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Discussion: All Topics in Geometry
Topic: Is a rhombus a kite?

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Subject:   RE: Is an integer rational? (was RE: Is a rhombus a kite?)
Author: gsw
Date: Jun 7 2005
On Jun  6 2005, Alan Cooper wrote:

> Well this makes me wonder what are
> *the* integers?
> Surely their image in the rationals does satisfy
> all of the axioms required of an instance of the integer number
> system.

In fact, as for any axiomatic system, even when the
> axioms define a structure that is unique up to isomorphism there may
> be many distinct instances of that structure. (There are at least a
> couple of models of the integers constructible within Zermelo-
> Frankel set theory and any of these could be used to generate a
> model of the rationals by taking ordered pairs, but there are also
> other models of both number systems that might be constructed from
> other choices of "fundamental" system.)

of course the subset of rationals that the integers map to is isometric to the
integers - or it wouldn't map. But that dont make integers rationals. You might
think I'm splitting hairs - but actually I'm being pedantic....

> To my mind, 'Mathman' is
> right, and if an embedding exists then it is correct to say that one
> structure "is contained in" (or even "is a subset of") the other.
> Alan

not an embedding, a mapping.

(back to Rhombuses and Kites
- well actually
> Squares and Rectangles)
gsw also wrote: > Kids always complain
> when you tell them that a square is a rectangle, and usually
> want
> to add an exclusive clause to the definition of rectangle when
> you point out to them that the normal definition includes squares.
> Has anyone studied or considered why kids naturally tend to do this
> even though they have no difficulty with the idea that a cat is an
> animal? Is it based on something fundamental about how we perceive
> shapes, is it because there is only one named subclass of
> rectangles, or is it just a result of how the material is first
> presented to them?

This is interesting, I think (unlike all that Foundations nonesense...)  I think
it's partly because the first activity kids do with shapes is categorizing them
- sorting them into piles - which is done in a way that implies exlusivity. And
there also seems to be some piece of wiring that needs to be reminded that "All
A's are B's" != "All B's are A's"

Mostimportantly, it ties into the social nature of language - if you ask someone
how much money they have on them and they say $10, that implies, given our
social habits, that they don't have $20. But of course, if they have $20 they do
have $10... If you ask someone to say what the design is on their new carpet and
they say "ellipses" you would reasonably be surprised if it turned out to be
circles - even though circles are ellipses. Describing a square as a rectangle
breaks the usual contract. It's important to explain this in a way that doesn't
make math seem just contrary and anti-social. (But hands up all those who were
attracted to math for that very reason?)

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