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==Note 20==
 
==Note 20==
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<pre>
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{| cellpadding="2" cellspacing="2" width="100%"
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You may be wondering what happened to the announced subject
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| width="60%" | &nbsp;
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of "Dynamics And Logic". What occurred was a bit like this:
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| width="40%" |
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the way of heaven and earth<br>
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is to be long continued<br>
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in their operation<br>
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without stopping
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|-
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| height="50px" | &nbsp;
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| valign="top"  | &mdash; i ching, hexagram 32
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|}
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We happened to make the observation that the shift operators {E_ij}
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The Reader may be wondering what happened to the announced subject of "Dynamics And Logic".  What happened was a bit like this:
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form a transformation group that acts on the set of propositions of
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the form f : B x B -> B.  Group theory is a very attractive subject,
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but it did not draw us so far from our intended course as one might
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initially think.  For one thing, groups, especially the groups that
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are named after the Norwegian mathematician Marius Sophus Lie, turn
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out to be of critical importance in solving differential equations.
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For another thing, group operations provide us with an ample supply
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of triadic relations that have been extremely well-studied over the
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years, and thus they give us no small measure of useful guidance in
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the study of sign relations, another brand of 3-adic relations that
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have significance for logical studies, and in our acquaintance with
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which we have scarcely begun to break the ice.  Finally, I couldn't
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resist taking up the links between group representations, amounting
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to the very archetypes of logical models, and the pragmatic maxim.
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Biographical Data for Marius Sophus Lie (1842-1899):
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We made the observation that the shift operators <math>\{ \operatorname{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think.  For one thing, groups, especially the groups that are named after the Norwegian mathematician [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842&ndash;1899)], have turned out to be of critical utility in the solution of differential equations.  For another thing, group operations provide us with an ample supply of triadic relations that have been extremely well-studied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3-adic relations that have significance for logical studies, and in our acquaintance with which we have scarcely begun to break the ice.  Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.
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http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
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</pre>
      
==Note 21==
 
==Note 21==
Jon Awbrey
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