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 Eilenberg.
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
