Re: real analysis - it is neither continuum nor discrete

Just my fuzzy thoughts -

The real analysis seems miserably stuck somewhere
between discrete and
the natural continuum analysis.

It can't calm to be continuum analysis because it
fails the preserve
the pattern of intervals and their bounds at the
fundamental level and
collapses both to same kind of points without proper
justification.

It also can't be discrete analysis because it
desperately depends on
the zero extent points to build up finite extent,
again without enough
reason. May be it is closer to discrete analysis than
to continuum
analysis, because the real line is a collection of
same kind of
objects with no extent and no bounds, thus loosing
the necessary
properties of continuum somewhere suddenly without
reason for the
change.

Instead, one could possibly have a continuum analysis
which preserves
the fractal pattern of intervals and their bounds
even at the
fundamental level and uses infinitesimal extents to
build finite
intervals of the continuum. This is what I call
natural continuum
analysis.

The logarithmic and linear distribution of the
continuum intervals
(and their bounds) are related to the two fundamental
attributes of an
interval - extent and bounds. Bounds trace the linear
distribution
while the extents can only be observed for scaling or
their geometric
expansion. Linear comparison for extents is as much
useless as the
"scaling" comparison for the marks of the bounds.

All this points to two kinds of fundamental elements
that make up
continuum. Limits and cuts. If one believes both are
of same kind,
then atleast we have two different paths to find them
which do not
merge into one.

-vr

To get grounded in how analysis works, you might
try reading Weyl's The Continuum, and Dedekind's
Essays on Theory of Numbers. (Both published by
Dover.) I don't find any fuzzy thinking there.

Tom
.

Relevant Pages