Pat, (01)
I agree with the following statement: (02)
PH> But as I said earlier, that does not imply that ontologies
> written in logics suffer from a "limitation" regarding their
> ability to describe continuity (which is what Jeffrey's email
> seems to suggest, and to which I was responding.) The point
> is that ontologies work by describing, and logically expressed
> ontologies are not limited to only describing finite or discrete
> structures or phenomena. (03)
But logic is more general than set theory. Whitehead and Russell
used logic plus set theory for their foundations of the first
three volumes of the _Principia Mathematica_. (04)
As I said in my previous note in this thread, Whitehead adopted
mereology for the projected fourth volume on geometry. Other
logicians, including Tarski and Lesniewski, also adopted mereology
for their foundations. And one of their reasons was the difficulty
of dealing with puzzles dealing with continuity -- e.g., the
question of what happens to the midpoint with you break a line
in two (hopefully identical) halves. (05)
Category theory is another choice for foundations of mathematics,
and CT supports different solutions to puzzles about continuity.
But logic can also be used to make statements about CT. (06)
In short, I agree that logic can be used to talk about and reason
about continuity and that any limitations of set theory should
not be considered limitations of logic. (07)
But by this statement, I am *not* claiming that people *think*
in terms of formal logic inside their heads. It is more likely
that the human brain uses various kinds of image-based reasoning,
including multimodal analogical reasoning that relates language-
like patterns and image-like patterns. But that is a separate
issue. (08)
John (09)
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