Rob, (01)
JFS>> ... no program can embody any "meaning" that goes
>> beyond or outside what is or can be specified in axioms. (02)
RF> 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: (03)
JFS>> 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. (04)
RF> 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. (05)
That is what I meant: There is nothing in the program or derivable
from the program that is not in or derivable from the axioms. (06)
RF> 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. (07)
That is no different from running a theorem prover on the axioms: (08)
If you have a program for which the halting problem is undecidable,
the program will loop forever. (09)
If you try to determine whether the program will halt by starting
with the axioms, your theorem prover will loop forever. (010)
That makes the two different specifications -- the axiomatic and
the procedural -- identical in meaning (at least for Turing machines
and today's digital computers). (011)
I'll avoid speculating about future systems that might incorporate
quantum computing or non-digital physical or biological processes. (012)
John (013)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (014)
|