Mitch, (01)
>On Monday, February 2, 2009 11:11 PM Mitch Harris wrote
>
>I don't know if this helps, but to me 'extensional' refers to
>examples, 'intensional' to a rule (which the examples satisfy).
> (02)
Yes, I understand the intuitive meaning of rule vs example. But I also believe
that 'intension' and 'extension' do exists objectively  not as platonic
notions. To me the 'extension' is sum of all 4D extents withing the scope of
interaction (you can call it 'perception horizon' or anything you want). It is
not fixed permanently, but must be common for all agents taking part in given
interaction in order for them to establish a 'common ground'.
Intenstions, on the other hand, do not have to be common (and arguably can not
be). It is sufficient for intensions to be grounded in common extension, or in
other words  to be projections of the same extension. (03)
> If you're working with subsets, it feels extensional to me (rather
>than rule based intensional).
>
> I can see composing projections (as specialized subset operation) as
>possibly a morphism, but its still kinda vague. Your specification of
>projection is not welldefined enough to guarantee associativity (or I
>just don't see it).
>
> maybe you're thinking of a given poset as a particular category
>where the elements of the poset are the category objects and the 'less
>than or equal' relation is the morphism?
> (04)
Do you see now what I am thinking? (05)
>
>>>> , and that there is a special intension serving as identity. The
>>>> later is simply a definition of identity  taking us to the point
>>>> where commonsense identity and categorical one meet (as I suggested
>>>> at the start of this thread)
>>>
>>>I fail to see how. In what sense is the commonsense notion of
>>>identity an intension? What has it got to do with relations in any
>>>sense at all?
>>>
>>
>> Relations map one 4D extent to another. But since relation must have a sign
>we need to compose projections and assign resulting composition to a sign.
>This looks like a categorical product (more specificaly  pullback, I think)
>
>I think you'd need to specify a whole lot more to really justify that leap. (06)
Can you help? (07)
>
>> To my nonmathematical mind it appears to be defining associativity of
>intensions.
>
>Associativity is one thing, commutativity is another. Associativity
>means that some compositions can be grouped however you like (as long
>as the sequence/order is preserved). Commutativity means that the
>sequential order doesn't matter. Matrix multiplication is associative
>but not commutative. An operation that is commutative but not
>associative ...is... now that's a hard one. (all I can think of is
>marriage in a family tree. any more mathematical ones that are
>natural?)
>
>As to 'intension'...maybe you do mean some kind of grammar/rule set.
>Is that where you're going?
>
>
>Mitchell A. Harris
>Research Faculty (Instructor in Computer Science)
>Department of Radiology
>Massachusetts General Hospital/Harvard Medical School (08)
> (09)
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