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: > > “so 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 dimensionality” (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 “The Law of Mind” [1892] Peirce uses the > example of a surface that is part red and part blue and asks the question, > “What, then, is the color of the boundary line between the red and the blue?" > [CP 6.126). His answer is “half red and half blue.”) 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 > > ----------------------------- > 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]<mailto:[email protected]> . To UNSUBSCRIBE, > send a message not to PEIRCE-L but to > [email protected]<mailto:[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 . > > ----------------------------- > 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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