Questions Concerning Certain Faculties Claimed For Signs
Let me sum up the main points of the above exchange before moving on.
Mathematics is useful in our present endeavor because it covers relations in general. In addition — and multiplication, too — mathematics is chock full of well-studied examples of triadic relations. When it comes to the job of analyzing sign relations and teasing out their relevant structures we could save ourselves a lot of trouble and trial and error by examining this record of prior art and adapting its methods to cover sign relations.
On the other hand, there are hints in Peirce’s work that triadic relations extend across a threshold of complexity, such that relations of all higher adicities can be analyzed in terms of 1-adic, 2-adic, and 3-adic relations. This is the point where the analogy with mathematical category theory both forms and breaks. In mathematics, category theory is largely based on the prevalence of functions in mathematical practice, and functions are dyadic relations. Still, triadic relations pervade the background of the subject, visible in the triadic composition relation and in the concept of what are called “natural transformations”, the clarification of which notion is one of the original motivations of the subject. Bringing the triadic roots of category theory into higher relief is one of my motives for bringing about an encounter with Peirce’s categories, an effort to which I have given not a few years of thought.
That brings us to the case of sign relations proper. I think it’s clear that these types of triadic relations form our first stepping stones and also our first stumbling blocks in the inquiry into inquiry, and I think I gave some indications already of why that might be true. I don’t know if I can do any better than that at this time, but I’ll think on it more after that all-essential secondness of caffeination.