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 & Mac Lane 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: > > “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] > . 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