Cologne, 17-Jun-2000
Ed Wall schrieb:
Betreff: Re: Cartesian Geometry
Datum: Fri, 16 Jun 2000 00:05:08 -0400
Von: Ed Wall <ewall@xxxxxxxxx>
> Hello Michael
>
> Thank you for your quick response. Definitely a bit of fresh air
> as I had been flailing around trying to be open to numbers, as we
> have been talking about them, being other than discrete - but I
> certainly don't have all the answers. I will try to comment briefly
> on what you have written and will sit down with Heidegger and
> Aristotle - amusingly I have a bookmark somewhat before the page you
> reference for Heidegger.
> >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.
>
> Yes, I agree with your characterization of real numbers. I am not so
> sure about points touching yet - my question is more about points
> than touching as it seems there is some baggage associated with point.
I'm wondering about Cartesian geometry and how it captures continuity, which is the
hallmark of sensuously given beings. The standard mathematical treatment of
continuity (quite late -- Dedekind?) is the epsilon-delta definition which says
roughly that any continuous point of a (real) function is surrounded by points
arbitrarily close to it. This always struck me as a round-about way of defining
continuity, and, if I recall correctly, Robinson proposed his non-standard analysis
to have a more elegant (simple) understanding of continuity (limits) in terms of
infinitesimals. What does Robinson's concept of continuity look like?
In Aristotle too, the ontological structure of continuity is complex, being built
up in several (six) steps. He has to formulate the notion of points on a line
touching each other at their extremities.
> >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.
>
> Smoothness is just that derivatives of all orders are continuous.
That is then a very complex notion, since continuity itself is already complex, let
alone derivatives.
> If
> one uses Robinson's infinitesimals to write this you essentially are
> saying, in a certain sense, that a type of weighted sum is continuous
> which in a certain sense encompasses the entire line (i.e. if you
> think of limits and as you go to higher orders). This probably wasn't
> why the name was coined, but often one uses weighted sums to smooth
> data. Anyway, for example, for a line having these sums (or
> derivatives) means, in a certain sense, you have put points into
> position.
I don't know if I get you here. Is it a line that has such sums or derivatives, or
rather a function? And would continuity of a function at certain co-ordinates mean
putting it into position? I am trying to preserve the distinction between the
arithmetical and the geometrical, because that is precisely what is at issue here
-- whether arithmetical structures (such as a continuous function) can be genuinely
regarded as a geometrical entity.
> Actually as I doddle, what might be interesting is
> smoothness on a monad and for monads to sort of touch - i.e. agree as
> to the values of all higher order derivatives above a certain order.
Well, that's getting beyond me, since I don't know what Robinson's monads look like
mathematically.
> There are some other possibilities here - one I didn't mention
> because of the way I misconstrued the conversation. Differential
> geometry provides some interesting possibilities. For, example, one
> might think of a line as composed of vectors, perhaps infinitesimal -
> these do seem to have the properties you want and, for me, always
> made more sense than cartesian geometry which was too much like
> arithmetic.
Are vectors all that different from Cartesian co-ordinates? Aren't they basically
just ordered pairs of Cartesian co-ordinates? And Cartesian co-ordinates just
ordered finite number sequences?
> Robinson's work seems to be largely misinterpreted. What he showed
> that you can take a mathematical model and extend it in ways that
> preserve (in a certain sense) it's traditional or standard
> properties, but introduces new entities. The model can be sets as
> well as number. The only real complaint is that so far you cannot get
> any results that are different from what you can get in the standard
> model (i.e. if you get a result in in the non-standard model and
> standardize it) although I haven't really paid much attention the
> last ten years. So he did far more than the usual calculus (although
> his interest was mathematical logic). A monad in the topological case
> is just what you might think it is, a point and all things
> infinitesimally close to it. What you gain, it seems, in such
> enlargements is an elimination of the limit arguments.
Yes, that seems sensible. The complexity of the monad in itself allows one to
dispense with a complicated notion of closeness between numbers as formulated in
limit definitions. The monad as a set with infinitely close proximity seems to be
like a hybrid geometrical-arithmetical construction.
> But it is
> certainly possible that someone has picked up on the differential
> geometry angle and done a type of merger. And, of course, there is
> point set topology (the only thing at this point in time I can
> remember well is that prof skipping down the sidewalk - I regret
> thinking he was weird and wishing I had gotten to know him better;
> skipping is a plus) - I should take a look.
Point set topology is beyond my ken, but no matter.
> >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).
>
> I think that is indeed the nature of the reals by, in a sense, design.
>
> >Number, of its nature, is discrete, and this it shares with _logos_.
>
> Yes, this is what I've always assumed about number and why I was
> struggling. I need to think about the language angle.
So do I, since it is one crux of the issue, if one is to characterize a digital
casting of being as 'arithmological'.
> >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
> >(_chorismos_) 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.
>
> You say a number of things here that I need to muse over. I do
> remember the time when battle between the digital and the analog
> computer was undecided (and still is for a very few) - it was not
> that long ago and I have/had been very much a part. But the word I
> might have used at that time, knowing and thinking little about
> metaphysics was mathematizing (for me a wrenching of mathematics from
> its context so as to fit it in the world of number). One's thinking
> about an analog program was quite different. I am definitely not sure
> given the breath of what mathematics may be, the extent to which this
> digitizing view has impinged - some places it seems none at all (and
> other places in very suspicious ways).
What does an analogue computer program look like? I thought that all computer
programming boiled down ultimately to binary difference.
Thanks,
Michael
_-_-_-_-_-_-_-_-_-_- artefact text and translation _-_-_-_-_-_-_-_-_-_
_-_-_-_-_-_-_-_-_-_-_-_-_-_-_- made by art _-_-_-_-_-_-_-_-_-_-_-_-_-_
http://www.webcom.com/artefact/ _-_-_-_-_-_- artefact@xxxxxxxxxx
_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_-
_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_
--- from list heidegger@xxxxxxxxxxxxxxxxxxxxxxxxxx ---
