Date: Mon, 30 Jun 2003 11:05:18 +0200
From: artefact-AT-t-online.de (Michael Eldred)
Subject: Re: The dimension of a manifold


Cologne 30-Jun-2003

Ed Wall schrieb  Sat, 28 Jun 2003 15:34:28 -0400:

> Michael
>
>     Apologies for my lateness in replying. Comments below.
>
> Ed Wall
>
> >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.
>
> My first reading of sensuous was in terms of a pleasuring of the
> senses. Although that idea is appealing, it was not what I was
> thinking about.

Ed,

Yes, both of the possible translations of German "sinnlich", viz. "sensuous" and
"sensible", have their ambiguities, even though both have the basic meaning of "Of
or pertaining to the senses; derived from, perceived by, or affecting the senses;
concerned with sensation or sense-perception." (OED)

> However, the idea of looking-at seems quite right.
> Consider the following right triangle inscribed between two parallel
> line (one through upper vertex and one through the base (which I
> haven't drawn)
>
> ------------------------
>              |\
>              | \
>              |  \
>              |   \
>              ------
>
> from plane geometry one knows (or assumes) that alternate interior
> angles cut by the sides of the triangle are equal. Hence, by
> inspection (by looking-at), one has the result that the two non-right
> angles sum to a right angle. One can obtain the same result by
> expanding sin (A + B) (i.e. the two non-right angles) giving (here  a
> and b ares sides opposite respective angles A and B, and c is the
> hypotenuse)
>
> sin (A + B) = sin(A)*cos(B) + sin(B)*cos(A)
>
>              = a/c*a/c + b/c*b/c
>              = (a^2 + b^2)/c^2
>              = 1
>
> While this may be pleasing, it is perhaps not grasped by the same
> kind of looking-at or, as the way a mathematician might put it, by
> inspection.

Yes, the equation does not have the direct sensuousness of a
'looking-at'/Anschauung/intuition.

>
>      One reason I find this business about looking-at interesting is
> because of a meeting I had in the mathematics department the other
> day. They have a two semester courses that, for the most part, in the
> first semester focus on visualization in 3D and in the second
> semester focus on numerical analysis. They are a little less
> concerned with the second semester (or so it seemed), but consider
> the first semester crucial for understanding. In a sense, they are
> asking how can they help students (a good many of which are
> engineering or science students) develop dispositions of looking-at.
>
> >  > 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"?
>
> Yes, my impression is that is what Eves was saying from his side.
> However, what I wonder about is what it looks like from Aristotle's
> side. Perhaps, in terms of the perceptual, the sphere version might
> be front/back, left/right, above/below, and perceived size.

"Perceptual" is ambiguous, since there is both intellectual perception and sensuous
perception. But I take your point. I would have to dig into Aristotle some more to
see more from his side.

> >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.
>
> The delta-epsilon business is a little misleading as it assumes, in a
> sense, a metric space. The topological notion (using, for example, a
> definition from Hall) is as follows: Let S and T be spaces
> [topological] and f:S->T a mapping. The f is said to be continuous at
> a point s [if one wants this can be interpreted as an element] of S
> if and only if, given any open subset G of T such that if s is a
> point of the inverse image under f of G [sorry, there is no nice way
> of doing this in email], there exists an open subset of V such that s
> is a point of V and V is contained in the inverse image under f of G
> [ well, a possible way might be f^-1(G)]

Do you mean "an open subset V" rather than "an open subset of V"?
My memory is a bit hazy here. Is an open subset of a topological space a subset in
which every element has a neighbourhood within the subset? And neighbourhood in a
topological space is an axiomatic notion of points close to a given point?

What you have provided is a definitional understanding of a function f being
continuous at a given point s, but isn't the notion of neighbourhood the site of
the more originary notion of continuity, of something 'hanging together'? Could it
be said that with the topological notion of neighbourhood the ontological problem
of closeness and continuity and connectednes is somehow swept under the carpet?

> Also there is perhaps an interesting definition for connected: A
> subset A of a space S is said to be connected if and only if there
> exists no continuous mapping f:A->R [R here stands for the reals, but
> think of it as an index set] such that f(A) consists of exactly two
> points.

This interestingly inverts Aristotle's sequence. For him, connectedness
(_echomenon_) is ontologically a more primitive notion than continuity
(_syneches_).

> And there is a theorem that might be relevant and that says if S is
> connected and f is continuous then f(S) is connected.

Intuitively (i.e. for a sensous looking-at), this is obvious (i.e. unconcealed).

> >"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)
>
> I like this, but I am not at all sure what you mean unwieldy or by
> calculation of a geometrical figure. Possibly you are thinking of
> something like my example.

I was thinking of something like the intersection of a plane with a cone. It is
easy to see this through a direct, sensuous 'looking-at', but it is more unwieldy
to work out the equation of the ellipse where the plane intersects the cone.

> However, the unwieldy part comes from my
> trying to do a proof. Calculations of geometric figures [and as I say
> I am not sure what you mean by calculation here] are not proofs and
> there are ways in which they are simple. Consider, for example, the
> ways in which the Greeks computed acreage (children often do
> something like this in the early elementary grades).

I agree that a calculation is not the same thing as a mathematical proof. This
distinction is important, say, in teaching engineers, who have to know how to
calculate without necessarily knowing the proofs which ultimately justify the
calculation.

Nevertheless, Heidegger deals with mathematical proof also as a kind of calculation
in which the steps in the proof are a kind of calculation:

"[The mathematical entity] 'm exists' means that, starting from a definite
starting-point in the calculation, this magnitude can be unambiguously constructed
using definite means of calculation. What is constructed in such a way is thus
demonstrated as something effective (Wirksam, real) within a connected argument of
the calculation, 'm' is such a thing one can reckon with and with which one has to
reckon un certain conditions." (Nietzsche II S.419)

This formulation covers not only calculations in the narrower sense (such as
solving a set of equations), but proofs such as showing that 'm exists' in the
sense of proving that a certain kind of set of differential equations has a
solution.


> >  > 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.
>
> Rereading that passage from the Sophist I am beginning to think that
> I had better take a look (smile).
>
> By the way have you looked at Hilbert's axioms for geometry?

Yes, but a long time ago. There would seem to be a clear parallel between Hilbert's
axiomatic formalization of geometry and Leibniz' metaphysics, based as it is on the
distinction between "originary"/axiomatic truths and deduced truths. Truth is
conceived of as identity between statements. The proof of the truth of statements
is either direct, if the identity is obvious and unconcealed, or it is indirect,
and the identity has to be brought to light through a chain of deduction.
Heidegger's Leibniz lectures in Gesamtausgabe 26 are very good on this.

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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