I think there is a very interesting question here.
In a non-technical sense, "congruent" means "agreeing; corresponding,"
and *as we would take in ordinary usage*, agreements and correspondences
are matters to be negotiated. In this case, Gauss proposes that the
criteria for correpondence is that the difference is measured by the
mudulus, but I can think of others, such as "a is relative to b iff a
and b have a common factor." (not an equivalence relation, of course)
There would have to be some discussion in the community as to whether
such correspondence is an interesting one to consider.
The interesting point, however, is that in producing meaning for a given
text we always do it in relation to what I call a "kernel" (objects and
locally undisputed truths about them), and those can be quite varied for
the same text. A lawyer would probably produce for "congruent" meanings
related to "incongruent," particularly having to do with arguments.
Sam's kernel had "figures" in it, but also "numbers," while Ken
Pledger's had "numbers" and had "dictionary."
What strikes me is that we are constantly discussing what people meant,
but very rarely we discuss it as meaning *the meanings those people
produced for a given text*, which would imply discussing those kernels
(as I call them).
As with the good old argument about the "existence" (or not) of a
geometric algebra in Euclid, I think there is a matter of agreement (or
not) in question: essence, plausibility or meanings *we* produce. What
degree of neutrality is required if we are to unveil what a person
*really* meant, and is this achievable?
Romulo Lins