To replicate the explanation i gave on IRC,
(to use subscripting in pseudo haskell)
for any type where we can define some sort of distance (induced via a norm via dist a b = norm (a -b ), ignoring overflow issues)
lets define a quantitative version of equality
a ==_{r} b = if dist a b <= r then True else False
then we use the triangle inequality (dist a c <= dist a b + dist b c)
to get the following quantitative analogue of transitivity
a ==_{r1} b && b ==_{r2} c IMPLIES a ==_{r1 + r2} c
this is a bit more general (and weaker) than the notion of equality that we're accustomed to, but still a pretty natural idea.
you can consider more general things than using the + function too, like min/max/sum of squares etc. But I leave that as a fun exercise for the reader.
this gets into talking about reasoning about things using tools from Analysis rather than Algebra, and that sort of modeling is pretty powerful.
I guess my point is Analysis is a very powerful far reaching mathematical tool, and only considering models that elide that is ... :)