Re: Epistemology 201: The Science of Science

From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 03/03/05 

Date: 2 Mar 2005 16:08:27 -0800

Lester Zick says...
> stevendaryl3016@yahoo.com (Daryl McCullough) in comp.ai.philosophy wrote:

>>>Omnidirectionality implies a privileged position in
>>>space whether we consider three dimensions or four.
>>
>>No, it doesn't. Go back to the case of a balloon inflating.
>>The *surface* of the balloon (that is, the 2D rubber ***)
>>has no center. The center of the sphere is not part of the balloon.
>
>Incorrect, Daryl. Spatial and temporal metric eccentricity is what
>defines directionality. A balloon certainly has a center of air and if
>it is inflated eccentrically it will not expand omnidirectionally with
>uniformity.

Okay, I'm not sure whether when you talk about the "center"
of the balloon, you mean the center of the *sphere* (which, as
I say, is not a part of the balloon), or the hole into which
air is blown. If you mean the air hole, you can get expansion
without such a hole. If you take a perfectly spherical rubber
ball filled with air up to 20000 feet, it will expand, with
no point on the ball that is the center of the expansion.

>I assume nothing from the outset except that spatial and temporal
>metrics are either concentric or eccentric and that if eccentric must
>result in anisotropic temporal dilation unless the eccentricity is
>radial.

Try reading a book on General Relativity. The case of an isotropic
expanding universe with no center is the simplest cosmological
solution.

I don't know what you mean by "the spatial and temporal metrics".
In relativity there is only one metric, that works for both time
and space. The metric is the function that determines the proper
distance between two spacetime points.

>If this is what you call Euclidean space I'm curious as to
>what alternative there is apart from simply calling the space non
>Euclidean. To me it seems more cartesian than Euclidean.

Cartesian applies to coordinate systems, while Euclidean applies
to geometry. But they are related, in that the geometry is Euclidean
if and only if it can be given a global Cartesian coordinate system.

>>For example, an intrinsic description of the surface of a 3D cube
>>consists of 6 squares A, B, C, D, E, F
>> __
>> |E |
>> __ __ __|__|
>> |A |B |C |D |
>> |__|__|__|__|
>> |F |
>> |__|
>>
>>with the stipulations that
>>1. The top edge of E is identified with the top edge of B.
>>2. The left edge of E is identified with the top edge of C.
>>3. The right edge of E is identified with the top edge of A.
>>4. The left edge of A is identified with the right edge of D.
>>5. The left edge of F is identified with the bottom edge of C.
>>6. The bottom edge of F is identified with the bottom edge of B.
>>7. The right edge of F is identified with the bottom edge of A.
>>etc.
>>
>>Much simpler, the surface of a 3D torus (or "doughnut") can be
>>defined by a single square, with the stipulation that the right
>>and edges are identified, and the top and bottom edges are identified.
>>
>>In a similar way, you can give intrinsic characterizations of
>>nonEuclidean 3D objects such as a Kline bottle or a hypersphere
>>such as the universe. There is no need for a higher dimensional
>>space to embed these objects.
>
>Well your first illustration suggests that you can model the surface
>of a 3D cube the way you indicate. Simply saying the top edge of E is
>identified with the top edge of B doesn't make it happen.

Sure, in the same way that saying that the northern border of the US
is the same as the southern border of Canada doesn't make it so. Geometry
is *descriptive* of the universe, and my point is that everything you
need to know about the geometry of the universe is conveyed by a
complete collection of maps for parts of the universe, together with
a specification of which maps are overlapping. There is no need to
postulate a higher dimension in order to describe a non-Euclidean
universe.

>A Kline bottle is a 3D object.

I don't know why you would say that. Its intrinsic geometry is
2D---it is a curved surface. A Kline bottle cannot
be embedded in Euclidean 3D, so I don't know in what sense you
would call it a 3D object.

Maybe you don't know what a Kline bottle. One way to describe it
is to take two Mobius strips, and glue their edges together. It
can't be done in Euclidean 3D.

>I don't see why you called it non Euclidean
>unless you're using some specialized definition for non Euclidean.

Euclidean geometry obeys Euclid's axioms for geometry. In particular:

   1. Given two points, there is exactly one line connecting them.
   2. Given a line, and a point not on that line, there is a second
   line that passes through that point that is parallel to the first.
   3. The sum of the interior angles of a triangle is 180 degrees.

These are not true of non-Euclidean geometry.

>As far as I am concerned Euclidean space is just the three dimensional
>manifold,

No, it's a three-dimensional manifold with specific connectivity
and geometry. The surface of a sphere is a 2D surface, but it
isn't a Euclidean 2D surface. 

--
Daryl McCullough
Ithaca, NY
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