File spoon-archives/heidegger.archive/heidegger_2003/heidegger.0306, message 70


Date: Wed, 18 Jun 2003 15:40:33 +0200
From: artefact-AT-t-online.de (Michael Eldred)
Subject: Re: The dimension of a manifold


Cologne 18-Jun-2003

Ed Wall schrieb  Sun, 15 Jun 2003 22:37:08 -0400:

> Michael
>
>     Thanks. I see the confusion. I would say Eves talk about the
> continuous labeling is somewhat misleading. That is just, so to
> speak, for bridging representations between, for example, analytical
> and plane geometry. What I had wanted to point to (smile) was Eves's
> comment:
>
> In short, dimension depends not only on the "space," but also upon
> the fundamental elements that make up the space.  For the first time,
> geometers had defined perceivable spaces of more than three
> dimensions.
>
> and I was thinking about the conversation you and I and Stuart had
> awhile back about Aristotle (and, to a degree, Heidegger's excursus
> in Plato's Sophist) views of point, line, plane, and solid.
>     First, about resistance to the reduction to tuples. This kind of
> reduction can be done in analytical geometry. However, and I'm not
> sure how to say this, the 'geometric' proofs become considerably
> complicated and non-intuitive.

Ed,
Hegel says that "geometry deals with abstract sensuous
representations/imaginations/ideas" (Vorstellungen; Enz. I Werke Bd. 8:67). This
implies that analytical geometry loses something, namely, the direct sensuous
Anschauung/intuition/looking-at, and is a numerical calculus based on equations. So
you are right about its "complicated and non-intuitive" character. But at the same
time, the mathematical entity becomes calculable and therefore also digitizable.

> There seems to be a way in which
> Aristotle's formulation - almost as if it over determines - somehow
> provides a little something extra (and this is tricky since things
> seem necessary and sufficient in both geometries). So there is a kind
> of resistance, but perhaps not of the kind you are indicating.
>     This seems to takes me to Eves's point about dimensionality of a
> manifold (this is something Heidegger also refers to - as I don't
> read German well I'm not sure however what he intended). Eves is,
> most likely, using that definitional bridge to say 4 with lines, but
> the interesting question is what does that imply for Aristotle as a
> solid is that which is divisible (in the English in the excursus
> Heidegger uses 'resolvable') in three senses (in the English
> Heidegger uses 'dimension'). Is there actually a perceivable four for
> Aristotle and geometers via Euclid?
>     Anyway, there is this odd bridge (I think you call it a Cartesian
> casting) that Eves wants to build between tuples and Euclid (and,
> thus, perhaps Aristotle) - this is, of course, an activity
> mathematicians engage in all the time. So I was wondering what four
> looked like from Aristotle's side of the bridge (and four is for both
> lines and solids).

Following up Eves', "space is three-dimensional in points, it may be shown that it
is four-dimensional in lines, and also in spheres" I thought, perhaps too
simple-mindedly, that the four-dimensionality of space in spheres would boil down to
4-tuples with three co-ordinates for the centre and one co-ordinate for the radius,
since that uniquely determines a sphere. Is that adequate? Does that count as a
"perceivable four"?

In lines, one could say that a line is uniquely determined by two points. That would
make 2x3 = six co-ordinates, but there is overdetermination (overkill) here, and one
could reduce to a standard co-ordinate representation of, say, the point of
intersection with the xy-plane (two co-ordinates), the angle of intersection with
the xy-plane and the angle made by the projection onto the xy-plane with the x-axis.
That makes four co-ordinates in all. Does that count as a "perceivable four"?

With regard to both line and sphere one requires a notion of continuity, and
Aristotle's seven-step build-up of an ontology of continuity (_syneches_ cf.
Heidegger's _Sophistaes_ lectures) and 'hanging together' (Zusammenhang,
_echomenon_) seems superior to the later analytical notions of continuity in terms
of delta-epsilon limit definitions precisely because the _mode of being_ of
continuity is seen as a problem by Aristotle.

"What is ontologically most complex, i.e. the geometric figures, is most simple for
sensuous perception, but is very unwieldy for calculation. And conversely: what is
ontologically more simple, i.e. the arithmetic entities, is not accessible to
sensuous perception but can be calculated without any difficulty." (Draft Casting of
a Digital Ontology)

> Also I have begun to wonder about some of what
> Heidegger was thinking about Weyl (Weyl was more or less of the
> Cartesian persuasion - I can't find my copy of his Space-Time-Matter
> though so am going by his Theory of Groups and Quantum Mechanics). I
> am beginning to have a suspicion he might approve of what is
> presently called differential geometry (smile).

Unfortunately I don't have Weyl either to have a look at him.

Michael
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>
>
> Ed Wall
>
> When the you and I and Stuart were discussing Aristotle's point
> >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
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> >_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_-
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> >
> >




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