Re: Category Theory & CSP

Irving:

On May 1, 2006, at 1:06 AM, Peirce Discussion Forum digest wrote:

>  A _category_ is the class of all members of some =
> kind of abstract mathematical entity (sets, groups, rings, fields  
> topologic=
> al spaces, etc.) and all the functions that hold between the class   
> mathema=
> tical entity or structure being studied.

I find category theory to be somewhat of a conundrum.

 From the perspective of language, how is it possible to  
conceptualize both the subject and the copula for a category?

If so defined, would you say that category theory is a sort of sortal  
logic over mathematical objects?  Even metaphorically?

Cheers

Jerry

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Re: Category Theory & CSP

>
> Irving:
>
> On May 1, 2006, at 1:06 AM, Peirce Discussion Forum digest wrote:
>
>>  A _category_ is the class of all members of some =
>> kind of abstract mathematical entity (sets, groups, rings, fields 
>> topologic=
>> al spaces, etc.) and all the functions that hold between the class 
>> mathema=
>> tical entity or structure being studied.
>
> I find category theory to be somewhat of a conundrum.
>
> From the perspective of language, how is it possible to  conceptualize 
> both the subject and the copula for a category?
>
> If so defined, would you say that category theory is a sort of sortal 
> logic over mathematical objects?  Even metaphorically?
>
> Cheers
>
> Jerry

Dear Folks,

Yes, this is what is puzzling me  -- seems that the fundamental rules or 
notions that relate the categories are in effect a definition of the 
categories themselves.  So for me the question becomes as I think Jerry is 

[Fwd: Re: Category theory and C. S. Perce]

List,

I sent John Sowa Irving Arellis' message of 4/30 asking him if he'd 
share his thoughts on category theory in relation to Peirce with the 
list. His remarks appear below.

Gary

Gary,

I would say that the description of category theory by
Irving A. is a reasonable explanation of the subject.

But category theory wasn't invented until about 40 years
after Peirce died.  Therefore, he wasn't aware of it.

On the other hand, I don't think that there's much point in
arguing "whether it can be connected to any part of the work
of Peirce in any significant way?"   He probably would have
approved of it, but so what?

There are other developments, such as DNA and Heisenberg's
uncertainty principle in quantum mechanics, which are much
closer to themes that Peirce had discussed.  Those could be
considered support for his positions, but I'd put category
theory into an area that is compatible with Peirce's views,
but not directly supportive of anything he said in particular.

John

Re: Category Theory & CSP

Re: Category Theory & CSP

Re: Category Theory & CSP

Re: Category Theory & CSP

List,

Comparing the attached two diagrams may assist in beginning a discussion 
of Robert Marty's use of elementary category theory and Rudolph Wille's 
lattices in the lattice structures Marty's developed in relation to 
Peirce's phenomenology.. Marty takes Terry Winograd's simple (but, 
admittedly somewhat confused) diagram and reworks it as a 
concept-lattice. These diagrams are taken from the outline in English 
previously posted and the commentary preceding and following the two 
diagrams will probably need to be read to contextualize the 
transformation of the first into the second.
http://www.univ-perp.fr/see/rch/lts/marty/semantic-ns/default.htm

Gary

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Re: Peirce and Prigogine

Helmut, thanks very much for this (quoted below) -- it definitively
answers the question of how Prigogine and Stengers managed to quote from 
an unpublished Peirce manuscript in their 1984 book.

Somewhat more vague, it seems to me, are the "similarities between his
approach in non-equilibrium thermodynamics esp. the order-creating power
of dissipative structures and Peirce's ideas about the constitutive
relation between chance-processes and the constitution of new habits."
I'm pretty well convinced now that this emphasis on the creative power
of chance, if i may call it that, is what Prigogine and Peirce had in
common. It also seems clear that the *dissipative* (order-destroying,
entropy-producing) aspect of the habit-taking process was *not*
emphasized (or perhaps even mentioned) by Peirce, so in that sense he
did not anticipate later developments in non-equilibrium thermodynamics.

