Math Reflections and Resonances - Part 6
The Idea of congruence
If two plane
figures are of the same size and have the same shape, the figures are said to be congruent.
More specifically, when one plane figure is congruent with some other plane figure, then,
each figure can be mapped into the other by means of a transformation which:
(a) moves
points;
(b) does not
affect the incidence relations existing between points and lines;
(c) permits
parallel lines to remain parallel; (d) leaves the areas of the figures intact;
(e) does not
change the length of line segments;
(f) does not
alter the angles which exist between the lines of the figure.
Two
congruent figures are said to be directly congruent when, as a result of their having the
same orientation in relation to a given fixed orientation of the plane, they can be
transformed into one another after one has performed a series of translations and
rotations of the plane.
Translations,
rotations and reflections are all examples of congruence transformations.
Such transformations can be used, singly or in combination, during the course of an
analysis of plane figures in order to establish whether or not any two figures found on
the plane are congruent.
Similarity
is a somewhat less stringent basis for comparison of plane figures than is provided for by
congruency. Two geometric figures are said to be similar when they have the same shape
even if they do not have the same dimensions.
As such, the
figures being compared do not have to be exactly the same as long as either: (a) the sides
or line segments of the respective figures maintain the same ratios with respect to each
other (this is known as the ratio of similarity); or, (b) all the corresponding angles of
the two figures are the same. Two figures which are similar can be transformed into one
another by means of any one-to-one geometric transformation that leaves the corresponding
angles of the figures intact.
Two
structures are said to be hermeneutically congruent if one structure can be mapped into
the other structure by means of a sequence of hermeneutical transformations that preserves
the following properties and conditions:
(a) the
ratio of constraints and degrees of freedom of the structure into which another structure
is being mapped is not altered on the level of scale into which the mapping is done;
(b) on any
given level of scale, the character of the phase relationships of the structure into which
it is being mapped are not changed;
(c) for any
given ontological point structure on a given level of scale, its hermeneutical image can
be shown to manifest some 'minimal' degree of adherency in its E-neighborhood in relation
to the ontological point structure to which identifying reference is being made.
What
constitutes a "minimal degree of adherency" will vary from situation to
situation. However, in all circumstances, the number of points of the E-neighborhood which
overlap with the ontological point structure must lend plausibility to the mapping and not
just possibility.
Something
considered to be a possible mapping (i.e., according to what the structural
character of a context may permit in the sense of being consistent with the character of
the structural context ) becomes plausible when one can point to at least several
pieces of evidence that support a given mapping possibility. In other words, in order to
be permitted to go from possibility to plausibility, one is being asked to produce a
mapping that goes beyond merely conforming to, or being consistent with, the structural
character of the latticeworks being connected through the mapping.
The more
pieces of evidence there are to support a given mapping, the more plausible the mapping
becomes. Obviously, some mapping proposals are more plausible than others are.
(d) there
must be a homeomorphic or bijective mapping capable of being established on any given
level of scale between primary or key point structures of the ontological latticework and
primary or key point structures of the hermeneutical latticework. This suggests that one
may have to distinguish between essential and peripheral congruence. Moreover, the
essential/peripheral congruency distinction connects up with what is meant when speaking
of a minimal degree of adherency since this would concern essential congruency, rather
than peripheral congruency.
Similarly,
plausibility concerns essential congruency among a given set of E-neighborhoods considered
to constitute key or primary point structures in the latticeworks being considered. At
some point, a mapping proposal goes from being highly plausible to being reflective when
both essential and peripheral congruency have been established.
(e) the
relationship between the ontological latticework and the hermeneutical latticework must
be, at a minimum, analogical in character;
(f)
obviously, there are degrees of congruency that can be affected by the extent to which any
given mapping can satisfy the foregoing conditions;
(g) the
congruency mapping which is established through a series of hermeneutical operator
transformations cannot leave an implausible tension between the interrogative imperative
and congruency relationships such that the former overshadow the latter (this is the
remainder theorem discussed in an earlier chapter). In effect, this means that although
some degree of this kind of tension can exist without undermining the claim for
congruency, if there are too many unanswered questions of an essential or key nature, the
claim for congruency becomes implausible, even if it subsequently turns out that in the
light of further experiential data, reflection and the use of a new sequence of
hermeneutical transformations, the two structures can be shown to be congruent.
(h) the
greater the number of levels of scale on which congruency mappings can be established, the
greater will be the congruency between a given hermeneutical structure and a given
ontological structure. This means that if one is comparing two sets of congruency mappings
in relation to one and the same ontological structure, that mapping which is effective
across a greater number of levels of scale will be the more congruent of the two mappings;
(i)
hermeneutical similarity requires a less exacting correspondence or set of mapping
functions between hermeneutical and ontological structures than is required by
hermeneutical congruence.
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