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Cardinality vs. Ordinality
Posted:
Jun 5, 2009 1:25 AM


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 handson and concrete (how did they ever teach this stuff precomputer? 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 mathteach 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



