Sung, I appreciate your sticking with this, as I think the potential connections between category theory,relation theory in general, and triadic sign relations in particular is a topic well worth pursuing. At present, however, I need to keep my focus on Peirce's 1870 Logic of Relatives or else give it up
for another lifetime.
So I'll create this separate thread for collecting the appropriate background material and whatever else may come to mind as time goes by. Regards, Jon Sungchul Ji wrote:
Jon, Can you clarify what you mean below ? I presume there is some discrepancy between our interpretions of the Peircean sign triad ? "..., no, I don't actually talk that way ..." With all the best. SungWell, no, I don't actually talk that way ... Looking back through the thread 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 (theobjects) 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 aclear 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
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