Regarding the rest of this thread -- i'm now back from the road trip
that took me out of the conversation, but it will take awhile for me to
catch up with it and come up with responses (if any are required).

        gary

----- Original Message ----- 
Sent: Saturday, April 29, 2006 9:18 AM

To everybody interested in the Prigogine / Peirce issue:
          in  1981 I visited Prigogine at the universite libre in
Bruxelles. I told about some similarities between his approach in
non-equilibrium thermodynamics esp. the order-creating power of
dissipative structures and Peirce's ideas about the constitutive
relation between chance-processes and the constitution of new habits. I
made accessible to him some of the unpublished MSs, esp. "Design and
Chance". And Prigogine wrote for me a preface to a German edition of
Peirce's writings on cosmology and philosophy of nature (title:
Naturordnung und Zeichenprozess). So from that time onwards he was
pretty well-informed about Peirce's thought.

Helmut Pape
Director
Peirce Edition Project Bamberg

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Re: Category Theory & CSP

Jim Piat a ¨crit :

>  
>  
> Grary Richmond wrote:
>  
> I agree with your assessment of the relational nature of Peirce's 
> categories at least in the sense that at least in 'genuine' 
> trichotomies that each of the three has a relation to the other two. 
> But in another sense your comments seem to me  to perhaps mix apples 
> and oranges.
>  
> As Bernard Morand pointed out in his message of 4/29:
> BM: As regards the relevance to Peirce one has to consider first that 
> the word category in mathematics has nothing to do with the same word 
> as it was used by Aristotle, Kant or Peirce.The mathematical category 
> is an abstract construct which has no denotation nor connotation in 
> itself.
>  
> Dear Gary, Bernard, Folks--
>  
> Thanks for the comments.   I don't know anything about mathematical 
> category theory but I wonder what sort of construct (abract or 
> otherwise) has no denotation nor connotation in itself.  Isn't a 
> construct's location in time/space in effect a self denotation?  And 
> isn't a constructs properties or form its self connotation?    Aren't 
> all constructs defined in terms of either their qualities or 
> locations.  My guess is that these so called mappings, transformations 
> and such of category theory are in some fashion an elaboration of the 
> meaning of such terms as connotation and denotation  -- or 
> alternatively form and location. The ways in which these categories 
> are preserved under various logical, syntactical or 
> mathematical operations.  I don't know the differences among these 
> operations but they seem related to me.
>  

Nice Jim! I had the feeling that I was blundering just at the time of 
writing that the categories in the sense of maths  have no denotation 
nor connotation .  However I could not see where the blunder was. So I 
decided to let the  idea as it was and see what will happen.
The underlying problem is I think the relationship between maths and 
other sciences, the most developed and interesting of them to observe 
being physical sciences. I suspect them to use mathematics as a 
convenient language in order to work physics but not for the very 
mathematical properties of this language. There is only a very basic 
arithmetics in the formula : e=mc2.  J. Chandler suggests a similar 
shortcut in a previous message for chemistry: "Suppose I construct an 
abstract algebra for chemistry / biology that  is not expressible in 
category theory". And this looks to be the problem of the admissibility 
of Gary's vectors too.
In this line of thought, I wanted to convey that mathematical theory of 
categories does not presuppose any arrangement of the real (no 
denotation) nor any purpose for its internal organisation. What would be 
added to this even if we were agreeing that  it is self denoting and 
connoting?
Now, the fact that such mathematical systems really tell something to 
us, and very accurately,  is always a divine surprise  to me.

