MyWikiBiz, Author Your Legacy — Monday December 23, 2024
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, 21:04, 8 June 2009
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| 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. |
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− | 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: |
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| + | {| 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>. |
| + | |} |
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| + | 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: |
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| {| align="center" cellpadding="6" width="90%" | | {| align="center" cellpadding="6" width="90%" |