Pat, (01)
I was hoping that you can fill the blanks and help me since math is not my
specialty. But it looks like you are saying that my attempts to apply your
specialty are beyond (or below) help. Let me make one last attempt to explain
what I mean in a plain English. (02)
>
>On Feb 2, 2009, at 8:20 PM, Len Yabloko wrote:
>
>> 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.
>>
>> 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"
>
>I see what they are trying to say, but its not a very good definition.
>The trouble is, the set of possible things is still a kind of
>extension. It would be more accurate to say that the intension of a
>phrase is its meaning or sense (in Frege's terminology) and the
>extension is the set of things which are described by the phrase.
>
>>
>> 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.
>>
>> Put aside relations for a moment. Consider extension of an object to
>> be its physical 4D extent.
>
>OK.
>
>> In absence of any observer the 4D extent of any object is infinitely
>> large
>
>Nonsense. What do extents of objects have to do with observers?
> (03)
How else can you separate things (if you object to 'object'). No more math on
my part - just recall famous phrase which slave philosopher used to say to his
master "...drink the see". In case you forgot that was a condition the master
made, thinking it was impossible. The philosopher then ask the master to
separate all the rivers from the see. (04)
>> (even if has a finite time dimension) because one object can not be
>> separated from the rest of the universe without observer 'drawing'
>> physical boundaries (a.k.a 'perception').
>
>I presume you mean that until an observer distinguishes one thing from
>another, there are no 'objects'. That claim is arguable, but I don't
>agree, myself.
>
>> Now, if you consider a set of observations, you can divide it into
>> any number of sub-sets (aka 'projections').
>
>I really have no idea what you are saying here, but it sounds wrong.
>(Why any number of subsets? Are you presuming that all sets of
>observations are infinite?) (05)
All I am saying is that observers decide which observations are defining the
thing. Since there are many observers only subset of all observations comprises
individual perception. (06)
>
>> 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).
>
>Well, no, its a set of possible objects, not a set of 4-d extents. The
>objects might be abstract, for example (think of the numbers.) (07)
I am talking only about observable objects which is what I suppose 4D extent is. (08)
>
>> Can we now say that these are morphisms:
>>
>> extension <--> projections <--> sign
>
>No. Or at any rate, I see nothing here remotely like a morphism. What
>do your arrows mean?
> (09)
You contradict yourself because 'arrow' is kind of 'morphism'
http://en.wikipedia.org/wiki/Morphism
"morphism is often thought of as an arrow linking an object called the domain
to another object called the codomain" (010)
>>
>>
>> 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.
>
>You havnt said what a projection is, let alone what composition means.
>
>> (011)
'Projection' stands for 'projection morphism'
http://planetmath.org/encyclopedia/ProjectionMorphism.html (012)
>>
>>>
>>>> , 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.
>
>No, they don't. (013)
What is the extent of relation if not a product of 4D extents? (014)
>
>> 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 specifically - pullback, I think)
>
>Again, I completely fail to follow you here. It would help if you had
>a simple example worked out in detail.
>>
>>
>>>> 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?
>
>I have no idea what you mean by 'projection'.
>
>>
See reference above. (015)
>>
>>>
>>>> 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?
>
>Grammars specify wellformedness of syntactic structures, which use
>signs.
>
>> According to general definition - intension as mapping from 4D
>> extents to a sign.
>
>That is not the general definition you cite above. Read it again. (016)
OK. "Intension refers to the possible things a word or phrase could
describe". If possible things are all 4D extents and 'describe' stands for
'map', then intension maps 'phrase' to 4D extents. Right? (017)
>
>> Therefore, composition of signs can be mapped to composition of
>> intensions.
>
>That doesn't follow at all, even if what you just said made sense.
>
>> 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.
>
>I'm afraid you appear to simply have no idea what the words mean. That
>isn't what context-free grammar means.
> (018)
Not exactly perhaps, but close enough for you to guess what I am trying to say.
This is what specialists in different fields have to do to make any progress. (019)
But let me say it in plain English. When we put signs together to communicate
(as I do now) we use some rules of composition (ie grammar). Some grammars are
easier to use for composing and parsing because signs of alphabet do not have
to be in the same order to convey the same information - only totality of signs
in a phrase comprises its intension. Moreover totality of phrases has the same
intension when put in any order. Do you have a better name for such grammar? (020)
>>
>
>Pat
>
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>
>
> (021)
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