Cologne, 15-Jun-2000
Ed Wall schrieb:
Betreff: Re: Cartesian Geometry
Datum: Wed, 14 Jun 2000 00:22:25 -0400
Von: Ed Wall <ewall@xxxxxxxxx>
> Hello Michael
>
> My apologies for taking so long to reply. You have set a
> fascinating question which I am sure I do not completely appreciate.
> So I will meander somewhat in the possibility that I might say
> something useful. Definitions of continuity, in general, go beyond
> the idea of just a sum or collection of points. One can, of course,
> have continuity in the usual sense or uniform continuity and even
> talk about smooth curves (i.e. c^infinity). In thinking of the latter
> case, there is a sense in which a line is characterized by how all
> its points, in a way, hang together. But it seems, in the way you've
> expressed it, that the crack is still there although perhaps somewhat
> different. Part of the problem may lie in how we have extended the
> rationals to the reals. [Amusingly Cohen and Ehrlich define the
> notion of gap in the rationals and show how the reals fill such
> gaps.] But there is a way in which the problem seems to lie back in
> the rationals and closure and order. That is as you essentially note,
> given any two rationals I can construct a third between the two (and,
> of course, that extends to the reals). How would Aristotle have
> handled that? That is, between any two points on a line it would seem
> that one could construct a midpoint (I am probably missing
> something). So a crack would be there but, it seems, the line would
> still be a geometrical unity.
Hi Ed,
It seems to me that there is a difference in ontological status between numbers and
geometrical entities such as the line. Geometrical entities are _aisthaeta_, i.e.
they are perceived/perceivable by the senses and are derived from physical bodies
by "lifting out" their _perata_, or limits. As such, geometrical entities are still
positioned (but have no _topos_ or place) and their parts have a positional
relationship to each other, an arrangement (_hexis_). They have an orientation,
e.g. a top and bottom, right and left.
Numbers, by contrast, are not positioned, nor are they objects of the senses, but
are abstracted not only from place, but from position/orientation as well. Numbers
arise in the first place by counting. They are the result of an iterative
procedure. This is the original Greek understanding of _arithmos_. The numbers are
thus discrete and sequential, and these properties are retained in the reals.
As I roughly outlined in my previous post (following Aristotle), the ontological
structure of geometrical entities is more complex than that of numbers since the
parts of the former 'hang together', i.e. they maintain a specific relationship of
orientation towards each other. This complexity is reflected in Aristotle's
stepwise construction of the continuity that characterizes geometrical entities.
If this is so, then it would seem that there is a fundamental ontological
difference between arithmetical and geometrical entities. The representation of the
real numbers by a line (a geometrical, sensuous entity) would then be questionable.
For, the points in a line touch each other, and, furthermore, the points in a line
are all the same, differing only in position, whereas real numbers do not touch
each other, nor are they the same but each differs from the other, nor do they have
a position or orientation at all.
> What Robinson does is construct an extension of the reals
> (actually an enlargement) which, among others, has the property that
> every mathematical "notion" (by this he seems to mean, at least,
> addition, multiplication, and order) which is meaningful for real
> numbers is meaningful for the enlargement. In addition there is at
> least one number that is greater than any integer. And, of course,
> by closure you get the infinitesimals. So already your crack is built
> in. [Of course, there are other enlargements one might construct
> which can be extensions of other mathematical objects]. Amusingly
> Robinson had a sense of humor and defined what he termed a monad
> (actually this is done both in an arithmetical and topological way) -
> given any real number a in the extension, the monad of a, mu(a) is
> the set of all real numbers in the extension which are infinitely
> close to a. One could, of course, define smoothness using
> infinitesimals and then a line's behavior on it's monads seems to
> become critical.
I don't know how smoothness is defined, nor what it means in connection with
numbers, but numbers which are infinitely close to a given real would touch it,
would they not? Perhaps you could say more about the notions of infinite closeness
and smoothness.
When you say that Robinson's notion of monad is both arithmetical and topological,
does this mean that he is attempting a kind of hybrid construction to bridge the
gulf between the geometrical and the arithmetical?
My question comes from an inkling that numbers cannot get any closer to the
outlines of sensuously given beings than indefinitely close approximations. There
is an essential ontological difference in the way of being (not to be confused with
_the_ OD).
Number, of its nature, is discrete, and this it shares with _logos_. The
fundamental sameness of number and language lies in the sameness of being, oneness
and something (on = hen = ti), i.e. every being is something and as something is
also a unity, a 'one'. The way beings are given to _nous_ in _logos_ is discrete.
This is the ontological origin of all digital break-up of beings.