Regards

Bernard

> In my view, following Peirce, there are three basic categories under 
> which all conceivable modes of being fall:    qualities or form,  
> otherness or location (others must occupy different locations) and the 
>  contrual of the two producing a third which is representation.   I 
> cant quite imagine operations on hypothetical categories that have 
> neither properties nor locations.  Categories whose specific 
> properties and locations are not at issue yes, but not categories 
> absent these relations.
>  
> Ah,  it finally occurs to me that this may be just what you and 
> Bernard mean by abstract categories.  Abstract categores are 
> those whose *particular* connotations and denotations are not at issue 
> -- not categories without qualities or locations per se.  Is this what 
> you mean?  However,  if that is your meaning then I would still 
> argue that the rules establishing how these categories relate to one 
> another are in effect definitions of the general properties of the 
> categories themselves.  And further, that Peirce's categories are 
> abstract or general in just that sense.
>  
> Which is to say that form, substance and function are inseparable 
> relations in the sense of being inextricable aspects of the same 
> thing -- being itself.  They are defined in terms of one another and 
> there is no way around it.  The most fundamental constituents of any 
> system must be all defined in terms of one another (all in terms of 
> all) or else they are not fundamental. 
>  
> I'm not sure how much sense any of this makes, Gary, but I've worked 
> too hard on it to just give it the heave.  So I'm posting it in hopes 
> someone might either agree or point out some problems with it  -- if 
> they have the time and inclination.  Thanks again for interesting and 
> helpful comments.  I too, btw, would like further discussion of Robert 
> Marty's work if others are interested.  I tried to follow it on my own 
> a few years ago but was unable to make much progress and need help.   
>  
>  
> Cheers,
> Jim Piat
>  
>  
>  
>  
>
>>      
>>      
>>      
>>     There has been the beginning of some discussion of category
>>     theory in relation to knowledge representation at ICCS the past
>>     few years and I have noticed that the mathematicians and
>>     logicians who attend the conference ( Bernhard Ganter, John Sowa,
>>     Rudolph Wille, etc.) do not conflate mathematical category theory
>>     with philosophical discussions of categories. In a certain sense
>>     this surprised me as these same folk at first resisted the use of
>>     'vector' to describe 'movement through' a trichotomy of Peircean
>>     categories--for example in evolution, sporting (firstness) leads
>>     to new habit formation (thirdness) leads to a structural change
>>     in an organism (secondness)--and there are both temporal and
>>     purely logical 'vectors' considered by Peirce.  Mathematicians
>>     especially would seem to get quite territorial as regards their
>>     terminology so that even  Parmentier's precedent use of 'vector'
>>     to describe the sort of 'movement' I just described had to be
>>     reinforced by arguments concerning the use of the term in
>>     biology, genetics, medicine, etc. for them to somewhat grudgingly
>>     accept it for trichotomic (as I use it in my trikonic project).
>>     But, again, this is because category theory (perhaps badly named)
>>     has no direct relation to the categories of Kant & Aristotle,
>>     etc. which philosophical categories are, of course, well-known.
>>
>>     But it seems to me that it indeed may be possible to use
>>     mathematical category theory as Marty has done to explicate the
>>     lattice structures underlying certain sign relations (this in a
