Changes

The genealogy of this conception of pragmatic representation is very intricate.  I'll sketch a few details that I think I remember clearly enough, subject to later correction.  Without checking historical accounts, I won't be able to pin down anything approaching a real chronology, but most of these notions were standard furnishings of the 19th Century mathematical study, and only the last few items date as late as the 1920's.

The genealogy of this conception of pragmatic representation is very intricate.  I'll sketch a few details that I think I remember clearly enough, subject to later correction.  Without checking historical accounts, I won't be able to pin down anything approaching a real chronology, but most of these notions were standard furnishings of the 19th Century mathematical study, and only the last few items date as late as the 1920's.

The idea about the regular representations of a group is universally known as Cayley's Theorem, usually in the form: "Every group is isomorphic to a subgroup of <math>\operatorname{Aut}(S),</math> the group of automorphisms of an appropriate set <math>S\!</math>". There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers.  The crux of the whole idea is this:

The idea about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:

 

{| align="center" cellpadding="6" width="90%"

| Every group is isomorphic to a subgroup of <math>\operatorname{Aut}(S),</math> the group of automorphisms of a suitable set <math>S\!</math>.

 

There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers.  The crux of the whole idea is this:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"