On Feb 3, 2009, at 9:02 AM, Len Yabloko wrote: Pat,
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.
And I will try to reply pointing out the problems as clearly as I can. But this correspondence has to end at some point.
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 extention 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?
How else can you separate things (if you object to 'object').
Your reply is a non-sequiteur. YOu seem to be confusing the notion of object with the ways we have of identifying objects. It is true, that if there were no observers in the universe, then there would be no divisions made between one object or another, because there would be no divisions made at all. Still, however, the things that we observers call 'objects' would in fact exist in this unobserved universe. They would be there, being objects, even if there was nobody to call them 'object'. Or at any rate, that seems clear to me.
Some philosophers have taken your position (if I follow you here), but IMO its a very hard road to hoe. You seem to have to say that when we observe the universe, our act of observation creates the objects we observe. Not just classifies them as being objects, but actually brings them into existence.
Or you may be intending to refer to what might be called quantum philosophy, arising from the Copenhagen interpretation of what constitutes an observation. If so, please say so clearly as I will then be more able to understand you. 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. (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').
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?)
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.
This sounds very like the Copenhagen idea, where we refuse to talk about the real world at all but only describe observations and their results. An experiment is then defined by the values of a certain number of observations, and we are allowed to speak of the world only by specifying experiments. The theory predicts the likelihood of observational values,, given the description of the experimental apparatus.
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.)
I am talking only about observable objects which is what I suppose 4D extent is.
But CT says nothing about observability, and intensions need not be observable.
Can we now say that these are morphisms:
extenstion <--> projections <--> sign
No. Or at any rate, I see nothing here remotely like a morphism. What
do your arrows mean?
You contradict yourself because 'arrow' is kind of 'morphism'
I don't contradict myself. This use of "arrow" is simply technical jargon in CT. I was using it in the everyday sense to refer to the symbols in your email. Just typing "<-->" does not make something into a category. 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"
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.
'Projection' stands for 'projection morphism' http://planetmath.org/encyclopedia/ProjectionMorphism.html
But then that entirely begs the question. You used the term before you had shown that anything actually was a category, and indeed you appealed to the properties of your projections to argue that your structure was a category. You can't then claim that what you were talking about was a category-theoretic notion in the first place.
, 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.
What is the extent of relation if not a product of 4D extents?
The extension of a relation is a set of tuples of elements from the set over which the relation is defined (often called the 'universe', but don't be misled by that technical term: it can be any nonempty set.) These sets need not even have a topology defined on them, let alone have anything to do with 4-d space.
<snip>
According to general definition - intenstion as mapping from 4D
extents to a sign.
That is not the general definition you cite above. Read it again.
OK. "Intension refers to the possible things a word or phrase could describe".
OK If possible things are all 4D extents
They aren't. and 'describe' stands for 'map'
That is OK, in a sense, but you must distinguish "describe" and "could describe". An interpretation of signs is a mapping from signs to things they mean/denote/signify: OK. But those denoted things are extensions, not intensions, and they are actually described, when the signs are interpreted this way. Other ways of interpreting them will make the signs denote different extensional meanings. In order to make sense of the phrase "could describe", you need to have one thing - the intension - which can stand for any of the things the sign might actually denote, in various interpretations. What this seems to be is a mapping from interpretations to things: what is often described as a mapping from possible worlds to things. So an intension is quite a complicated beast. Its certainly not anything as simple as a 4-d extent.
, then intension maps 'phrase' to 4D extents. Right?
Wrong. But in any case, what you said was that the mapping went the other way, from 4D extents to signs.
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.
Not exactly perhaps, but close enough for you to guess what I am trying to say.
No, really, which is why I gave up at that point. I have absolutely no idea what you are trying to say, but it sounds wrong. Look, grammars describe syntactic structures of sentences, which are linear sequences of signs. Of course the order of the signs is significant and cannot be permuted ad lib: "john loves mary' doesn't mean the same as "mary loves john". Being context-free means that there is some limitation on what a grammar is able to describe, but it still allows a large number of very complex structures to be described, eg virtually all programming languages have context-free grammars. Certainly there is no justification for thinking that the strings of a language with a context-free grammar have any kind of commutative or associative property. This is what specialists in different fields have to do to make any progress.
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).
Right. 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.
Wrong. Moreover totality of phrases has the same intension when put in any order.
Wrong again. Do you have a better name for such grammar?
If it allows phrases to occur in any order, it is a trivial grammar. In EBNF it might be the grammar
sentence ::= word+
which is hardly worth calling a grammar in any but a technical sense.
Pat Hayes
Pat
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