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[ontolog-forum] Ontology and Category Theory

To: "ontolog-forum" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Len Yabloko" <lenya@xxxxxxxxxxxxx>
Date: Thu, 29 Jan 2009 19:25:42 +0000
Message-id: <W5711914940210671233257142@webmail17>
Pat,     (01)

Thank you for explaining to me and other unqualified to speak about Categories 
- what they really mean.    (02)

But I am still not on the same page with you, and I suspect I am not alone on 
this forum. I believe the ultimate value of expertize is in making difficult 
things simple.    (03)

>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.    (04)

I feel stupid at this point (and may be I am). But I don't think that 
mathematics in CT in particular were invented to turn ordinary notions into 
schizophrenic image of it. There has to be a level at which identity is ... 
well identity, at least for all involved in conversation. From this level it 
can split infinitely and reach any kind of absurd and degeneration. But we need 
first to establish that point before descending to feeling stupid or 
schizophrenic.     (05)

>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.    (06)

I thought that was one of the objectives of Category theory.    (07)

 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.    (08)

Perhaps, to the same extend as worlds can have nothing to do with meaning.    (09)

>One detailed point, morphisms in category theory aren't operations  
>which can be performed. They are just abstract mathematical mappings.      (010)

I think abstract mathematical mappings are actually real physical mappings in 
your brain at the time of thought. There is no reason for mappings in CT to 
remain abstract while someone applies it to specific objective at hand.    (011)

>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) .    (012)

This is 'degenerate case' of identity, however correct it may be for task at 
hand. And it may not be 'null procedure' for other task.    (013)

>>> 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.    (014)

If you and I are not things that can be in one of categories(I doubt that is 
true), then may be a new category is required to operate on real world things. 
If CT can be stated in common Logic and the later can operate on you and me, 
then how is it possible that CT can't operate on things like you and me?     (015)

>> , 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
>>>>>> object.
>>>>> 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.    (016)

I am not ready to accept the thesis of mathematical sense vs common-sense.    (017)

>> 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.
>>>>> -Ed
>>>> 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.    (018)

I had impression from reading about 'symbolic grounding' as applied to robots 
for instance, that was never achieved to any useful extend. Did any machine 
ever pass Turing test (that would require some form of grounding)?    (019)

>> (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
>>>> grounding.
>>> ------------------------------------------------------------
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>> Len
>IHMC                                     (850)434 8903 or (650)494 3973
>40 South Alcaniz St.           (850)202 4416   office
>Pensacola                            (850)202 4440   fax
>FL 32502                              (850)291 0667   mobile
>phayesAT-SIGNihmc.us       http://www.ihmc.us/users/phayes
>    (020)

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