Jeff D, Jerry, Jon, List,
Jeff D, List,
In [biosemiotics:5311] posted on March 9, 2014, which is partially
reproduced below, I was let to conclude that
The Peircean sign is a mathematical category. (043014-1)
To the extent that (043014-1) turns out to be true, I wonder if it would
be logical to conclude that
The category theory of Eilenberg and McLane (043014-2)
is a mathematical version of semiotics,
mathematical semiotics, or 'mathematicized
semiotics.
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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(A apartial reproduction of [biosemiotics:5311] dated March 9, 2014)
A sign is anything, A, which, (030914-14)
(1) in addition to other characters of its own,
(2) stands in a dyadic relation Þ, to a purely active correlate, B,
(3) and is also in a triadic relation to B for a purely passive
correlate, C, this triadic relation being such as to determine C
to be in a dyadic relation, µ, to B, the relation µ corresponding
in a recognized way to the relation Þ.
------ SS. pp. 192-193, ca. 1905
The supplementary triadic relation intrinsic to the definition of signs
given in (030914-13) and (030914-14) can be represented diagrammatically as
shown in Figure 2:
Þ x
B --------- > A ---------- > C
| ^
| |
|__________________________________|
µ
Figure 2. The Peircean sign as a mathematical category consisting of
three objects, A (i.e, sign or representamen), B (or object), and C (or
interpretant), and structure-preserving mappings (also called
morphisms), Þ, x, and µ ,that are related by commutativity in the
sense that the combination of Þ and x has the same effect as the relation,
µ, where x is the relation postulated to be missing in Peirces statement
above.
It is evident that Figure 2 is isomorphic with (i.e., exhibits the same
principle as) Figure 1, leading to the conclusion that
The Peircean sign is a mathematical category. (030914-15)
which can be translated into an algebraic expression based on the
definition of supplementarity given (030914-1):
Sign = Object + Intepretant (030914-16)
This statement seems to contradict my conclusion published elsewhere [4] that
Peircean signs are gnergons.
(030914-17)
where gnergons are defined as discrete units of Gnergy which is in turn
defined as as the complementary union of information /form (denoted by
gn-) and energy/matter (denoted by --ergy) [5], which can be
transformed into another algebraic expression based on the definition of
complementarity given in (030914-2):
Sign = Form^ Matter (030914-18)
The apparent contradiction between (030914-16) which embodies
supplementarity and (030914-18) embodying complementarity can be avoided
if it is assumed that the Peircean sign is a complete triad, not just a
supplementary nor a complementary triad. That is, the Piercean sign
may satisfy the triadic closure condition, (0309140-3), and hence
constitutes a complete triad:
Sign = (Object + Interpretant) & (Form^Matter) (030914-19)
If this statement turns out to be valid, the following conclusion would
hold:
Peircean signs can be represented as mathematical (030914-20)
categories that embody three principles
SUPPLEMENTARITY(symbolized by +), COMPLEMENTARITY
(symbolized by ^), and TRIADIC CLOSURE symbolized by &).
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> Jerry, List,
>
> I was not trying to argue for the claim. Rather, I was only reporting
> what some who work on category theory say. Here, for instance, is John
> Baez: "The point is, that a category is really a generalization of a
> group." (http://math.ucr.edu/home/baez/categories.html)
>
> As far as I am able to see, Klein's aim in the Erlanger program and Mac
> Lane's aim in his early work were quite similar. Here is how it is
> explained in the Stanford Encyclopedia article: "Category theory reveals
> how different kinds of structures are related to one another. For
> instance, in algebraic topology, topological spaces are related to groups
> (and modules, rings, etc.) in various ways (such as homology, cohomology,
> homotopy, K-theory). As noted above, groups with group homomorphisms
> constitute a category. Eilenberg & MacâLane invented category theory
> precisely in order to clarify and compare these connections. What matters
> are the morphisms between categories, given by functors. Informally,
> functors are structure-preserving maps between categories."
