Re: An uncountable countable set
- From: Lester Zick <dontbother@xxxxxxxxxxx>
- Date: Fri, 03 Nov 2006 15:49:32 -0700
Well, since no one chooses to reply or even acknoweldge the following
I think I'll add a few remarks myself.
On Wed, 01 Nov 2006 12:50:10 -0700, Lester Zick
<dontbother@xxxxxxxxxxx> wrote:
On Tue, 31 Oct 2006 10:30:08 -0500, Tony Orlow <tony@xxxxxxxxxxxxx>
wrote:
[. . .]
(Tony, since we have an already established audience on this thread
I'm piggybacking what amounts to a new line of reasoning to your post
to which I've already replied in detail. I hope the following explains
exactly the origins and significance of mathematical and arithmetic
infinities in purely mechanical terms.)
Real Theory
~v~~
I'm now of the opinion that there is a specific reason why modern math
and set analysis are wrong in fundamental mechanical terms. The
difficulty has to do with what are thought of as real number lines and
their supposed characteristics as lines. In effect if we use the Peano
axioms and the suc( ) axiom to generate the naturals, we lock into a
system of straight line segments which never correspond to curves and
transcendental numbers and infinities drawn in terms of those curves.
In other words it is quite possible to generate straight lines in such
terms but there can never be an exact equivalence between those lines
and any kind of transcendental infinity. Thus we can treat arithmetic
infinites which exist in terms of infinitesimal subdivision of
straight lines such as irrationals like the square root of two but not
those which exist in terms of transcendental infinities such as pi.
On the other hand, however, we can proceed in the opposite direction
quite easily by generating straight lines as tangents through Newton's
calculus and his method of tangents. And in so doing we can develop
all possible reals through the mere assumption of curves instead of
straight line segments and infinitesimal subdivision instead of Peano
axioms and the suc( ) axiom.
The problem is and always has been that mathematics in general is not
arithmetic in particular. And we can always generate straight lines in
terms of curves through tangency but not vice versa because in terms
of form there is only one straight line but infinite kinds of curves
to which there are straight line tangents and we can't proceed from
any straight line tangent backwards to any specific curve.
In effect then arithmetic theoreticians have straight lines but they
cannot deduce curves from those straight lines and are forced to
imagine such collateral forms of infinity either superimposed on
straight lines themselves, such as imaginary real number lines, or in
some other group altogether. But in neither case can they deduce the
existence and properties of such transcendental infinities from the
existence and properties of straight lines produced by arithmetic
axioms except through approximation with straight line segments.
Mechanical Implications
~v~~
Although our primary interest here is mathematical there is a good
deal more significance than mere conventional mathematics would
suggest or imply. We and all forms of being operate and think in terms
of curves or at least in non straight line forms through tautological
negation which is demonstrably true. However at the conceptual level
we communicate with one another through straight linear forms. This is
only true and possible because there is only the one form of straight
line but an unlimited number of curves.
Thus each of us operating at the level of abstract thought has to
reduce curvilinear tautological results to straight line tangents in
order to compare to and communicate thoughts of one ontological
individual with those of another. And this process is exact through
tangency with those curvilinear tautological results. However then we
are left to ponder the origin of those exact results because we cannot
reverse the process in exhaustive mechanical terms to determine with
which curvilinear tautological form the tangency originated.
~v~~
In mathematics it is often assumed that one can do whatever one
chooses through the definition of axiomatic assumptions. Possibly the
reasonableness of this idea can be established historically by
earliest forms of geometric analysis where the existence of points,
straight lines, and curves were simply taken for granted because
commensuration of such elements seemed a more urgent problem.
And yet one of the primary functions of mathematics is the integration
of all conceptual realms and not merely a wishing away of differences
through facile assumptions. Apparently we have no choice but to infer
the existence of curves in various forms. But we have no corresponding
need to infer the existence of straight lines as well because these we
can derive through Newton's method of drawing tangents to curves.
In other word one cannot integrate what one has not disintegrated. In
mathematical parlance this means one cannot just willy-nilly integrate
what has not first been differentiated because we have no derivative
to indicate direction, no limits to indicate boundaries, and no
functions to indicate variations. We can assume all these things but
when we do there is no limit to our imagination and we risk giving the
impression that precedence is the other way around, that somehow we
can proceed from straight lines to curves when causation is actually
in the opposite direction.
Of course for purposes of commensuration the residuum is the straight
line and its subdivisions. But the ancestry and lineage in tangency of
straight lines must never be forgotten and we must always recognize
transcendental curves as mathematically primitive to straight lines.
~v~~
.
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- Re: An uncountable countable set
- From: Tony Orlow
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