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Topic: Cardinality vs. Ordinality
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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Cardinality vs. Ordinality
Posted: Jun 5, 2009 1:25 AM
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A topic mathematicians could weigh in on, re our digital
math track, is what segments they'd like to see (think of
Sesame Street clip length) on "Cardinality versus Ordinality".

Take as a given that we want this distinction. Cardinality
relates to naming, therefore distinguishing one from another,
so you could talk about equal versus not equal, but in
pure form there's no ordering, or the ordering isn't
by simple ranking.

Ordinality relates to sorting, including alphabetically,
also indexing, and concretely in number types where > and
< are defined, not just == and !== (not equal).

In computer languages, we often distinguish data structures
depending on whether they include a concept of ordering
or partial ordering (or no ordering), which makes this
stuff more hands-on and concrete (how did they ever teach
this stuff pre-computer? In lots of good ways I'm sure).
Like in Python we have the cardinal dictionary, the
ordinal list (3.1 has a new kind of ordinal dictionary
I gather). We also have the option, when defining types
of our own, to define __lt__, __gt__, __eq__ -- or not,
as the case may be (for operators <, > and == respectively).

Remember, we get into "operator overloading" quite early
in DM, don't do like some CS curricula and wait until
2nd year, because "object oriented" is itself considered
"advanced". In chronological terms, "object oriented"
came later, but its purpose was to be more primitively
conceptual i.e. as a philosophical logic, we want to start
with OO, now that we've got it, not "build up to it"
over a long tedious period (recapitulating every develop-
ment in temporal order -- the historical order -- is
*not* necessarily an intelligent way of connecting the
dots in curriculum writing, I hope many here would agree).

We want more emphasis on cardinality in part because of
Supermarket Math, but labeling vertices of polyhedra or
polygons is a fine example of labeling without ordering
in a typically linear fashion ("addressing" and "ordering"
are related but not identical concepts).

I'm not saying math-teach is the best place to organize
this structured discussion. We have lots of other tools
in place, are setting up workflow. However, this is a
good example of the sort of base conceptual level at
which green field development needs to occur. Midhat
Gazale's book 'Number' might be a good source if you're
looking for more background.

Kirby



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