>
> In addition to remarking on the general similarity of their aims, I was
> also making a historical suggestion. The point is nicely stated by Mac
> Lane and Eilenberg in "General Theory of Natural Equivalences, so I'll
> refer to what say about the inspiration for the development of category
> theory: "This may be regarded as a continuation of the Klein Erlanger
> Programm, in the sense that a geometrical space with its group of
> transformations is generalized to a category with its algebra of mappings
> ([74], 237).
>
> For the sake of applying these 20th century ideas in mathematical category
> theory to the interpretation of Peirce, I do think it is helpful to
> consider the genealogy of those ideas--especially when they can be traced
> back to conceptions that Peirce was explicitly using. As a separate
> matter, I wonder how much difference there really is between Klein's
> geometrical approach to the use of group theory and MacLane's algebraic
> approach to the use of category theory. Peirce suggests that, in many
> respects, geometry and algebra are just two different ways of expressing
> the same basic mathematical ideas.
>
> --Jeff
>
>
>
> Jeff Downard
> Associate Professor
> Department of Philosophy
> NAU
> (o) 523-8354
> ________________________________________
> From: Jerry LR Chandler [[email protected]]
> Sent: Tuesday, April 29, 2014 8:59 PM
> To: Jeffrey Brian Downard
> Cc: Peirce List
> Subject: Re: [PEIRCE-L] category theory in math
>
> Jeff, List:
>
> Category theory is a generalization of several mathematical structures:
> sets, groups, rings, vector spaces, topologies and numerous other
> structures of mathematics. So what are you really trying to say, Jeff?
>
> See: Mathematical Structures and Functions by Saunders Mac Lane for
> several detailed maps of relations between mathematical structures.
> Saunders Mac Lane is one of the originators of Category Theory, which is a
> special form of associative logic.
>
> Your key sentence: "As such, we should not assume that the general idea of
> what is involved in the conception of mathematical category is entirely
> foreign to Peirce."
>
> I do not concur with your sentence.
>
> The key concept of category theory was not formulated until 1941 by Mac
> Lane and Eilenberger.
> This is a issue of fact, not philosophical judgment.
>
> By the way Jeff, given you general interest in Mathematics, the book by
> Mac Lane is a masterpiece as a coherent overview of most branches of
> mathematics. His diagrams of relationships among the many branches of
> mathematics including groups theory, was a real eye-opener to me at the
> time (mid-90s.?)
>
> I stopped by his office at U. Chicago and spent a couple of hours with
> him, discussing his philosophy of mathematics. An amazing man.
>
> Cheers
>
> jerry
>
>
>
> On Apr 29, 2014, at 9:48 AM, Jeffrey Brian Downard wrote:
>
>> Jerry, List,
>>
>> We might add that category theory, as it has been developed in
>> mathematics in the 20th century, is a generalization upon the conception
>> of a mathematical group. Peirce was quite familiar with Klein's use of
>> group structure as a basis for exploring the relations between different
>> areas of mathematics. As such, we should not assume that the general
>> idea of what is involved in the conception of mathematical category is
>> entirely foreign to Peirce.
>>
>> Jeff
>>
>>
>> Jeff Downard
>> Associate Professor
>> Department of Philosophy
>> NAU
>> (o) 523-8354
>> ________________________________________
>> From: Jerry LR Chandler [[email protected]]
>> Sent: Tuesday, April 29, 2014 7:38 AM
>> To: Peirce List
>> Cc: Michael Shapiro
>> Subject: Re: [PEIRCE-L] continuing the discussion re Structuralism
>>
>> List, Michael:
>>
>> Further comments on your unusual posts concerning your linguistic
>> perspectives on CSP's "structuralism".
>>
>> 1. Do you accept the fact that the concept of continuity is a geometric
>> concept, as in CSP's example of the LINE and the separation of the line
>> from the surface? Continuity as a concept is historically grounded in
>> Greek philosophy, isn't it?
>>
>> 2. Category theory is an algebraic theory. It originated about 1941. It
>> is exceedingly abstract form of infinities of associative relations on
>> graphs. Linguistically, the terms are clearly separate and distinct,
>> are they not? In addition, many. many philosophies of categories have
>> been described.
