🚀 go-pugleaf

On Thu, 03 Apr 1997 17:55:00 GMT, pdx4d@teleport.com (Kirby Urner)
wrote:

[snip]

>  Yet even conceptual
>points appear to occupy space, and in computer
>ray-tracing, have definite existence, just like
>everything else.  

I don't see what you're trying to say here.  First, I don't see why a
conceptual point - I take it that you mean here the abstract,
dimensionless entity - should appear to occupy space.  I can see
regarding it as part of the fabric of some space, or as an address,
but not as something that takes up space.  And I don't see how any of
these views precludes its definite existence.  (Yes, Virginia, there
is a (1, 2, 3).)

>Then we go on to define lines as 'one dimensional' 
>phenomena made of countless gazillion 'zero dimensional'
>points,

I hope not; 'infinitely many' is both more accurate and more concise!

> planes as stacks of 'one dimensional lines'
>and cubes as stacks of planes (well, not cubes 
>exactly, but some abstract infinite expanse we
>call volume, and then try to suggest is cubical,
>given our XYZ paradigm of 'three dimensional').

No, we don't *try* to suggest that it's cubical.  That's an unintended
consequence of the use of rectangular coordinates.  If we began with
cylindrical coordinates, we'd unintentionally suggest that space was
cylindrical.

But it seems to me that you're conflating two quite different things
here: Euclidean geometry, and Cartesian geometry.  A great deal of the
former can be done without any notion of dimensionless points, etc.
(And if the use of logic is the primary goal, some finite geometries
might be a better place to start anyhow, since they *do* permit all
sorts of different but understandable interpretations.)  As for the
latter, one need only take as an axiom that there is a 1-1
correspondence between points and addresses (coordinates) in order to
get all of the desired properties.

>All of this rhetoric contradicts the ray tracing
>paradigm, wherein relatively planar objects are 
>simply flatter, but not 'infinitely flat' -- and
>no concept of 'infinity' is required (computer
>chips don't take well to such an uncomputable 
>concepts).

What's the contradiction?  All of this is handled within the
traditional framework.  And no concept of infinity is *required* in
order to do traditional geometry.

>My point is that maybe some kids feel a kind of
>intuitive hostility to Euclideanism because it
>begins with this indoctrination in a bunch of
>unbelievable concepts ('outright garbage', 
>some of the less conformist kids may be 
>thinking), and then goes on to demand all this
>rigour and step-by-step logic.  Seems kind of
>hypocritical, doesn't it??

Not really.  We know that these 'unbelievable' concepts are useful,
and it is precisely those concepts that are counter-intuitive that
demand the most rigor and care in handling in order to avoid error.
This itself is a lesson that could usefully be taught.

Brian M. Scott