Pat, (01)
>
>On Jan 30, 2009, at 3:56 PM, Len Yabloko wrote:
>
>> Mitch,
>>
>>> On Fri, Jan 30, 2009 at 12:36 PM, Len Yabloko <lenya@xxxxxxxxxxxxx>
>>> wrote:
>>>
>>>> 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
>>> are:
>>>
>>> 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?
>> (02)
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.
Following the execise that you kindly provided above I have to show that the
intension composition is associative, 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)
The former requires me to establish such rules of composition for intensions
which would ensure their associativity. 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. (03)
>>
>>>
>>> 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? (04)
Sure. Let me quote from http://www.entcs.org/files/mfps19/83018.pdf (05)
"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." (06)
>
>> , 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. (07)
Yes. In the following paper Robert E. Kent proposed a direct connection
(interestingly enough he starts with references to you and John Sowa)
http://suo.ieee.org/IFF/metalevel/lower/namespace/type-language/version20021205.pdf (08)
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." (09)
In another paper http://www.ontologos.org/Papers/ISKO6/ISKO6.pdf he suggests
following translation (010)
INTUITIVE | MATHEMATICAL
----------+-------------
context | category
passage | functor
invertible| adjoint
sum | coproduct
quotient | ?
fusion | pushout (011)
>
>> What am I missing?
>>
>>>
>>>> I believe this objective to be a paramount to making it useful
>>>> beyond calculation and in the real of reasoning. If I am wrong and
>>>> CT is not attempting to do that, then some other theory should.
>>>> And my observation is that neither classical Logic nor Ontology as
>>>> discipline are adequate to this goal if they can't "nail down' the
>>>> identity.
>>>
>>> There's all sorts of discussion within the ontology community about
>>> identity. Whether the unique name assumption (UNA) holds, inferring
>>> subsumption of one concept by another.
>>>
>>>
>>>> Again, I not talking about absolute and universal identity (I
>>>> don't know what it is), but about sufficient level of
>>>> identification required for business transactions.
>>>
>>> If you're concerned that a particular formalism might be
>>> inappropriate
>>> for business transactions, then CT is definitely it. Even
>>> well-educated and, separately, intelligent people have difficulties
>>> with even boolean logic.
>>>
>>
>> I always thought that you don't need to be a specialist to apply
>> someone else specialty to your work. Isn't it the job of specialist
>> to make it possible?
>
>No. The job of the specialist is to specialize. :-)
>
>Pat
>
>
>> (012)
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