File spoon-archives/heidegger.archive/heidegger_2003/heidegger.0306, message 50


Date: Wed, 11 Jun 2003 22:42:47 +0200
From: artefact-AT-t-online.de (Michael Eldred)
Subject: Re: The dimension of a manifold


Cologne 11-Jun-2003

Ed Wall <ewall-AT-umich.edu> schrieb Tue, 10 Jun 2003 21:18:56 -0400:

> It is clear that somehow I assumed more was obvious in my reference
> than there actually was. Apologies. I take Eves as saying something
> very different than a reduction of the world to numbers or tuples -
> in fact, just the opposite. My understanding, and I could be
> incorrect, is that when he speaks of geometers he is not speaking of
> analytical geometers, but of geometers more or less in lineage of
> Euclid. For them such space is not given by a 3-tuple and, hence,
> four dimensions are not given by a 4-tuple. However, space is somehow
> known and navigated, for example, via solids, planes, and lines. This
> is still a way of thinking and doing mathematics and is understood,
> among most mathematicians, as not something that reduces or should
> reduce to arithmetic. What I thought interesting (and I seem to be
> alone in this thought - smile) is that the dimensionality, in a
> sense, increases when space is characterized in terms of lines or
> spheres.

Ed,
It seems that I have missed something in your original post.

If "a set of geometrical elements ... can be labeled with a real, continuous
coordinate system, [and] .... the dimension of the manifold is the number of
coordinates needed to determine a general element of the set... " (Eves), then
what prevents a geometric space of, say, spheres from being represented by
quadruples? Isn't it possible to make such a co-ordinate reduction?

I take your point that "dimension depends not only on the "space," but also upon
the fundamental elements that make up the space"  (Eves), and this is certainly
in line with Aristotle's understanding of _topos_ as the place occupied by a
being (here a geometrical entity) (and thus "space" cannot be considered as a
homogenous space of points), but does this offer essential resistance to a
co-ordinate reduction (to an n-tuple)?

Regards,
Michael
_-_-_-_-_-_-_-  artefact text and translation _-_-_-_-_-_-_-_-_-_
_-_-_-_-_-_-_-_-_-_-_-_- made by art  _-_-_-_-_-_-_-_-_-_-_-_-_-_
http://www.webcom.com/artefact/ _-_-_-_-artefact-AT-webcom.com _-_
_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_-
_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_


>
>
> The other thing and this is not meant as a criticism: is that your
> mapping to the reals has some interesting problems. Consider (and, of
> course, there is the interesting question if .9 (repeating) is what I
> say it is - smile).
>
> 1, 1, 1 = .9 (repeating), .9 (repeating), .9 (repeating) ->
> .9(repeating) = 1 -> 0, 0, 1
>
> Here  -> corresponds to your map. One can, of course, quibble about
> what .9 (repeating) represents, but it is clear that your
> digitization of space (smile) is more arithmetical than mathematical.
> Having said all this let me note that your method nicely epitomizes
> something that seems mathematically other than a Cartesian reduction.
> There is an interesting way in which it has, one might say,
> non-Euclidean aspects (smile). However, it is a reduction that is not
> atypical (for a number of interesting reasons) of the zero/one world.
>
> Ed Wall
>
> >However, I am not sure about where the decimal point
> >>  goes. For example does 1,0,0 correspond to 100 and 0, 0, 1 correspond
> >>  to 1?
> >
> >x=3.141529 y=2.718281 z=1.414213
> >
> >give the number r=321.174411184522281913
> >and for x=13.141529 we would have r=100321.174411184522281913
> >
> >and so on, for if any number (or so) can be written (with approximation) in
> >binary, then our number is a complicated way of counting three numbers in
> >one, in the base 1000.
> >
> >It is a simple way of putting R and R^3 in a bijective relationship. So, it
> >seems that all our digital casting does, is to reduce the qualities of our
> >world to numbers, and this is made possible by the fact our world is
> >one-dimensional (yet tri-drectional spatially plus Minkowskian spatialized
> >time, four directions) and its dimension is distance, which is being put in
> >a bijective relationship with R (the set of real numbers).
> >
> >Therefore, one may play all his life with multi-directional spaces, yet not
> >know how a (really) two-dimensional or three-dimensional space looks like.
> >But, we have a hint in our thoughts and emotions, to which we cannot simply
> >attribute distantial numbers. Of course, one could measure their intensity,
> >yet the same way a computer makes no sense of a Charlie Chaplin DVD (though
> >it is expressed in numerical quantities), but we do make sense of it through
> >watching it, the same way would he be unable to say (while having access to
> >such numerical input) what is really going on in one of us, who is being
> >measured. It takes awareness to make sense of awareness.
> >
> >Gigantomachia peri tes ousias!
> >
> >Tudor Georgescu
> >
> >http://intellect-club.nl.eu.org
> >
> >Fax +1-775-245-5922





     --- from list heidegger-AT-lists.village.virginia.edu ---

   

Driftline Main Page

 

Display software: ArchTracker © Malgosia Askanas, 2000-2005