My contribution using category theory : http://hardsemiotics.net/HARD/MISCELLANEOUS/algebraic%20formalization%20of%20semiotics.doc
Robert Marty ----- Original Message ----- From: "Jon Awbrey" <[email protected]> To: "Peirce List 1" <[email protected]> Sent: Friday, March 21, 2014 3:10 PM Subject: [PEIRCE-L] Category Theory, Relational Arrows, Sign Relations > 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. >> >> Sung >> >> >>>> Well, 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 (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 >>>> > > -- > > academia: http://independent.academia.edu/JonAwbrey > my word press blog: http://inquiryintoinquiry.com/ > inquiry list: http://stderr.org/pipermail/inquiry/ > isw: http://intersci.ss.uci.edu/wiki/index.php/JLA > oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey > facebook page: https://www.facebook.com/JonnyCache > -------------------------------------------------------------------------------- > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] > . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] > with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at > http://www.cspeirce.com/peirce-l/peirce-l.htm . > > > > >
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