Re: Epistemology 201: The Science of Science

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 02/27/05 

Date: Sun, 27 Feb 2005 15:51:29 -0500

Albert wrote:
> robert j. kolker wrote:
>
>>
>>
>> Lester Zick wrote:
>>
>>> You know, Wolf, I've never seen an explanation for BB that doesn't
>>> first regress the analysis to two dimensions and then tell the reader
>>> to generalize it to three dimensions. Why don't we just begin where
>>> the action actually takes place: three dimensions. Show me how all the
>>> raisins in an expanding raisin pudding expand omnidirectionally in
>>> equal proportion to distance away from any pov except the center.
>>
>>
>>
>> The analogy illustrates that an unbounded closed manifold does not
>> have a center. Any old point will do. He is not generalizing from a
>> two deminsional closed unbounded manifold, he is using it as an example.
>
>
> If the balloon is not a "two dimensional closed unbounded manifold" as
> you say then it is not an example at all, being not even analogically
> accurate. Perhaps you can offer a /proper/ analogy.

[...]

The surface of the balloon is an unbounded 2D manifold. The spheroid
enclosed by it is not.

NB that "unbounded" only means that it has no boundaries - ie, if you
move about within the manifold (that means on the surface, not inside
the spheroid enclosed by it) you will never cross a boundary. ** Not
that a two sided surface, such as is illustrated by a *** of paper,
may have a boundary - the one between its two sides. The two sides of
the surface of the balloon are each unbounded, though, as the only way
to get from one side to the other is to punch a hole thorugh the surface
- which automatically creates a boundary (and deflates the balloon raher
suddenly, but we'll assume and "ideal" balloon that won't deflate when
we do this, or else all sorts of irrelevancies will be introduced into
the discussion.)

Is it possible to have a one-sided surface? Sure.

Like I said, imagining an unbounded 3D manifold is a little harder than
seeing an unbounded 2D manifold right before your eyes. I seem to be
vindicated by several reactions to my attempt to make a moderately
difficult (because unintuitive) concept a little easier to understand. :-)

** The unwillingeness to accept this elementary mathematical notion as
it applies to the surface of the spehroid on which we live has caused a
great deal of bloodshed.

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