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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 
> 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.
P.S.
(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 antisocial. (But hands up all those who were
attracted to math for that very reason?)
 
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