[CG] Re: Precursors Of Category Theory

Precursors Of Category Theory • 4 • https://inquiryintoinquiry.com/2024/05/28/precursors-of-category-theory-4-a/ C.S. Peirce • “Prolegomena to an Apology for Pragmaticism” (1906) ❝I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates. ❝That wonderful operation of hypostatic abstraction by which we seem to create “entia rationis” that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think “through”, into being subjects thought of. We thus think of the thought‑sign itself, making it the object of another thought‑sign. ❝Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions. Does this series proceed endlessly? I think not. What then are the characters of its different members? ❝My thoughts on this subject are not yet harvested. I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being: Actuality, Possibility, Destiny (or Freedom from Destiny). ❝On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being. Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into “Realms” for the different Predicaments.❞ (CP 4.549). The first thing to extract from the above passage is that Peirce's Categories, for which he uses the technical term “Predicaments”, are predicates of predicates. Considerations of the order Peirce undertakes tend to generate hierarchies of subject matters, extending through what is traditionally called the “logic of second intentions”, or what is handled very roughly by “second order logic” in contemporary parlance, and continuing onward through higher intentions, or higher order logic and type theory. Peirce arrived at his own system of three categories after a thoroughgoing study of his predecessors, with special reference to the categories of Aristotle, Kant, and Hegel. The names he used for his own categories varied with context and occasion, but ranged from moderately intuitive terms like “quality”, “reaction”, and “symbolization” to maximally abstract terms like “firstness”, “secondness”, and “thirdness”. Taken in full generality, k‑ness may be understood as referring to those properties all k‑adic relations have in common. Peirce's distinctive claim is that a type hierarchy of three levels is generative of all we need in logic. Part of the justification for Peirce's claim that three categories are necessary and sufficient appears to arise from mathematical facts about the reducibility of k‑adic relations. With regard to necessity, triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates. With regard to sufficiency, all higher arity k‑adic relations can be analyzed in terms of triadic and lower arity relations. Reference — • Peirce, C.S. (1906), “Prolegomena to an Apology for Pragmaticism”, The Monist 16, 492–546, CP 4.530–572. Resources — Precursors Of Category Theory • https://oeis.org/wiki/Precursors_Of_Category_Theory Propositions As Types Analogy • https://oeis.org/wiki/Propositions_As_Types_Analogy Survey of Precursors Of Category Theory • https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-… Regards, Jon cc: https://www.academia.edu/community/LY0pam cc: https://mathstodon.xyz/@Inquiry/112502692087155139