Re: Category Theory & CSP
Bernard Morand <morand <at> iutc3.unicaen.fr>
2006-05-02 07:47:39 GMT
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
> 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
Now, the fact that such mathematical systems really tell something to
us, and very accurately, is always a divine surprise to me.
> 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.
> 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:
> *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.
> Jim Piat wrote:
>>> 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?
>> 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 . . .
>> Jim Piat
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