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Revision as of 22:04, 1 December 2015
, 22:04, 1 December 2015→Operational Representation: + a few more paragraphs
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call “multiplying on the left”.
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call “multiplying on the left”.
In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>i\!</math> the relate and <math>j\!</math> the correlate, the elementary relative <math>i\!:\!j\!</math> now means that <math>i\!</math> gets changed into <math>j.\!</math> In this scheme of reading, the transformation <math>a\!:\!b + b\!:\!c + c\!:\!a\!</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,\!</math> or what we would now call the set <math>\{ a, b, c \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
{| align="center" cellpadding="6"
|
<math>\begin{Bmatrix}
a & b & c
\\
b & c & a
\end{Bmatrix}\!</math>
|}
This is consistent with the convention that Peirce uses in the paper “On a Class of Multiple Algebras” (CP 3.324–327).
We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the ''operationalization'' of ideas. The basic idea is to replace the question of ''What it is'', which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of ''What it does'', which most people know at least a modicum about.
In the case of regular representations of groups we found a non-plussing surplus of answers to sort our way through. So let us track back one more time to see if we can learn any lessons that might carry over to more realistic cases.
Here is is the operation table of <math>V_4\!</math> once again:
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Klein Four-Group}~ V_4\!</math>
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" |
<math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{e}\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{f}\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{g}\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{h}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| <math>\mathrm{e}\!</math>
| <math>\mathrm{f}\!</math>
| <math>\mathrm{g}\!</math>
| <math>\mathrm{h}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\mathrm{f}\!</math>
| <math>\mathrm{e}\!</math>
| <math>\mathrm{h}\!</math>
| <math>\mathrm{g}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\mathrm{g}\!</math>
| <math>\mathrm{h}\!</math>
| <math>\mathrm{e}\!</math>
| <math>\mathrm{f}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\mathrm{h}\!</math>
| <math>\mathrm{g}\!</math>
| <math>\mathrm{f}\!</math>
| <math>\mathrm{e}\!</math>
|}
<br>
==Logical Cacti==
==Logical Cacti==
