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Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Len Yabloko" <lenya@xxxxxxxxxxxxx>
Date: Tue, 03 Feb 2009 02:20:06 +0000
Message-id: <W239012377330941233627606@webmail10>
LY>> No. Earlier in this thread 
>>  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 
"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)
>> John returned the favor in this very relevant thread 
>> 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.
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>    (015)

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