[CG] Re: Peirce's 1870 “Logic of Relatives”

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.15 https://inquiryintoinquiry.com/2014/05/15/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.15 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, I’m going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving mappings, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce’s “number of” function on logical terms. The “structure” preserved by a structure-preserving map is just the structure we all know and love as a triadic relation. Very typically, it will be the type of triadic relation that defines the type of binary operation that obeys the rules of a mathematical structure known as a “group”, that is, a structure satisfying the axioms for closure, associativity, identities, and inverses. For example, in the case of the logarithm map J we have the following data. • J : Reals ← Reals (properly restricted) • K : Reals ← Reals × Reals where K(r, s) = r + s • L : Reals ← Reals × Reals where L(u, v) = u ⋅ v Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in brackets as a name for the triadic relation that defines the corresponding group, we have the following set-up. • J : [+] ← [⋅] • [+] ⊆ Reals × Reals × Reals • [⋅] ⊆ Reals × Reals × Reals It often happens that both group operations are indicated by the same sign, usually one from the set { ⋅ , ∗ , + } or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like one of the following two variants. • “The image of the sum is the sum of the images.” • “The image of the product is the sum of the images.” Figure 50 presents a generic picture for groups G and H. Figure 50. Group Homomorphism J : G ← H https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-group-homom… In a setting where both groups are written with a plus sign, perhaps even constituting the same group, the defining formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the shape J(u + v) = Ju + Jv, which looks analogous to the distributive multiplication of a factor J over a sum (u + v). That is why morphisms are regarded as generalizations of linear functions and are frequently referred to in those terms. Regards, Jon