http://hardsemiotics.net/HARD/MISCELLANEOUS/algebraic%20formalization%20of%20semiotics.pdf
I hope that it is good... It is a work in progress, but near its end..
Best regards
Robert Marty
----- Original Message -----
From: "Jeffrey Brian Downard" <[email protected]>
To: "marty robert" <[email protected]>
Sent: Friday, March 21, 2014 8:07 PM
Subject: RE: [PEIRCE-L] Category Theory, Relational Arrows, Sign Relations
Hello Robert Marty,
I opened the .doc file that contains "Algebraic formalizations of the
Peircean phaneroscopy and semiotics," and quite a bit of the formal
notation on every page has been lost. The diagrams appear to be intact,
but the algebraic formulae embedded in the text have quite a number of
black squares and rectangles. The version of Word on my computer is quite
recent, so you might want to see what needs to be done to preserve the
special notation you're using. One option would be to convert it into a
pdf file.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: marty robert [[email protected]]
Sent: Friday, March 21, 2014 7:27 AM
To: Jon Awbrey; Peirce List 1
Subject: Re: [PEIRCE-L] Category Theory, Relational Arrows, Sign Relations
My contribution using category theory :
http://hardsemiotics.net/HARD/MISCELLANEOUS/algebraic%20formalization%20of%20semiotics.doc
Robert Marty
----- Original Message -----
From: "Jon Awbrey" <[email protected]<mailto:[email protected]>>
To: "Peirce List 1"
<[email protected]<mailto:[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/
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