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.
http://en.wikipedia.org/wiki/Normed_vector_space and the associated page on
Hilber tSpaces can be a useful starting point i guess.
I guess my point is Analysis is a very powerful far reaching mathematical
tool, and only considering models that elide that is ... :)
On Sat, Sep 27, 2014 at 1:12 PM, Albert Y. C. Lai
On 14-09-27 10:02 AM, Carter Schonwald wrote:
Its approximately transitive. Distances always obey the triangle Inequality. Good enough for geometry. Also the emphasis is on the geometry / distance.
You've got me curious. How do we define "approximately transitive"?
(I am not one of those who want to rid floating point of Eq and Ord, or rid of floating point altogether.)
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