Re: Epistemology 201: The Science of Science

From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 03/04/05 

Date: 4 Mar 2005 04:37:27 -0800

Lester Zick says...

>>Asking where (spatially) the Big Bang took place is like asking what
>>is the longitude of the South Pole. The South Pole doesn't have a longitude,
>>or it has all possible values of longitude, because the South Pole is a
>>single point.
>
>I don't care where BB took place. I just care whether it had some
>origin

What do you mean by "had some origin"? The Big Bang was an *event*.
What does it mean to say the origin of an event.

>and whether we are assumed to be concentric with that origin

I don't know what you mean by "concentric with", either. Once again,
consider the 2D spacetime model of a sphere. The Big Bang is the
South Pole, and the spatial universe at a given moment is a line of
lattitude. Every point along a line of lattitude is as close to the
South Pole as any other point.

>Okay, Daryl. I offer my example to explain the geometric anomaly in
>omnidirectional cosmic red shifts and you reject it and insist on your
>own example to explain my thinking.

Well, the model I offered is exactly what is going on with
an expanding universe, except that the dimensionality of space
has been decreased from 3 to 1 for ease of visualization.

>You don't understand my example and I don't understand yours.
>It's another pitfall of reasoning by example

My model is not reasoning by example, it is reasoning by
suppression of unimportant details. The number of dimensions
of space are not relevant to the question of whether it is
possible to have an isotropic, homogeneous expansion.

>>You can't "look" in the direction of the past or the future. Any
>>direction you can turn is pointing in a spatial direction.
>
>And any direction you look you look into the past.

In the same sense, every direction you look, you are looking
toward the Big Bang.

>>That's not quite right. In non-Euclidean geometry, a "line"
>>is *defined* to be the shortest path between two points,
>>but the path must be *in* the space. So considering the
>>surface of the Earth to be a non-Euclidean 2D geometry,
>>the shortest distance between two points on the surface
>>of the Earth is a "great circle" connecting those points.
>>Great circles are the lines for the Earth.
>
>And straight lines are still the shortest distance between points in
>Euclidean and non Euclidean geometries alike even if you can't
>traverse them.

If the universe is non-Euclidean, then there *are* no
paths other than those through the space. You can't traverse
them because they don't exist.

>>That's pretty much right. Every non-Euclidean geometry can be
>>built up out lots of little pieces, each of which is approximately
>>Euclidean.
>
>Not at all what I had in mind. Every non Euclidean geometry has to be
>built up on the properties of straight lines plus further assumptions.

Why do you say that?

>>Basically, what you are asserting is that space *must* be Euclidean.
>>But why must it?
>
>Correct because straight lines are the shortest distance between
>points

That's true for non-Euclidean geometries as well. That's the *definition*
of a "straight line".

>>Why is a higher order space required? It's true that if you want
>>to embed the non-Euclidean geometry inside Euclidean geometry,
>>you have to use a higher-dimensional Euclidean geometry to do
>>the embedding. But why must it be embedded in Euclidean geometry?
>
>Because space is common to all geometries and Euclidean assumptions
>regarding straight lines are common to all points in any space. This
>is the sense in which I meant Euclidean geometry represents the limit
>for non Euclidean geometries in general.

That is not the way that geometry is defined. I sketched the way it
is defined these days: A manifold consists of a collection (possibly
infinite) of patches, each of which is like a little of Euclidean
space, together with connectivity rules (which patches connect to
which other patches). Euclidean geometry is involved in the very
small (every little patch is approximately Euclidean) but there is
no larger Euclidean space containing the whole collection.

>>>Your explanation for
>>>contiguous shapes seems to require a higher dimension than the one
>>>postulated for the shapes themselves.
>>
>>Only if you want to put together the object inside Euclidean
>>space.
>
>Which space do you prefer and continue to describe it in terms of
>dimensionality defined in Euclidean spatial terms. Otherwise there is
>no reason to use the term dimension at all.

Dimension refers to the number of independent direction vectors.
On the surface of the Earth we have two: North-South and East-West.
So the surface of the Earth is 2D. Space is 3D, because we can add
another direction: up-down.

>>The hole is only necessary if you try to embed a Kline bottle in
>>Euclidean 3D space. It's kind of like trying to draw a knotted rope
>>on a 2D piece of paper. You can't actually draw it without cutting
>>holes in the rope for it to pass through itself. But in Euclidean
>>3D you can have a knotted rope that doesn't have any holes. In
>>Euclidean 4D, you can have a Kline bottle that doesn't require any
>>holes.
>
>So my idea of a Kline bottle in 3D space is correct?

No. A Kline bottle has nothing to do with 3D. It's 2D, and it
requires 4D for embedding in Euclidean space.

>>The Kline bottle is 2D, but it requires 4D if you want to construct
>>it in Euclidean space. Similarly, a Mobius strip is also 2D, but
>>it requires 3D if you want to construct it in Euclidean space.
>
>Well here you use a different definition of dimensionality than the
>idea of embedding space.

Dimensionality is always the number of independent direction
vectors.

>>A line being the shortest distance between two points is true by
>>definition.
>
>It's true by assumption and exhaustion of curved alternatives as
>longer and not definable between points at all. You can never define a
>circle or any other curve or non straight line between points. If so
>you could square the circle.

I have no idea what that paragraph means.

>>No, it doesn't. The dimensionality of a space is the number
>>of independent directions in the space. On the surface of a
>>sphere, there are two independent directions: East-West, and
>>North-South. So it's two dimensional. 3D comes into play only
>>if you want to consider the sphere embedded in a higher-dimensionality
>>Euclidean space.
>
>Well I had this discussion at length a couple years back. Since the
>spatial dimensionality I'm interested in comprehending is defined in
>terms of mutually orthogonal straight lines,

That's the same definition I'm using, except that a direction
vector is not really the same thing as a "line" except in Euclidean
space. A line has length, while a direction vector only has direction.

>I expect we'll just have
>to disagree although I see no point to using the term dimension to
>refer to anything other than embedding space.

Why? It's a perfectly clear concept. If something is 2D, then it
takes 2 coordinates to specify a point. If it is 3D, then it takes
3 coordinates. Whether the space is Euclidean or not is irrelevant
to the dimensionality.

>>In other words, *if* you insist that everything is embedded in
>>Euclidean space, *then* what you say follows.
>
>Until someone shows me some non Euclidean space where straight lines
>aren't the shortest distance between points and curves are definable
>between points, I guess Euclidean space will just have to do.

We just did: the surface of a sphere. The shortest path between
two points on the Earth is the great circle route. 

--
Daryl McCullough
Ithaca, NY

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  • Re: Epistemology 201: The Science of Science
    ... The South Pole doesn't have a longitude, ... >Euclidean and non Euclidean geometries alike even if you can't ... But why must it be embedded in Euclidean geometry? ... >dimensionality defined in Euclidean spatial terms. ...
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    ... The South Pole doesn't have a longitude, ... >Euclidean and non Euclidean geometries alike even if you can't ... But why must it be embedded in Euclidean geometry? ... >dimensionality defined in Euclidean spatial terms. ...
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