3 May 2016
Jeff, all:
In my March 24 post I expressed interest in your remarks about
Peirce's mathematics and theory of the self. With a hint from your
March 29 response I have located what I'm curious about in your post
dated 4 July 2014 from the De Waal Seminar. You write:
"My first suggestion is that we start by thinking about the self as a
relationship between representations. In Euclidean geometry, for
instance, we say that one figure is congruent with another if we can
move one around and superimpose it on top of another. The basic ways
in which figures can be moved in a diagram is by translation, rotation
and reflection. It turns out that reflection is the fundamental kind
of symmetry on a Euclidean surface because we can achieve the same
effect as translation by reflecting over two lines. The same is true
of rotation (but the lines cross).
[quotation continued:] Before we introduce any ideas from projective
geometry, how might we use this geometric notion of reflection to
think about the symmetries that hold between one representation and
another representation that is, in some sense, 'congruent' with it?
Or, if you prefer a weaker relation, how is one representation similar
to another? In what ways might one representation mirror another?"
In my rudimentary understanding of reflection as an operation upon
graphs on a Euclidean surface, certain facts about points on the
reflected graph necessitate certain facts about points on the
reflecting graph. For example, given a coordinate system with the
graph A of a 1-1 function, there is a reflecting graph B such that for
any point on A specified by the x coordinate _a_ and the y coordinate
_b_, there is a point on B specified by the x coordinate _b_ and the y
coordinate _a_. This gives reflection about the graph of the function
y = x. Graphs of other relations give other reflections, e.g.,
reflections about the x axis, y axis or the origin.
There are conditions on whether there is a reflection and if so what
it is like.
First, there are conditions on the algebraic properties of that which
is being graphed. It matters whether it is a relation, whether the
relation is a function, what results from negating the function, or
negating the function's argument, and whether the function is many to
one or one to one.
There are corresponding conditions on graphic representation, which
include provision for a coordinate system with an x and y axis, their
intersection at a point of origin, points specifiable by x and y
coordinates, and the possibility of various symmetries between graphs
with respect to this system.
Extending these rudimentary remarks, I suppose reflection _might_ be
best understood as conditioned by the movement of a point from one
place to another, represented graphically by having a point on the
graph maintain its identity through changes in facts about it, such as
having one or both its coordinates change value from positive to
negative or vice versa, or having its x coordinate become its y
coordinate and vice versa. More needs to be said to account for
preservation of identity through such changes.
It also seems important to be more careful than I am above in keeping
straight whether one is speaking of a graph as a legisign or a
sinsign; it should be recognized that reference to "points" as part of
a sinsign may be misleading.
Are we on the same page so far?
Best,
Charles
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