Date: Fri, 25 Jul 2003 17:53:55 +0200
From: artefact-AT-t-online.de (Michael Eldred)
Subject: Re: The dimension of a manifold


Cologne 25-Jul-2003

 Ed Wall <ewall-AT-umich.edu> schrieb Wed, 23 Jul 2003 15:39:03 -0400:

> Michael
>
>       My apologies for not replying sooner and then in such a brief
> manner. My impression from what you have written that for both
> Aristotle and Euclid the mathematical move was a kind of calculative
> (in the sense, I think, that you are using it and, it seems, in the
> way Heidegger seems to use it - but again I am reading him in
> English) move. That is, given certain mathematical or logical
> conditions, there are certain mathematical or logical results. These
> can be 'computerized' as there are, for example, theorem provers and
> the successors of Boole's logical calculus. The digital computer is,
> in a sense, designed around the if-then, the go-to, and the
> true-false.
>      Hilbert was writing at a time when it was believed that, in a
> sense, all mathematics could be reduced to something more or less
> computational - Principia Mathematicia is, in a sense, the crowning
> example - however, Godel and others put a stop to that both in
> mathematics and for computers. But for most mathematicians these are
> theoretical results (smile) and, in practice, somewhat ignored. I had
> mentioned his axioms as he makes explicit mention of Aristotle.

Ed,

What Goedel put a stop to was also a metaphysical dream, clearly formulated by
Descartes and moreso by Leibniz, of knowledge being deducible from axioms in a
logically consistent way. And the casting of knowledge goes back to Aristotle, who
precasts what Descartes and Leibniz take up later under a more explicitly
mathematical bent.

>      Anyway my reading of Heidegger's reading of Aristotle (and Stuart
> nice clarifying) and my reading of Aristotle seems to suggest that,
> for example, the line aspect of geometry is built, to an extent,
> around the idea of line segment (one can do something similar for
> surfaces and solids). So if I consider a line to be a series of n
> line segments, these segments seems to possess many of the qualities
> necessary. They are oriented, they touch, they are successive, an end
> of one is the beginning of the other. Hence, according to Robinson
> (and one needs to be a little careful here), one can extend this to
> an infinite integer n and get a set of things of which Aristotle
> might approve (smile). Note there may be items in between these
> things, but they are not of the same character (i.e. belonging to the
> same infinite integer). They are also not Cartesian points. However -
> within the right framing - they are close to Cartesian points and
> this Cartesian reduction, so to speak, gives the usual analytical
> geometry. The unanswered mathematical question since Robinson is does
> this reduction lose a mathematical something. For example, are there
> mathematical ways of knowing lines that are captured in the
> extension, but not in the reduction.

What work of Stuart's is this?
I think I get your line of reasoning here. The Cartesian reduction to points
'forgets' the ontological question of being of the line as continuous. Aristotle
sees, as you point out, continuity as a breakdown into segments where the extremes
of consecutive segments coincide. With the Cartesian reduction to points, how the
points hang together to make a line is left unanswered.

I'd be interested in what you come up with in your investigations of whether
Robinson has some epistemic surplus-value.

> I am unclear about the following:
>
> >Connected means no gaps or holes, but still distinct segments.
> >E.g. a chain is connected (a concatenation of contiguous links). A
> >cable is not
> >only connected, but continuous.
> >E.g. a mosaic is connected when the individual stones/tiles
> >touch/are contiguous,
> >but a slab of marble is continuous.
> >
> >How is this Aristotelean difference captured in mathematics?
>
> as a mathematization (in terms of the geometry) could capture the difference.

You seem to have answered my question above.

Do you know Leibniz' "Initia rerum mathematicarum metaphysica" ("Initial
Metaphysical Grounds of Mathematics" 1715)? It must be interesting for a
philosophically inclined mathematician. The editor of the German edition writes:

"On the basis of the distinction between quantity and quality, Leibniz demonstrates
and comments once more on his method of tracing back all the data given by what is
in space and time ("verites de fait") to ultimate truths founded in reason ("verites
de raison"). These latter can be won exclusively from definitions and identical
statements, i.e. such statements whose subject contains the predicate wholly or at
least partially, and with the aid of the principle of non-contradiction. The text is
simultaneously the attempt to integrate the infinitesimal calculus cast during his
stay in Paris from 1672 to 1675 into the 'mathesis universalis', that is, of a
universal calculus as a means of deduction in the 'characteristica universalis'
which Leibniz in turn understands as the adequate logical expression of the world
order."

Here one gets the flavour of the totalizing nature of metaphysical castings.

Cheers,
Michael
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