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Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Mitch Harris <maharri@xxxxxxxxx>
Date: Mon, 2 Feb 2009 23:11:46 -0500
Message-id: <553db06d0902022011n1dce802cib5b376bdf13a187f@xxxxxxxxxxxxxx>
On Mon, Feb 2, 2009 at 9:20 PM, Len Yabloko <lenya@xxxxxxxxxxxxx> wrote:    (01)

> Pat,
> 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.
> 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"
> 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.    (02)

I don't know if this helps, but to me 'extensional' refers to
examples, 'intensional' to a rule (which the examples satisfy).    (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').    (04)

OK. That's a pretty counterintuitive stipulation. Something people
call an object is usually separated by (arguably) perceptual
boundaries. And it seems to me that by your idea, there's only one
object the universe.    (05)

> Now, if you consider a set of observations, you can divide it into any number 
>of sub-sets (aka 'projections').    (06)

I think Id just call that a subset. A projection takes an observation
and 'forgets' one or more salient properties, where a subset is just
an arbitrary subset.    (07)

>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:
>  extenstion <--> projections <--> sign
> I believe this fits the definition of intension which I quoted above.
>>> 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?
> Can we consider composition of projections above to be a composition of 
>intensions? I think so.    (08)

- If you're working with subsets, it feels extensional to me (rather
than rule based intensional).    (09)

- I can see composing projections (as specialized subset operation) as
possibly a morphism, but its still kinda vague. Your specification of
projection is not well-defined enough to guarantee associativity (or I
just don't see it).    (010)

- 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?    (011)

>>> , 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?
> 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)    (012)

I think you'd need to specify a whole lot more to really justify that leap.    (013)

>>> 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.
> Do projections qualify as natural transformations?    (014)

- 'natural transformation' is a technical concept. I think Pat was
using 'natural' in the sense 'motivated'    (015)

- I think you'd need to define project much more explicitly and then
see if they satsiy what's necessary.    (016)

>>> 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?
> Grammars operate on phrases (more generally signs) - right?    (017)

strings (sequences) over a given alphabet.    (018)

> According to general definition - intenstion as mapping from 4D extents to a 
>sign.    (019)

What's a 'sign'?    (020)

>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.    (021)

No. Grammars (in the formal language/automata sense) are rule systems
that recognize/generate languages (which are sets of words (which are
sequences of alphabet items)). A sequence is an ordered object, a
member of a semigroup, which is associative, but -not (necessarily)
commutative-.    (022)

The order of composing signs (I'll take it that your word 'sign'
corresponds to 'word')...hm...first what does it mean to compose
signs? I think, 'naturally', that would mean concatenating words. And
order is significant in concatenation. or the order of concatenation
-does- change the result.    (023)

> To my non-mathematical mind it appears to be defining associativity of 
>intensions.    (024)

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?)    (025)

As to 'intension'...maybe you -do- mean some kind of grammar/rule set.
Is that where you're going?    (026)

Mitchell A. Harris
Research Faculty (Instructor in Computer Science)
Department of Radiology
Massachusetts General Hospital/Harvard Medical School    (027)

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