ONT Re: Differential Logic -- Series A
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note A20
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| the way of heaven and earth
| is to be long continued
| in their operation
| without stopping
|
| i ching, hexagram 32
You may be wondering what happened to the announced subject of "Differential Logic".
If you think that we have been taking a slight excursion my reply to the charge of
a scenic rout would be both "yes and no". What happened was this. We chanced to
make the observation that the shift operators E_ij form a transformation group
that acts on the set of propositions of the form f : B^2 -> B. Group theory
is a very attractive subject, but it did not have the effect of drawing us
so far off our initial course as one might at first think. For one thing,
groups, in particular, the special family of groups that have come to be
named after the Norwegian mathematician Marius Sophus Lie, turn out to
be of critical importance in the solution of differential equations.
For another thing, group operations afford us examples of 3-adic
relations that have been extremely well-studied over the years,
and thus they supply us with no small bit of guidance in the
study of sign relations, another class of 3-adic relations
that have significance for logical studies, in our brief
acquaintance with which we have scarcely even begun to
break the ice. Finally, I could not resist taking up
the connection between group representations, which
constitute a very generic class of logical models,
and the all-important pragmatic maxim.
Jon Awbrey
Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o