John, (01)
Some comments on your responses to Pat C, which may not relate
directly to Pat C's arguments, but which I think are important. (02)
On Sun, Feb 7, 2010 at 8:14 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
> ...
> PC> I have often (when trying to be careful) used the term "intended
> > meaning" to emphasize that the "meaning" of an ontology term should
> > be what the ontologist intended it to be (unless s/he made a logical
> > error -- which should be detectable by testing).
>
> I agree. But computer algorithms only deal with formal symbols,
> and it's irrelevant whether those symbols came from pure or applied
> math. The computer has no access to what's in the head of the
> programmer or ontologist. (03)
I agree completely. The results showing no theory can explain all
theories apply to manipulations of symbols, and are quite general. (04)
I don't think Pat C's attempt to keep some kind of FO project alive by
limiting its domain has much traction. (05)
But now it gets more interesting... (06)
> ... no program can embody any "meaning" that goes
> beyond or outside what is or can be specified in axioms. (07)
I have to quibble with this. Words are slippery. At best I think
anyone reading it could be easily confused. We have to remember, as
you said: (08)
On Wed, Jan 20, 2010 at 4:57 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote: (09)
"For some pairs of (M,T), the predicate Will_Halt can be
determined by a proof in FOL. But for others, the theorem
prover will loop forever. But any (M,T) that is undecidable
in FOL will be just as undecidable in English or any other
language, formal or informal." (010)
If you define "meaning" to be what is specified by axioms, it may be
technically true that 'no program can embody any "meaning" that goes
beyond or outside what is or can be specified in axioms.' But we have
to remember there are at least some properties of computable processes
which are not specified, other than by running the process itself. (011)
Whether you choose to interpret that to mean computable processes are
limited, or whether you choose to interpret it as I do to mean
computable processes are more powerful, it is undeniable that some
things about programs can only be specified by the program itself. (012)
> ... any meaning that cannot be specified in axioms cannot
> be programmed on a computer ... (013)
Same thing. All the steps of a program my be individually specifiable
with axioms, I grant you, but it is fundamental to the nature of
computation that some aspects of computation, like halting, will not
be known until after the program is run. (014)
I won't go on to argue at length again here what I think that means
for knowledge representation: essentially that a learning process,
because it is a process, may be the only complete representation. But
at the very least I think we need to keep these aspects of the
relationship between what can be expressed in axiomatization and
computation in mind when we discuss what "meaning" can be expressed in
computer programs. (015)
-Rob (016)
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