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Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Sun, 1 Feb 2009 20:57:39 -0600
Message-id: <1878749B-FC81-4EB9-A209-732B2C4803AF@xxxxxxx>

On Jan 31, 2009, at 8:01 PM, Len Yabloko wrote:


On Jan 30, 2009, at 3:56 PM, Len Yabloko wrote:


On Fri, Jan 30, 2009 at 12:36 PM, Len Yabloko <lenya@xxxxxxxxxxxxx>  

my impression of CT remains to be as of an attempt to make  
abstract reasoning in general (not only mathematical form of it)  
more precise.

If anything, the executive summaries of logic and category theory  

logic is the study of reasoning.

category theory is the study of transformations.

Reasoning and transformations are closely related to each other,  
even in strict mathematical sense- I believe.

Its hard to say whether this is true or not  until you say what you  
mean more precisely what you mean. There are certainly connections  
between logics and CT, which is hardly surprising given that CT is  
such a very general theory. One connection is that formal reasoning  
systems - more exactly, formal proof systems - form a natural category  
in which the sentences are the objects and the proofs are the  
morphisms: a proof with A as premis and B as conclusion is the  
'mapping' from A to B.  (Exercise for the reader: show that this is a  
category. You have to show that the morphism composition is  
associative and that there is an identity morphism. Hint: its so  
simple you might find it hard to see that there is anything to do.)   
So that is one relationship between reasoning and transformations,  
yes. Is that the one you had in mind?

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.

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. 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. 

Following the execise 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? 

, 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?

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. 

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? 

Sure a gross oversimplification, but I think in the appropriate
direction for each, and it allows meaningful distinction and
comparison. It might be difficult to extract the above from wikipedia
or other easy online sources, but still it's a start.

It is not that difficult to extract since 'morphisms' are at the  
very definition of Category. What is difficult is to understand what  
it is 'appropriate direction' for each. I dawned on me long time ago  
that both complement each other

Can you elaborate on this insight? In what sense do they complement  
each other?

Sure. Let me quote from http://www.entcs.org/files/mfps19/83018.pdf

"Category theory (CT) has provided a unified language for managing concep-
tual complexity in mathematics and computer science." and little later
"...CT mandates that concept structures should be looked at collectively as a whole, with appropriate morphisms relating one individual structure to another. It can be seen as a universal object-oriented language."

, but I still can't figure out why everybody insists on keeping them  
separate (and seemingly as far from each other as possible).

They clearly are separate. They are different topics, with different  
motivations and different methods and topics of interest. It isn't at  
all remarkable that they are separate. They do have connections, cf.  
the above, which have of course been thoroughly explored.

Yes. In the following paper Robert E. Kent proposed a direct connection (interestingly enough he starts with references to you and John Sowa)

John returned the favor in this very relevant thread http://ontolog.cim3.net/forum/ontolog-forum/2007-08/msg00411.html
by saying (I hope you don't me quoting you John)
"For the standard, I believe that we should build on existing
standards, such as the Metadata Registry, and on well defined
mathematical systems.  Robert Kent's IFF system, for example,
has been suggested, and I believe that it would be an excellent
basis.  However, the full details of category theory, etc., are
more than even IT specialists should have to learn."

Kent's idea is the use CT as a metatheory of ontologies. This is very much in line with Goguen and Burstall's work, and other approaches (such as the elaborate system work coming out of Kestrel, which I referred to earlier in this thread) which all use CT to describe very high-level relationships between theories, programs, formal languages, type structures and suchlike things. This is exactly where one would expect to use CT, to describe relationships between complex, abstract (and mathematically well-defined) structures. But this is all going in the opposite direction from the one you suggest. None of these are remotely commonsensical, everyday kinds of thing.  


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