Date: Sun, 02 Aug 1998 23:24:02 +0300
From: Yair Mahalalel <yairm-AT-tabs.co.il>
Subject: Re: god and gramm


malgosia askanas wrote:
> 
> Steve wrote:
> 
> > Peirce asks the question whether a line is composed of the points along
> > that line, and answers that that cannot be the case, for even if you have
> > an infinite number of points, there would still be gaps between the points.
> > Therefore the line must be something other than the points (even infinite)
> > that are arrayed along it.
> 
> This somehow keeps haunting me, so let me yield and go back to it.  Could
> you say more about it?  I think that the haunting has to do mainly with
> the fact that both my history of math and my history-of-math books are
> in storage.  So for example, is the idea of "gaps" consistent with Euclidean
> geometry as it was in Peirce's time?  Can the Euclidean plane (as it was
> axiomatized then) admit of two kind of entities -- points and gaps -- without
> contradictions such as, for example, that one would then be able to deduce
> the existence of two non-parallel lines that do not intersect at any point?
> Is the notion of "continuity" ensured only by putting forth the mutual
> mappability between the Euclidean line and the real numbers?  Is this
> putting forth what constitutes the Peircean "something other"?  _Why_ did
> Pierce think that there would be gaps between the points?  It seems much
> easier to _not_ think that.
> 
> -m

Even in the times of Euclid, the distinction between the scattered real events
we absorb through our senses and the continuous "objects" that constitute our
perception of the world was already clear. Take for example the complete ante
world Plato has built just to host all these lines, triangles and white crows.

All forms of continuity cannot be verified empirically (and even intrinsically
banned by modern chemical and physical theories) - These are purely artifacts of
our mind which defend us from the insufferable chaotic flow of events that reach
us from the "Real World".

Euclid was fully aware of this when he build his system. Geometry is a
mathematical system, and as such it does not require any input from the physical
world, and cannot supply any new information about it. Such a system is composed
of elements which all reside in the realm of abstractions, which do not "exist",
which cannot be perceived through the senses.

The Euclidean "point" is a dimentionless, sizeless location, separable from any
other identical point only by its location. A line is thus defined as the
shortest path (in Euclidean space, we'd say today) between two such points.

But these postulates (It is not precise to call them 'axioms', since they do not
assert anything "real") are only propositions from which a coherent is built.
This system cannot say anything about reality since nobody has ever seen an
Euclidean point or line, much as nobody has ever seen the number "4" walking
down the street.

The sparse events that our senses collect are further broken by the discrete and
partial way they operate. The firing of neurons and the harsh selection of input
done by all levels of signal processing in the brain reduce our contact with the
world only to a narrow band of events which conform to a very large extent with
our expectations.

Since such constructions have no bearing in reality, any statement we make which
contains a hidden assumption of continuity smells of "metaphysics", but without
it, not only all forms of grammar will be forbidden, but also all forms of
seeing, willing and even life itself.

This admittedly extreme argument serves only one purpose - To show that
"delusions" and "metaphysics" are the foundations of our very existence. Not
only can't we free ourselves from them, but there will probably never be a
situation in which we'll want to free ourselves from all these hidden
presuppositions.

The wise way of conduct should be to be aware of these assumptions and to choose
the most appropriate system to tackle each problem. If N. claims that the our
focusing on opposites blinds us to the charms of continuity, so be it. It may
increase our freedom, our power, but we can never expect, nor want, to become
infinitely free or powerful.

Molecules of graphite may land on the piece of paper so that those of two non
parallel lines never come in contact one with each other, but still two non
parallel lines on a Euclidean plane always intersect on one and only one
Euclidean point.

Yair.


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