Well, no, I don't actually talk that way ...
Looking back through the theread in question ...
SJ:http://web.archive.org/web/20140311032523/http://permalink.gmane.org/gmane.science.philosophy.peirce/12127
JA:http://web.archive.org/web/20140311054002/http://permalink.gmane.org/gmane.science.philosophy.peirce/12131
SJ:http://web.archive.org/web/20140311205639/http://permalink.gmane.org/gmane.science.philosophy.peirce/12134
JA:http://web.archive.org/web/20140311211000/http://permalink.gmane.org/gmane.science.philosophy.peirce/12141
SJ:http://web.archive.org/web/20140313032831/http://permalink.gmane.org/gmane.science.philosophy.peirce/12147
JLRC:http://web.archive.org/web/20140313201902/http://permalink.gmane.org/gmane.science.philosophy.peirce/12156
JA:http://web.archive.org/web/20140313201602/http://permalink.gmane.org/gmane.science.philosophy.peirce/12157
JLRC:http://web.archive.org/web/20140314020223/http://permalink.gmane.org/gmane.science.philosophy.peirce/12158
JA:http://web.archive.org/web/20140314020207/http://permalink.gmane.org/gmane.science.philosophy.peirce/12159
JBD:http://web.archive.org/web/20140314213801/http://permalink.gmane.org/gmane.science.philosophy.peirce/12170
SJ:http://web.archive.org/web/20140315024804/http://permalink.gmane.org/gmane.science.philosophy.peirce/12171
JA:http://web.archive.org/web/20140315040001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12175
JA:http://web.archive.org/web/20140315190001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12177
SJ:http://web.archive.org/web/20140316030412/http://permalink.gmane.org/gmane.science.philosophy.peirce/12180
JA:http://web.archive.org/web/20140316031001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12181
JA:http://web.archive.org/web/20140316160001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12183
SJ:http://web.archive.org/web/20140316203648/http://permalink.gmane.org/gmane.science.philosophy.peirce/12184
JA:http://web.archive.org/web/20140316210002/http://permalink.gmane.org/gmane.science.philosophy.peirce/12185
JA:http://web.archive.org/web/20140317002001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12186
SJ:http://web.archive.org/web/20140317212640/http://permalink.gmane.org/gmane.science.philosophy.peirce/12188
JA:http://web.archive.org/web/20140317220001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12190
SJ:http://web.archive.org/web/20140318000002/http://permalink.gmane.org/gmane.science.philosophy.peirce/12191
JC:http://web.archive.org/web/20140318155646/http://permalink.gmane.org/gmane.science.philosophy.peirce/12198
SJ:http://web.archive.org/web/20140318160003/http://permalink.gmane.org/gmane.science.philosophy.peirce/12199
SR:http://web.archive.org/web/20140318160209/http://permalink.gmane.org/gmane.science.philosophy.peirce/12200
JA:http://web.archive.org/web/20140318203010/http://permalink.gmane.org/gmane.science.philosophy.peirce/12205
JA:http://web.archive.org/web/20140318233001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12215
JA:http://web.archive.org/web/20140319180002/http://permalink.gmane.org/gmane.science.philosophy.peirce/12220
It looks like Sung is referring to what I said here:
http://web.archive.org/web/20140316160001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12183
Correcting some typos:
Mathematical category theory is abstracted from the properties of structured sets (the objects) and
structure-preserving mappings between them (the arrows or morphisms). Examples of structured sets
are things like plain old sets (with no extra structure), groups (with one binary operation), rings
(with two binary operations), a host of different ordered sets (with their various order relations),
topologies, metric spaces, and so on. In each case there is a set of arrows Arr(X, Y) defined for
each pair of objects X, Y in a given category C.
Where does the triadicity come in? In the prototypical category of sets and mappings between them
the sets are monadic, that is, associated with monadic predicates, and the mappings between sets are
dyadic, since functions are special cases of dyadic relations. But the structures added to the sets
can be richer, typically beginning with the triadic relations that we know as "binary operations".
However, this appearance of triadicity comes optional with the objects taken up and is not mandated
by the definition of a category itself.
Working in that spirit, however, one application of category theory to semiotics would be to take
the category of triadic sign relations and appropriate structure-preserving mappings between them.
Still, the arrows are always defined on pairs of objects, and therefore retain a dyadic character.
The place where triadicity really comes in as an integral part of the category concept is with the
composition operation, like all "binary operations" defining a triadic relation, in this case on
composable pairs of arrows, ∘ : Arr(C) × Arr(C) → Arr(C).
Consequently, in a commutative diagram like the following:
` ` ` ` ` ` ` ` `
` ` f ∘ g = h ` `
X-------------->Z
`\` ` ` ` ` ` `^`
` \ ` ` ` ` ` / `
` `\` ` ` ` `/` `
` f \ ` ` ` / g `
` ` `\` ` `/` ` `
` ` ` \ ` / ` ` `
` ` ` `v`/` ` ` `
` ` ` ` Y ` ` ` `
` ` ` ` ` ` ` ` `
The arrows f, g, h are in a triadic relation, namely, f ∘ g = h, but the objects X, Y, Z are not.
That is to say, the objects X, Y, Z are freely chosen and there are no non-trivial relations of
determination among any of the objects X, Y, Z.
The point that's being missed here I think is this: It's "putting the carte before the territory"
to speak of structure-preserving maps between sign relations before one has a clear idea what the
structure of a sign relation actually is. That seems to be the main source of confusion in all of
these recent (and not so recent) discussions of semiotics.
Regards,
Jon
Sungchul Ji wrote:
Edwina wrote:
". . . my contention that the semiosic Sign is (5469-1)
a single set of three interactive Relations, while
you consider that it is a 'single triadic relation'."
Jon Awbrey and I recently came to the agreement that
"The Peircean sign can be viewed as a mathematical (5469-2)
category characterized by the composition condition
symbolically represented as Object|Sign = Interpretant,
meaning that a sign is determined by its object and
determines its interpretant so that the latter is indirectly
determined by the object."
A corollary of (5469-2) would be that
“The Peircean sign cannot be represented as a single (5469-3)
set of three relations, i.e., the R-R, R-O, and R-I
relations, since these dyadic relations alone,
without any specific composition condition of a
mathematical category, cannot represent a triadic
relation.”
With all the best.
Sung
__________________________________________________
Sungchul Ji, Ph.D.
Associate Professor of Pharmacology and Toxicology
Department of Pharmacology and Toxicology
Ernest Mario School of Pharmacy
Rutgers University
Piscataway, N.J. 08855
732-445-4701
www.conformon.net
--
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