>>     formal sense). Bernard writes:
>
>>     MB: [Y]ou can use the category theory, not in order to put to the
>>     test the caenopythagorian ones (which would have no sense at
>>     all), but in order to express the formal internal relations
>>     between, say,  the 10 or 66 classes of signs for example. This
>>     has already been done by Robert Marty all along the 400 pages of
>>     his Algebre des signes (John Benjamins Publishing Company, 1990).
>>     He shows there that the formal structure that lies behind the
>>     sign classes is a lattice and he argues for an algebraic approach
>>     in terms of a precisely defined communication language between 
>>     the people involved into this inquiry. I think it is a very
>>     strong point but I will let Robert speak for himself if he is
>>     following the thread.
>
>     I don't recall Marty's concept-lattices being much discussed here
>     and would welcome his participation on the list to explicate them
>     in relation to category theory and as used in semeiotic, etc. My
>     own study of Marty's work has been hampered by my primitive grasp
>     of French (as has my study of Bernard Morand's work for that
>     matter). For those not fluent in  French see. See:
>     *http://www.univ-perp.fr/see/rch/lts/marty/semantic-ns/default.htm
>
>     *Finally, Ben Udell's comment distinguishing set theory from
>     category theory suggest that this might provide an interesting
>     approach to Peirce's ideas concerning continuity, infinitesimals,
>     etc. He wrote:
>
>>     BU: There are, between category theory & set theory, various
>>     differences which sound philosophically interesting. One of them
>>     is that category theory gets away from the idea of everything's
>>     coming down to sets of discrete things.
>
>     I myself have too little knowledge of category theory to know to
>     what extent this is so, but I might have made Ben's comment in
>     relation to mereology rather than category theory. But, again, I'd
>     hope the logicians here might enlighten us in these matters.
>
>     Gary
>
>     Jim Piat wrote:
>
>>
>>
>>>
>>>     Irving:
>>>
>>>     On May 1, 2006, at 1:06 AM, Peirce Discussion Forum digest wrote:
>>>
>>>>      A _category_ is the class of all members of some =
>>>>     kind of abstract mathematical entity (sets, groups, rings,
>>>>     fields topologic=
>>>>     al spaces, etc.) and all the functions that hold between the
>>>>     class mathema=
>>>>     tical entity or structure being studied.
>>>
>>>
>>>     I find category theory to be somewhat of a conundrum.
>>>
>>>     From the perspective of language, how is it possible to 
>>>     conceptualize both the subject and the copula for a category?
>>>
>>>     If so defined, would you say that category theory is a sort of
>>>     sortal logic over mathematical objects?  Even metaphorically?
>>>
>>>     Cheers
>>>
>>>     Jerry
>>
>>
>>     Dear Folks,
>>
>>     Yes, this is what is puzzling me  -- seems that the fundamental
>>     rules or notions that relate the categories are in effect a
>>     definition of the categories themselves.  So for me the question
>>     becomes as I think Jerry is asking  -- how do we have both
>>     entities and relations.  Seems to me that one or the other is not
>>     fundamental.  I think the Piercean approach that all being is
>>     merely relations is more satisfying.  Some of these relations (of
>>     relations) we relate to as objects, collateral objects, etc. The
>>     fundamental categories are themselves relations.  I take that to
>>     be one of Peirce's main contributions to the theory of categories.
>>
>>     Sort of . . .
>>
>>     Cheers,
>>     Jim Piat
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Re: Prigogine, Rosen, Pattee, category theory and biosemiotics

