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→‎Note 18: edit + markup
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==Note 18==
 
==Note 18==
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<pre>
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'''Obstacles to Applying the Pragmatic Maxim'''
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| Consider what effects that might 'conceivably'
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| have practical bearings you 'conceive' the
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| objects of your 'conception' to have.  Then,
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| your 'conception' of those effects is the
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| whole of your 'conception' of the object.
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|
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| Charles Sanders Peirce,
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| "Maxim of Pragmaticism", CP 5.438.
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Obstacles to Applying the Pragmatic Maxim
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; Obstacle 2
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: Applying the pragmatic maxim, even with a moderate aim, can be hard.  I think that my present example, deliberately impoverished as it is, affords us with an embarrassing richness of evidence of just how complex the simple can be.
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Obstacle 2.  Applying the pragmatic maxim, even with a moderate aim, can be hard.
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All the better reason for me to see if I can finish it up before moving on.
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I think that my present example, deliberately impoverished as it is, affords us
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with an embarassing richness of evidence of just how complex the simple can be.
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All the better reason for me to see if I can finish it up before moving on.
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Expressed most simply, the idea is to replace the question of ''what it is'', which modest people know is far too difficult for them to answer right off, with the question of ''what it does'', which most of us know a modicum about.
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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.
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Expressed most simply, the idea is to replace the question of "what it is",
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Here is is the operation table of <math>V_4\!</math> once again:
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which modest people know is far too difficult for them to answer right off,
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with the question of "what it does", which most of us know a modicum about.
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In the case of regular representations of groups we found
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<br>
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a non-plussing surplus of answers to sort our way through.
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So let us track back one more time to see if we can learn
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any lessons that might carry over to more realistic cases.
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Here is is the operation table of V_4 once again:
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{| 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%"
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|+ <math>\text{Klein Four-Group}~ V_4</math>
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|- style="height:50px"
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| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{e}</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{f}</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{g}</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{h}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{e}</math>
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| <math>\operatorname{e}</math>
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| <math>\operatorname{f}</math>
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| <math>\operatorname{g}</math>
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| <math>\operatorname{h}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{f}</math>
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| <math>\operatorname{f}</math>
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| <math>\operatorname{e}</math>
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| <math>\operatorname{h}</math>
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| <math>\operatorname{g}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{g}</math>
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| <math>\operatorname{g}</math>
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| <math>\operatorname{h}</math>
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| <math>\operatorname{e}</math>
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| <math>\operatorname{f}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{h}</math>
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| <math>\operatorname{h}</math>
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| <math>\operatorname{g}</math>
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| <math>\operatorname{f}</math>
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| <math>\operatorname{e}</math>
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|}
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Table 1.  Klein Four-Group V_4
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<br>
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o---------o---------o---------o---------o---------o
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|        %        |        |        |        |
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|    ·    %    e    |    f    |    g    |    h    |
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|        %        |        |        |        |
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o=========o=========o=========o=========o=========o
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|        %        |        |        |        |
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|    e    %    e    |    f    |    g    |    h    |
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|        %        |        |        |        |
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o---------o---------o---------o---------o---------o
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|        %        |        |        |        |
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|    f    %    f    |    e    |    h    |    g    |
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|        %        |        |        |        |
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o---------o---------o---------o---------o---------o
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|        %        |        |        |        |
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|    g    %    g    |    h    |    e    |    f    |
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|        %        |        |        |        |
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o---------o---------o---------o---------o---------o
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|        %        |        |        |        |
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|    h    %    h    |    g    |    f    |    e    |
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|        %        |        |        |        |
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o---------o---------o---------o---------o---------o
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A group operation table is really just a device for
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A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.</math>
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recording a certain 3-adic relation, to be specific,
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the set of triples of the form <x, y, z> satisfying
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the equation x·y = z where · is the group operation.
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In the case of V_4 = (G, ·), where G is the "underlying set"
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In the case of <math>V_4 = (G, \cdot),</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h} \},</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G</math> whose triples are listed below:
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{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
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whose triples are listed below:
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<pre>
 
|  <e, e, e>
 
|  <e, e, e>
 
|  <e, f, f>
 
|  <e, f, f>
Jon Awbrey
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