Linear Thinking LO23024

Winfried Dressler (winfried.dressler@voith.de)
Wed, 27 Oct 1999 12:35:19 +0200

Leo shared with us:

>My puzzle is: is an emergence a change from a nD-world towards a
>(n+1)D-world?

Dear Leo,

on my meandering path of thoughts, I came accross two thoughts years ago
those thoughts up as you may pick up peculiar stones on a walk. I didn't
"tame" those thoughts so they are not truely mine. May be they are useful

1.) During my studies of physics I read a book by a (Hans?) Reichenbach.
Sorry I don't remember the title. He pondered the following question: in
order to understand a curved n dimensional space, is it necessary to view
this space embedded in a n+1 dimensional flat space?

For example in order to explain an expanding universe often a balloon
which is being filled with air is taken. You are asked to think of the
surface of the balloon as a two dimensional model of the universe. Or
gravitation is explained as the geometrical impact which has a mass putted
on a previously flat surface.

In both cases, the deformation is realized while looking at it from the
outside - a flat three dimensional space.

Reichenbach argues that such a higher dimensional flatness is not
necessarily required to perceive and understand the topological properties
of our universe - It need not be thought of as embedded in higher
dimensional flatness. Or as I would interpret in the context of our
discussion here: In order to live in nonlinearity one need not to assume
this nonlinearity to be covered by a higher order linearity.

2.) The topological properties of a n-dimesional space depend very much on
n. So changes from 2->3 or from 3->4 cannot be summerized as change from
n->n+1

The most striking example for me was to see, that ONLY in three
dimensional space + one dimension time, the (Newtonian) differential
equation of motion and the (Hamiltonian) integral equation are equivalent.
While Newton's force F is the CAUSE of motion (define the starting
conditions and in principal, the motion is determined), the motion in
Hamiltons view will be such as to minimize "Wirkung" (EFFECT) S. In any
other dimensionality, it could be decided from the motion of particles
whether the driver is causal or teleological. Not so in our worlds
dimensions.

What does this mean? Sorry, no idea. But somehow it has its beauty for me.
And the dimesionality in which we live is not arbitrary.

Liebe Gruesse,

Winfried

--

"Winfried Dressler" <winfried.dressler@voith.de>

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