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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.
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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