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 _-_-_-_-_-_-_- artefact text and translation _-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_- made by art _-_-_-_-_-_-_-_-_-_-_-_-_-_ http://www.webcom.com/artefact/ _-_-_-_-artefact-AT-webcom.com _-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_- _-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ > > > 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 > >_-_-_-_-_-_-_- artefact text and translation _-_-_-_-_-_-_-_-_-_ > >_-_-_-_-_-_-_-_-_-_-_-_- made by art _-_-_-_-_-_-_-_-_-_-_-_-_-_ > >http://www.webcom.com/artefact/ _-_-_-_-artefact-AT-webcom.com _-_ > >_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_- > >_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ > > > > --- from list heidegger-AT-lists.village.virginia.edu ---
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