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Re: [ontolog-forum] Ontology and Category Theory

To: "Len Yabloko" <lenya@xxxxxxxxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Thu, 29 Jan 2009 11:40:19 -0600
Message-id: <D88E6B8D-CFEC-4DE8-8B15-B147F44B1696@xxxxxxx>

On Jan 29, 2009, at 9:37 AM, Len Yabloko wrote:


There is no reliable way in classical Logic to establish and confirm
the identity of any object outside of specific context.

[PH]>I did not comment on this at the time as it didnt seem like this
thread was likely to go anywhere,

Now I am not so sure it was a bug :-)

but it needs to be clarified. First,
classical logic does not even refer to contexts, so let us put that
particular issue to one side for now. So the claim is there is no way
in classical logic to 'establish the identity' of an object. I think
that is correct, but first I want to know what it is supposed to mean.  
What does 'establishing the identity' amount to? Can anyone give an
example of this hypothetical process being carried out successfully?
Suppose I tried, but failed, to establish an identity: how would I
know that I had failed?

In ordinary life their are plenty of examples of 'establishing identity' from simple "hello" to solving and prosecuting crime. But formal meaning of identity in CT is (in my non-mathematical mind) a any procedure that performs identity morphism according to definition of Category (cited below).

Hmm. I feel schizophrenic at this point. We seem to be talking about two different things at the same time. There is the ordinary every-day pre-formal notion of establishing identity, which we do when someone calls us on the phone and we say "who is this?" This means something like "figuring out who or what some unknown thing or person is", where to know "what something is" has never, AFAIK, been fully analyzed by linguistics and never formalized, but seems to mean something like having enough information about a thing to be able to mentally distinguish it from other similar things, or maybe having a description of the thing which is adequate for the purposes of holding a conversation, or some such. OK. But we are also talking about categories, as in category theory, which is an abstract, algebraic, theory of mathematical structures. On the face of it, these two topics would seem to have absolutely nothing to do with one another, so I am having trouble seeing how you subsume them under one heading. 

One detailed point, morphisms in category theory aren't operations which can be performed. They are just abstract mathematical mappings. For example in the Topos category, morphisms are continuous mappings between topological spaces. I don't  think it makes sense to speak of a procedure to perform the identity morphism, therefore. In any case, if there were such a procedure, it would be the null procedure which does nothing, since the whole point of the identity morphism is that it applies to anything and has that same thing as value: written as a function it would be   (lambda (x) x) .

What is correct is that there is no way in pure logic to write axioms
which guarantee that a given name refers to a particular thing (except  
possibly certain very abstract Platonic kinds of 'thing' such as the
property of being an Abelian group, but that is cheating since these
'things' are themselves defined only relative to logically expressible  
axioms.) For example, the fact the the name "Len Yabloko" refers to
you, the actual living breathing Len, cannot be captured by logical
axioms.  This is often referred to as the "grounding problem" in
discussions of knowledge representation in philosophical AI.

However, if this is what you are referring to, Category theory is no
help, as the fact that your name denotes you cannot be specified in
category theory either. In fact, it cannot be specified in any purely
mathematical theory or framework. So I am left wondering if indeed
this is what you are referring to, or whether you are talking about a
different notion altogether.

Again (in my engineering mind) if "I" is defined as object that belongs to some Category (along with "You" and every other participant in this forum)

You and I aren't the kinds of thing that can be in a category, though (what would be the morphisms of this category?). I guess you could say that our unit sets are elements in the category of sets. 

, then performing identity morphism procedure on any 'composition' of objects (such as this forum is) will have the same effect as 'composition' of separately morphed and then composed objects. This (I believe) is what axiom of associativity suggests (please correct me if I am wrong)

No, it just says that morphism composition is associative, which means roughly that if you have a series of them, it doesn't matter which order you think  of them as being done in. LIke addition between numbers. The identity is the one which when you compose it with any other, you get the same one you started with; zero (strictly, the operation of adding zero) is the identity for addition.

In what sense is this forum a composition? 


I understand that not every operator needs to associative. But according to definition of Category there must be some (at least one - identity) morphisms that are associative.

In a category, morphism composition is required to be associative. This doesn't mean that all mappings are associative, however. 

Those and only those define Category by providing 'law' of composition that always preserves identity and other properties entailed.

CT, on the other hand, includes identity in the very definition of  

Citation please.  This sentence means nothing to me.

As stated it means nothing, but I think what was meant is that CT
assumes that all 'objects' (in its highly technical sense of 'object',  
which might be glossed in English as 'mathematical object') have an
identity morphism defined on them. The identity morphism is a
foundational part of each category of objects. However, CT's notion of  
morphism should not be confused with any philosophical or even common-
sense notion of "identity".

I think the intention (not intension) of the term "identity" in CT is the same as in any philosophical or even common-sense notion of "identity".

I don't think so. In mathematics, "identity mapping" simply means the trivial mapping which takes a thing into itself. Another name for it is the 'constant function'. In systems of mappings, where the mappings themselves are the 'objects' (which is what CT is), identity mappings in this sense play the role of zero (for addition) and one (for multiplication) in arithmetic, which are also often called "identities". This usage of the word really has nothing at all to do with its common-sense meaning in "establishing identity" in ordinary language.

However, unlike the later it can be formally proven, and therefore "grounded" in quite ordinary sense of having solid ground under(attached to) it.


But identity is an interesting problem in logical theories, and it is
possible that this bit of the discussion is actually going somewhere.


The direction I would like it to go is actually back to the question  
of theory grounding,

Again, I need to know a lot more about what this is supposed to mean.
I don't think it means what is often referred to as "symbol
grounding", ie how to associate names with their intended denotations.  
(Is it??)

No. My understanding of "symbol grounding" is of attempt to turn symbols into Category with associativity guarantied by whatever grammar is used to compose expressions. This attempt had failed

I don't even know what attempt you are referring to, or what exactly was being attempted. Its easy to define categories of, say, grammars and languages described by grammars. They aren't particularly interesting from a CT point of view, but there are no deep problems in defining them. 

(as fa as I know) for reasons that no grammar serves that purpose (speaking informally)

which I see as the law of identity preservation.

I find this remark wholly opaque.

I hope I made it more transparent in my comments above.

Thanks for trying, but I'm afraid not :-)

Pat Hayes

Pat Hayes

I believe that in this sense CT provides a natural framework for

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