa wrote:
>
> Irrelevant. Your definition merely expresses your
> intentions, but there is nothing in the axioms and
> definitions that can in any distinguish terms that refer
> to the real world from those that are purely mathematical.
>
It's true that nothing in the symbols themselves
distinguishes a mathematical theory from a physical theory
-- from the viewpoint of a theorem prover. But if
we have both mathematical theories (no reference to the
real world) and physical theories (intended to describe the
real world), if both are logically self-consistent, nothing
can prove the mathematical theory wrong but the physical
theory can be proven wrong because the computer can interact
with the real world it is intended to model. This is a
difference which I believe is well worth recognizing.
[John F. Sowa]:
>
> But we must distinguish form (its syntax) and
> content (what it is talking about in the real world).
> From the computer's point of view, it's all mathematical,
> and the computer can't tell what the terms refer to.
>
This isn't completely true, because the computers
aren't totally deaf, dumb, and blind. They have inputs
and outputs which depend on objects in the real world.
If we have a theory of the (complete) syntax
of a natural language, and a disk file is asserted to be
an instance of a sample of that language, the failure of a
sentence in that file to conform to the syntax proves
that the language theory is not true for all sentences
of the language. There are other possible inputs.
A simple light sensor and an assertion that the time
of day can be found at the NIST clock site can prove
or disprove the assertion that the sky is considerably
brighter at noon than at midnight (on average). A
robot with mechanical movement and touch sensors can
test a wide variety of physical theories, many of
which may be perfectly mathematically consistent but
wrong in describing the physical world.
But my focus has been on ways to create a lattice of
physical theories that isolates and minimizes the differences
between the theories. Then the essential distinguishing
points of each theory can be summarized more succinctly
so as to provide an index that will help users find the theories
that they want. I think this would assist in forming the
kind of module metadata hierarchy that Leo was suggesting.
The point I was trying to make in discussing real-world
instances is that the most likely source of contradiction
between theories comes when two theories have the same real-world
object as instances of contradictory assertions. Without the
common real-world objects, the theories might appear
mathematically consistent (e.g. if they have different internal
labels for the real-world object). This point has no
mathematical significance, but I believed that it could help
in focusing on how we might try to recognize and minimize
differences between theories so as to maximize
their interoperability.
[John F. Sowa]:
>
> Bottom line: The lattice of all possible theories will
> include all those that are true of the real world and
> all those that aren't. Distinguishing one from the other
> is a job for science and engineering. It can't be done
> by the computer by itself.
>
Do we really want a lattice of all possible theories, or
just those reasonable enough to be possibly useful? and
preferably testable?
John--
One thing I am still not certain of -- do you prefer
to have an explicit subsumption lattice of theories, or
is this something that you feel can be derived from a
more general representation of the lattice?
Pat Cassidy
=============================================
Patrick Cassidy
MICRA, Inc. || (908) 561-3416
735 Belvidere Ave. || (908) 668-5252 (if no answer)
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internet: cassidy@micra.com
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