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", usually in the form:  "Every group is isomorphic to

a subgroup of Aut(S), the group of automorphisms of an appropriate set S".

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:

| Consider the effects of the symbol, whose meaning you wish to investigate,

| as they play out on "all" of the different stages of context on which you

| Consider the effects of the symbol, whose meaning you wish to investigate, as they play out on all the stages of context where you can imagine that symbol playing a role.

| can imagine that symbol playing a role.

This idea of contextual definition is basically the same as Jeremy Bentham's

This idea of contextual definition is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, p.&nbsp;216).  Today we'd call these constructions ''term models''.  This, again, is the big idea behind Schönfinkel's combinators <math>\operatorname{S}, \operatorname{K}, \operatorname{I},</math> and hence of lambda calculus, and I reckon you know where that leads.

notion of "paraphrasis", a "method of accounting for fictions by explaining

various purported terms away" (Quine, in Van Heijenoort, page 216).  Today

we'd call these constructions "term models".  This, again, is the big idea

behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,

and I reckon you know where that leads.

==Note 11==

==Note 11==