Pat,
>
LY>> No. Earlier in this thread
http://ontolog.cim3.net/forum/ontolog-forum/2009-01/msg00523.html
>> I already proposed category in which extensions are objects and
>> intensions are morphisms.
>
PH>Hmm. Im afraid this simply does not make sense to me. First, we have
>to find out what you mean by 'extension' and 'intension'. In my
>language these are usually used in the adjectival mode, to refer to
>ways of understanding relations. (01)
This is the most general definition I could find
http://en.wikipedia.org/wiki/Intension
"Intension refers to the possible things a word or phrase could describe. It
stands in contradistinction to extension (or denotation), which refers to the
actual things the word or phrase does describe" (02)
PH>The extension of a relation is simply
>a set (of tuples, those of which the relation is true). The
>extensional view of relations identifies a relation with its
>extension; in contrast, the intensional view of relations treats a
>relation is an abstract object sui generis, one which has an
>associated extension but is not identified with it.
>
>Now, in this picture there is no such entity as an intension, speaking
>strictly: there are simply relations, understood intensionally. So I
>am left wondering what you can possibly mean by morphisms being
>intensions. (03)
Put aside relations for a moment. Consider extention of an object to be its
physical 4D extent. In absence of any observer the 4D extent of any object is
infinitely large (even if has a finit time dimension) because one object can
not be separated from the rest of the universe without observer 'drawing'
physical boundaries (a.k.a 'perception'). Now, if you consider a set of
observations, you can divide it into any number of sub-sets (aka
'projections'). On the other hand, according to definition I quoted above
intension is a set of all possible 4D extents mapped to a phrase (or any other
sign). Can we now say that these are morphisms: (04)
extenstion <--> projections <--> sign (05)
I believe this fits the definition of intension which I quoted above. (06)
>
>> Following the exercise that you kindly provided above I have to show
>> that the intension composition is associative
>
>First you have to define it. What is the composition of two intensions? (07)
Can we consider composition of projections above to be a composition of
intensions? I think so. (08)
>
>> , and that there is a special intension serving as identity. The
>> later is simply a definition of identity - taking us to the point
>> where common-sense identity and categorical one meet (as I suggested
>> at the start of this thread)
>
>I fail to see how. In what sense is the common-sense notion of
>identity an intension? What has it got to do with relations in any
>sense at all?
> (09)
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) (010)
>> The former requires me to establish such rules of composition for
>> intensions which would ensure their associativity.
>
>Quite. BUt as a matter of methodology, one would expect that CT would
>be of most value when the morphisms of a proposed category arise
>naturally from some natural motion of mapping, rather than having to
>invent an artificial notion of morphism composition purely in order to
>have something you can call a category. (011)
Do projections qualify as natural transformations? (012)
>
>> Let me borrow from someone else specialty and venture to submit to
>> you that associativity of intensions roughly corresponds to context-
>> free grammar for symbolically grounded language.
>
>You have now completely lost me. Grammars describe syntactic
>structures. What have grammars (context-free or not) have to do with
>composition of mappings?
> (013)
Grammars operate on phrases (more generally signs) - right? According to
general definition - intenstion as mapping from 4D extents to a sign.
Therefore, composition of signs can be mapped to composition of intensions.
Context-free grammar (if I am not mistaken) defines a way to compose signs in
which the order of composition does not change the result. To my
non-mathematical mind it appears to be defining associativity of intensions. (014)
>>
>>
>>>>
>>>>>
>>>>> Sure a gross oversimplification, but I think in the appropriate
>>>>> direction for each, and it allows meaningful distinction and
>>>>> comparison. It might be difficult to extract the above from
>>>>> wikipedia
>>>>> or other easy online sources, but still it's a start.
>>>>>
>>>>
>>>> It is not that difficult to extract since 'morphisms' are at the
>>>> very definition of Category. What is difficult is to understand what
>>>> it is 'appropriate direction' for each. I dawned on me long time ago
>>>> that both complement each other
>>>
>>> Can you elaborate on this insight? In what sense do they complement
>>> each other?
>>
>> Sure. Let me quote from http://www.entcs.org/files/mfps19/83018.pdf
>>
>> "Category theory (CT) has provided a unified language for managing
>> concep-
>> tual complexity in mathematics and computer science." and little later
>> "...CT mandates that concept structures should be looked at
>> collectively as a whole, with appropriate morphisms relating one
>> individual structure to another. It can be seen as a universal
>> object-oriented language."
>>
>>>
>>>> , but I still can't figure out why everybody insists on keeping them
>>>> separate (and seemingly as far from each other as possible).
>>>
>>> They clearly are separate. They are different topics, with different
>>> motivations and different methods and topics of interest. It isn't at
>>> all remarkable that they are separate. They do have connections, cf.
>>> the above, which have of course been thoroughly explored.
>>
>> Yes. In the following paper Robert E. Kent proposed a direct
>> connection (interestingly enough he starts with references to you
>> and John Sowa)
>>
>http://suo.ieee.org/IFF/metalevel/lower/namespace/type-language/version20021205.pdf
>>
>> John returned the favor in this very relevant thread
>http://ontolog.cim3.net/forum/ontolog-forum/2007-08/msg00411.html
>> by saying (I hope you don't me quoting you John)
>> "For the standard, I believe that we should build on existing
>> standards, such as the Metadata Registry, and on well defined
>> mathematical systems. Robert Kent's IFF system, for example,
>> has been suggested, and I believe that it would be an excellent
>> basis. However, the full details of category theory, etc., are
>> more than even IT specialists should have to learn."
>
>Kent's idea is the use CT as a metatheory of ontologies. This is very
>much in line with Goguen and Burstall's work, and other approaches
>(such as the elaborate system work coming out of Kestrel, which I
>referred to earlier in this thread) which all use CT to describe very
>high-level relationships between theories, programs, formal languages,
>type structures and suchlike things. This is exactly where one would
>expect to use CT, to describe relationships between complex, abstract
>(and mathematically well-defined) structures. But this is all going in
>the opposite direction from the one you suggest. None of these are
>remotely commonsensical, everyday kinds of thing.
>
>Pat
>
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>
>
> (015)
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