In his message of April 22, Jerry LR Chandler wrote in part:

[[ Finally, I would note a critical logical and philosophical
distinction.  Robert Rosen and Howard Pattee (a biophysicist) reject the
Prigogine approach and developed a school of writings that uses the
mathematics of category theory to construct a narrative.  My French
colleague, Professor Ehresmann also uses category theory to construct
narrative about living systems.  (  see:  Memory Evolutive Systems
http://perso.wanadoo.fr/vbm-ehr/ ) ... ]]

I'd like to expand and comment on this a bit, as a way of groping toward
a nonspecialist answer -- for those who, like me, are not
mathematicians -- to the question of how all this (including category
theory) relates to philosophy in general and Peirce in particular.
Related bibliography and links can be found at
http://users.vianet.ca/gnox/meanlist.htm#biosys . In that
section of my annotated resource list, i've lumped Pattee, Rosen,
Prigogine, Salthe, Ulanowicz and others into a category called "life,
systems and complexity". To the extent that the nature and origin of
life is a matter of philosophical interest -- and it certainly was to
Peirce -- Pattee, Rosen and Prigogine are all interesting. Prigogine
approached to the problem through nonequilibrium thermodynamics, Rosen
through mathematical modeling, Pattee through epistemology and
semiotics.

Yesterday i looked through the works of Rosen and Pattee that i have at
hand, and although they both mention Prigogine in passing, i haven't
found any "rejection of the Prigogine approach" in either of them. I
would surmise that Jerry means that their choice of mathematical tools
differed from Prigogine's. This is certainly true of Rosen, and i'll get
to his comments on category theory below. As for Pattee, though, i
haven't seen any use of category theory in the dozen or so articles of
his that are accessible to me. His connection with Peirce lies more in
the domain of biosemiotics. One of his recent articles begins with this
paragraph:

[[[ The concept of Biosemiotics requires making a distinction between
two categories, the material or physical world and the symbolic or
semantic world. The problem is that there is no obvious way to connect
the two categories. This is a classical philosophical problem on which
there is no consensus even today. Biosemiotics recognizes that the
philosophical matter-mind problem extends downward to the pattern
recognition and control processes of the simplest living organisms where
it can more easily be addressed as a scientific problem. In fact, how
material structures serve as signals, instructions, and controls is
inseparable from the problem of the origin and evolution of life.
Biosemiotics was established as a necessary complement to the
physical-chemical reductionist approach to life that cannot make this
crucial categorical distinction necessary for describing semantic
information. Matter as described by physics and chemistry has no
intrinsic function or semantics. By contrast, biosemiotics recognizes
that life begins with function and semantics. ]]]

Pattee's usage of "categories" here seems more in the philosophical
tradition than the mathematical. By the way, his contribution to the
development of biosemiotics was recently recognized in a special issue
of _Biosystems_ (link to this on my resource list, URL above).

Turning to Rosen, we shift our attention to what he called the "modeling
relation", which (for me at least) is related to the use of what Peirce
called "diagrams" in the sciences, including philosophy. Much of Rosen's
investigation concerned how science models the objects of its
theorizing, but he also considered the modeling relation to be essential
to life itself. In other words, living systems are those which model
their worlds internally and thus function as "anticipatory systems":
their actions into the world are guided by their internal models, and
these models are self-modifying whenever there is a mismatch between the
anticipated results and the perceived results of their own behavior. It
follows that any mathematical technique adequate to the task of modeling
such systems must be capable of self-reference or meta-modeling. This
seems to be the key to Rosen's interest in Category Theory. It may be
helpful here to quote a couple of remarks from Rosen's section on this
in _Life Itself_ (pp. 143-151):

[[[ What is in many ways most instructive and illuminating about it is
its history, which illustrates better than anything else how mathematics
models itself and how it seeks to generalize by a process of forgetting
about the system of referents that produced the model in the first
place.
... Category Theory is quite unique among mathematical formalisms; it
comprises within itself a general theory of modelling per se and indeed
a flexibility remarkably approaching that of natural language. ]]]

This flexibility, according to Rosen, is due to the ability to iterate a
procedure capable of self-reference. "In a certain sense, then, Category
Theory can talk about itself, or describe itself, in ways more nearly
akin to natural languages than to the formal systems that normally
constitute mathematics" (p. 149). This characteristic of
self-referential recursion is also related to those forms of "closure"
which seem definitive of life. That is, living systems, while they must
be open to flows of energy and information, must also embody some kind
of logical closure. Pattee referred to this as "semantic closure" (or,
more recently, "semiotic closure"). Rosen famously refers to organisms
as being "closed to efficient causation" -- and indeed this is where
final causation enters the Rosen model. As for Prigogine, while he
doesn't deal explicitly with this kind of closure (as far as i know), it
seems implicit in his concept of self-organization. Thus i see the
various approaches of Pattee, Rosen and Prigogine as complementary
rather than conflicting. All three are explicit in their rejection of
mechanistic models of life; and all three are related to Peircean
semiotic logic, though in different ways.

I've only had time to glance at Ehresmann's work on the website, just
enough to see its affinity with Rosen's work. I hope to look into it
further when i have more time.

}The aspects of things that are most important for us are hidden because
of their simplicity and familiarity. [Wittgenstein]{

gnusystems }{ Pam Jackson & Gary Fuhrman }{ Manitoulin University
         }{ gnox <at> vianet.ca }{ http://users.vianet.ca/gnox/ }{

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