If this is so, then a fundamental ontological distinction has to be made between
the 'outline' of beings given to the _nous_ (mind) as _idea_ and the 'outline' of
beings given sensously to geometry as geometrical figures. For example, the
'outline' sight of a house, it's idea, is captured in the word 'house', whereas
geometrically, the outline of a house, when its _perata_ (limits) are lifted out of
it, may be, say, some kind of pentagonal solid.
Western metaphysics is based on _logos_ and _arithmos_, which break down
(diairesis) beings into calculable, discrete chunks which are independent
(_choriston_) of the physical presence of the beings to which they refer. Modern
(today: digital) science and technology, in turn, are only possible within the
dimension opened up by the metaphysical understanding of _logos_ and _arithmos_ in
their sameness, even though the sciences need know nothing explicitly about
metaphysics.
> I will think about this some more, but perhaps I need to do some
> reading in Aristotle first.
Have a look at the Physics, Book V, Chap. 3 and also H. Gesamtausgabe Band 19,
_Sophistaes_, pp.100ff.
Michael
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_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_-
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> >Cologne, 13-Apr-2000
> >
> >Ed Wall schrieb:
> >
> > > MIchael
> > >
> > > It might be useful to note that, in a way (not necessarily
> > > historical), the rationals are an extension of the integers and, in a
> > > way, the reals are an extension of rationals. Thus, in a way, certain
> > > arithmetic operations are 'extended.' In a way Robinson did this with
> > > the reals to construct (in a particular application of what is termed
> > > model theory) what he called a non-standard model of arithmetic.
> > > Infinitesimals sort of lie in the cracks of this extension - as might
> > > other things - and take on an extended character somewhat as reals
> > > and rationals do. Thus, the arithmetic, in a way, is usual (e.g.1/0
> > > is still undefined). All this may bear on notions of calculability
> > > (although not necessarily historically) as, this is, in a sense,
> > > something Robinson - an accomplished mathematical logician - was
> > > concerned with.
> > >
> > > But all this is somewhat off topic - I just thought you might find
> > > it of possible interest.
> >
> >Ed,
> >I'm interested in those cracks, since they have to do with the
> >difference between
> >continuity and discreteness, analogue and digital.
> >
> >Aristotle shows that a line (a geometrical entity) cannot be
> >regarded as the sum or
> >collection of its points, since there is an ontological structure that makes a
> >line a line. He calls this _syneches_, 'holding itself together', or
> >connected (in
> >a special way). A continuous line is characterized by having its
> >adjacent points
> >touch each other at their extremes (_eschata_), which are also "one
> >and the same"
> >_tauto kai hen_.
> >
> >The real numbers as an extension of the rationals are supposed to constitute
> >continuity, but there is a problem because there is a succession
> >(_ephexaes_) of
> >numbers, even real numbers, and there is always a real number in between two
> >unequal real numbers. That is, _ephexaes_ has to be distinguished as
> >a way of being
> >from _syneches_.
> >
> >So my question is: Does Robinson's non-standard model provide for
> >numbers touching
> >(_haptesthai_) each other end-to-end in an identical point? Or, put
> >another way:
> >What is the trick in filling in the cracks?
> >
> >Michael
>
> > >
> > > >Cologne, 12-Apr-2000
> > > >
> > > >Ed Wall schrieb:
> > > >
> > > > > Hello Michael
> > > > >
> > > > > A comment. Abraham Robinson proposed a number of years ago an
> > > > > alternative development of the infinitesimal calculus (Non-standard
> > > > > Analysis) which was somewhat more 'algebraic' (perhaps in the sense
> > > > > of Leibnitz) as opposed to Newton's formulation (limits do play a
> > > > > role in Robinson's version although to a certain extent more for a
> > > > > matching up the approaches). Are you familiar with Robinson's work
> > > > > and how might that affect that part of your argument below dealing
> > > > > with the formation of the infinitesimal calculus? However, perhaps,
> > > > > you are speaking in an historical sense and what Robinson did is
> > > > > quite tangential.
> > > >
> > > >Ed,
> > > >I do remember Robinson's Non-standard Analysis from my days at
> > > >Sydney University.
> > > >At the moment I am tapping in the dark, holding only the thread of
> > > >the "ontological
> > > >primordiality" of number and logos vis-a-vis continuous
> >entities. As I vaguely
> > > >recall, Robinson introduces an infinitesimal number with its own
> > > >'arithmetic' as a
> > > >way of circumventing limit formation with the ('unaesthetic') delta
> > > >and epsilon
> > > >formulation.
> > > >
> > > >I need to do more reading on the character of infinitesimal
> > > >calculus, i.e. how the
> > > >continuum is broken down into arithmetic (in the broad sense)
> > > >calculability to try
> > > >and get a better view of what is going on with the (mathematical)
> > > >casting of the
> > > >being of (natural) beings.
> > > >
> > > >Thanks,
> > > >Michael
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