List: Again, it is obvious that monadic/dyadic/triadic relations are not the *only* mathematical categories. However, I found myself wondering if they might in some relevant sense be the only *irreducible* mathematical categories. Further online research turned up a 1992 paper <https://core.ac.uk/download/pdf/82132443.pdf> by Robert Marty stating that "using Category Theory, we will build the formal framework for dealing with the phaneron (phenomenology), that is to say the structure of all that can be present to the mind" (p. 685). He continues, "It is easy to establish that every relational structure of a type equal to 3 or smaller than 3 is relatively irreductible within any domain," while any "relational structure of type *n* > 3 ... is relatively reductible on a set of relational structures of type 3" (ibid). The upshot is as follows.
RM: Now, the reduction theorem allows us to describe every relational structure on *X* of type *n* by means of the relative product of relational structures of type 1, 2 or 3, called elementary relational structures. We call "monads" the elements of relational structures of type 1 (1-tuples); we call "dyads" the ones of type 2 (2-tuples) and "triads" the ones of type 3 (3-tuples). Each monad corresponds to a simple "quality of feeling," each dyad to an existent individual or a fact, and each triad to a "concept, law or something expressible by universal proposition." This is the empirical decomposition of the phaneron into indecomposable elements many a time described by Peirce. (p. 686) Returning to the thread topic, Robert later adds the following. RM: [T]he phenomenological structures represent the objects of the world (present to the mind), whereas the phenomenological morphisms represent the relations between these objects, that is to say the modes of being. There are six fundamental classes of modes of being: Γ3, Γ'3, and Γ''3 corresponding respectively to Authentic Thirdness, Degenerate Thirdness at the first degree, Degenerate Thirdness at the second degree; Γ2 and Γ'2 corresponding respectively to Authentic Secondness and degenerate Secondness; Γ1 corresponding to Firstness. (p. 687) This all seems to confirm and extend my understanding of "the conceptual relationship between Peirce's trichotomic category theory and contemporary mathematical category theory." Consistent with Goguen's "first dogma," 1ns/2ns/3ns qualify as *mathematical* categories because their respective objects have monadic/dyadic/triadic structure and their morphisms preserve these structures. Specifically, their *phenomenological* morphisms are the six modes of being, which are even more clearly distinguished in Robert's new podium diagram; I have proposed to call them reality (Γ3=3), persistence (Γ'3=2/3), diversity (Γ''3=1/2/3), existence (Γ2=2), inherence (Γ'2=1/2), and essence (Γ1=1). I have come to prefer "diversity" to "governance" for doubly degenerate 3ns because it reflects how there are three different modes of being for a quality--in itself (essence), in individual things (inherence), and in a continuum of qualities (diversity). Regards, Jon S. On Sat, May 9, 2020 at 7:35 PM Jon Alan Schmidt <[email protected]> wrote: > John, List: > > I agree that Peirce's categories are *much broader* from a philosophical > standpoint than modern mathematical categories, and obviously > monadic/dyadic/triadic relations are not the *only *mathematical > categories. However, the question on the table, as originally stated by > Gary R., was about "the conceptual relationship between Peirce's > trichotomic category theory and contemporary mathematical category theory > if any." Goguen's "first dogma" provides an answer that I find > helpful--monadic/dyadic/triadic relations are three different species of > mathematical structure, objects having each structure belong to the > corresponding category (1ns/2ns/3ns), and morphisms of each category > preserve that structure. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Professional Engineer, Amateur Philosopher, Lutheran Layman > www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt > > On Fri, May 8, 2020 at 11:03 PM John F. Sowa <[email protected]> wrote: > >> Jon, >> >> Peirce was using the word 'category' in rhe tradition from Aristotle to >> Kant. That tradition is still alive and well in philosophy. >> >> It's unfortunate that the 20th c mathematicians used the same term for a >> different kind of mathematical theory. But as Robert M. hass been saying, >> it's possible to apply the mathematical category theory to analyze Peirce's >> theories. >> >> JAS> "A Categorical Manifesto" provides the kind of clear and succinct >> definition that I have been seeking. >> >> JG: To each species of mathematical structure, there corresponds a >> category whose objects have that structure, and whose morphisms preserve >> it. (p. 2) >> >> That simple statement may look clear and succinct, but underneath >> there's the kind of complexity that Robert was talking about. >> JAS> Would it then be accurate to say that Peirce's categories >> (1ns/2ns/3ns) are mathematical categories in the sense of corresponding to >> the structures of the three irreducible forms of relation >> (monadic/dyadic/triadic)? >> Short answer: No. >> The longer answer would be along the lines that Robert was talkinng about. >> John >> >
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