# Thematics Of Order

Jon Awbrey jawbrey at att.net
Fri Apr 15 20:14:12 CEST 2005

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TOO.  Note 2

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Re: TOO 1.  http://stderr.org/pipermail/inquiry/2005-April/002541.html

The idea of "reflective order closure" (ROC),
whereby a "statement about the order" (SATO)
becomes a "statement in the order" (SITO),
arose while exploring the lambda point
between algebra, geometry, and logic,
where algebra includes things like
group theory and number theory,
and where geometry includes
things like graph theory.

Many of the things that puzzle us the most in number theory
have to do with our lack of knowledge about the connections
between additive properties and multiplicative properties
of A000027, also known as the sequence of natural numbers.

For instance, if we take a very nice representation of natural numbers
in multiplcative terms, namely, their primes factorization expressions,
and try to add two numbers in this form, then we have no direct way of
doing this that does not "cheat" by resorting to their additive forms.
In particular, we do not know how to add 1 to a primes factorization
and get the primes factorization of its successor in any direct way.
And this amounts to saying that we do not know how to order the
primes factorizations in their usual additive order simply by
looking at their multiplcative representations.

Reflections like these eventually bring us to the question:

How much of the additive, linear, or total order in the
sequence of natural numbers is "purely mutiplicative",
that is, how much of the order can be determined
solely by inspection of primes factorizations?

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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