>>
>>
>> Michael, when you write:
>>
>> I can't say anything about mathematical category theory,
>>
>> are you referring as well to the algebraic form of continuity intrinsic
>> to category theory?
>>
>> In other words, are you excluding the Peircian logic of continuity and
>> representing continuity from set theory as your basis for interpreting
>> of your linguistic "habits"?
>>
>> Cheers
>>
>> Jerry
>>
>>
>>
>> On Apr 29, 2014, at 3:38 AM, Michael Shapiro wrote:
>>
>> Jerry, List,
>>
>> I can't say anything about mathematical category theory, but I would
>> certainly advocate applying Peirce's categoriology to the structure of
>> the syntagm. Apropos of the latter, in what sense do you mean that my
>> understanding of the syntagm is "artificial?"
>> M.
>> -----Original Message-----
>> From: Jerry LR Chandler
>> Sent: Apr 28, 2014 7:44 PM
>> To: Peirce List
>> Cc: Michael Shapiro
>> Subject: Re: [PEIRCE-L] continuing the discussion re Structuralism
>>
>> List, Michael
>>
>> A brief comment, the purpose of which is to sharpen the differences
>> between scientific structuralism and your usage of the term with respect
>> to linguistic continuity.
>>
>> On Apr 28, 2014, at 8:21 AM, Michael Shapiro wrote:
>>
>> âso space presents points, lines, surfaces, and solids, each
>> generated by the motion of a place of lower dimensionality and the limit
>> of a place of next higher dimensionalityâ (CP 1.501).
>>
>> This quote is not a purely mathematical notion.
>> This quote infers that the concept of "motion" is necessary for shifting
>> (transitivity) between lower and higher dimensions.
>> The notion of motion infers changes of positions with time, a
>> progression of durations. This is a physical concept, independent of
>> mathematical systems of axioms and of formal symbolic logics.
>> This quote excludes the notion of an icon as a real dimensional object -
>> for example, a molecule or the anatomy of our bodies.
>>
>> "Every element of a syntagm is to varying extents both distinct
>> (bounded) and conjoined with every other. (In âThe Law of Mindâ
>> [1892] Peirce uses the example of a surface that is part red and part
>> blue and asks the question, âWhat, then, is the color of the boundary
>> line between the red and the blue?" [CP 6.126). His answer is âhalf
>> red and half blue.â) With this understanding we are reinforced in the
>> position that the wholes (continua, gestalts) of human semiosis are
>> simultaneously differentiated and unified."
>>
>> This is a brilliant example of the conundrum of continuity as it relates
>> to the logic of relatives and the individuality of "real" objects in the
>> "real world". CSP ducks the basic issue by asserting that it is "half
>> red and half blue"
>> The scientific approach to this conundrum is to label a real object
>> (that which is presented to our senses) as an individual, and to give
>> the identity of this separate and distinct object a name that
>> distinguishes it from other objects.
>> Philosophically, scientific realism demands this. Thus it is the
>> concept of identity that clearly separates the presentative image of a
>> part from the entire image of the whole blackboard.
>>
>>
>> "To conclude and sum up, this is the kind of structuralism I mean when I
>> speak of "structuralism properly understood" and impute it, moreover, to
>> Peirce."
>>
>> It appears to me that your conclusion is not about structuralism as in
>> the sense of anatomy or chemistry, but about the continuity of a meaning
>> of a progression of symbols that you wish to give meaning to.
>>
>> I do not find this view of Peircian rhetoric to be consistent with CSP's
>> notion of a medad as a central concept of his logic of relatives. The
>> chemical concept of structuralism forms an exact spacial progression
>> (topological) that generates a smooth transfer of meaning from atoms to
>> molecules and to higher order structures, such as human anatomy.
>>
>> BTW, would you extend this analysis of Peircian rhetoric about
>> continuity to mathematical category theory? To any of the several
>> philosophical theories of categories?
>>
>> It is not that I disagree with your artificial understanding of the
>> concept of "syntagm", rather it is the representation of the signs that
>> you choose to represent the continuum.
>>
>> Cheers
>>
>> Jerry
>>
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>>
>>
>>
>
>
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