To Floor van Lamoen and John Conway especially, and to all who contributed questions, opinions, complaints, concerns,
This discussion of exclusive/inclusive definition has been - is - highly interesting and educational.
Now: it seems undisputable that the concept of counting numbers precedes the concepts of rationals, integers, and so on both in individual development and in social/mathematical history, but what about the geometric figures?
I think that a circle comes first (and I see it not as a disk although that's what's usually presented to toddlers being prematurely tutored in "shapes") both for an individual and society? Then what?
Mary Krimmel email@example.com
you several wrote what's below and much more:
>Hi, > >My actual problem with those definitions is not that they are difficult >to deal with. In my honest opinion these definitions are unmathematical >in the sense that mathematics generalizes things. > >The trapezoid is a weaker form of a rectangle (which is a weaker form of >a square), and as such theorems on geometric properties of trapezoids >naturally include rectangles (and squares). I am afraid that if one >teaches pupils to be precise on these exclusive definitions, one teaches >them to focus on the wrong things, and perhaps forget the important >concept of generalization. > >We wouldn't like to use exclusive definitions for number sets like >Natural numbers, Integers, Rational numbers, Real numbers and Complex >numbers, do we? It's so good that those include each other! The use of >exclusive definitions of - for example - trapezoids, is rather the same. > >Kind regards, >Floor van Lamoen. > >John Conway wrote: >> >> On Tue, 8 Aug 2000, Floor van Lamoen wrote: >> >> > No, No!! >> > >> > One must call it the "Trapezoid-Rectangle-or-Square Rule" of course, if >> > one really wants to use exclusive definitions. >> >> Thanks, Floor! This just goes to illustrate my point that it's >> so hard to work with the exclusive definitions that even the best of >> us (as I modestly term myself) can't actually manage to do it! >> >> John Conway > > > >