Re: The Numerical and the Intuitive Continuum

On Dec 10, 8:39 am, LauLuna <laureanol...@xxxxxxxx> wrote:
In the thread 'Non-zero Gaps between Real Numbers' I wrote repeatedly
that the perceptual continuum and the set of reals are disparate,
incommensurable structures.

Let me bring in a pair of passages from Weyl's Das Kontinuum cited by
John Bell athttp://publish.uwo.ca/~jbell/Hermann%20Weyl.pdf

[QUOTE]... the conceptual world of mathematics is so foreign to what
the intuitive continuum
presents to us that the demand for coincidence between the two must be
dismissed as
absurd.
(...)
Exact time- or space-points are not the ultimate, underlying atomic
elements of the duration
or extension given to us in experience. On the contrary, only reason,
which thoroughly
penetrates what is experientially given, is able to grasp these exact
ideas. And only in the
arithmetico-analytic concept of the real number belonging to the
purely formal sphere do
these ideas crystallize into full definiteness. [END OF QUOTE]

I'd put it this way. The conceptual and numerical analysis of the
perceptual continuum yields the reals. But the reals are dissimilar in
nature to the phenomenal continuum they try to analyze, since they are
not intuitive or perceptual objects but objects of reason.

Real numbers correspond to points or positions on the spatial line;
but extension is not made up of positions. It's quite another thing.

From the assumption that e.g. the line consists of positions Zeno-
style paradoxes arise; consider:

1. Any arrow flying along a line has to reach an infinity of positions
before reaching any new position, since there is an infinity of reals
between any two reals.

2. For how long does the arrow occupy each position?

etc.

If continuous motion is reduced to successive occupation of successive
positions irreducible nonsense is likely to appear.

This is not to say that there must be gaps between the reals. This
again conflates what is distinct. But in the analysis we often take
the limit of a decreasing sequence of distances to be a point, a
position, an object with zero extension. This provokes no numerical
error but is logically dubious. It makes us consider the slope of a
curve on a point or the change in an instant, which is logically no
better than the talk of infinitesimals.

Regards

Consider the one-dimensional line, and identifying a point on the line
in this manner: a directional arrow with a tip to the point and the
tail then away from the point, where the arrow can only be on the
line. Now, if this arrow is by itself, then it would push the point
along, where in a restriction to a one-dimensional line, generally
diagrammed horizontally, the point is basically free to move to the
right or left. Then, to imbue a corresponding opposite "force" on the
point to keep it fixed and non-moving, it takes two arrows to indicate
a fixed point on the line ("fixed" "point" not necessarily "fixed
point.") So, the definite point has two sides, on the line. In
removing one of the arrows, then there is only one side of the point,
but in removing the block, without notion of overshoot backwards and
so on (the arrow can only be "picked up" by its tail), in removing the
block then there are multiple one-sided points that are the
immediately adjacent neighbors of the two-sided point, in the line.

Then, in two dimensions, there is some question as to how many arrows
it takes to establish an equilibrium of force on the point to fix it,
given various possible perturbative forces on the point. With two
antiparallel arrows, the point would need only any force normal/
perpendicular to them to squeeze the point out of the force
equilibrium, and three arrows, the tripos, would better contain/
confine the point, so maybe the two-dimensional fixed point has three
sides, for a particularly stable configuration that is frugal with
arrows. In an arrow's direction, a line, the adjacent individua of
the continuum are again one-sided.

Then, in three dimensions, four (or five) arrows would be a
particularly stable yet economical configuration in definition of the
point.

Then, there is a parallel between the 1-D case and Vitali's non-
measurability (in two parts) and the 3-D case and Banach-Tarski ball
decomposition (in four or five parts). As well, there is some
consideration of the allegory to measurement, i.e., physical
measurement, with some presumed natural continuum.

Ross

--
Finlayson Consulting
.

Relevant Pages

  • The Numerical and the Intuitive Continuum
    ... In the thread 'Non-zero Gaps between Real Numbers' I wrote repeatedly ... that the perceptual continuum and the set of reals are disparate, ... perceptual continuum yields the reals. ...
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  • Re: Why real numbers and points cant model continuum
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  • Re: CH yet again.
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  • Re: Interval in a continuum is a fractal
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  • Re: Wheres respect? was Re: Corrective interpretation of real numbers
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    (sci.math)