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.
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 will think about this some more, but perhaps I need to do some
reading in Aristotle first.
Ed Wall
>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
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>_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_-
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>
> > Ed
> >
> > >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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