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Re: Continuous and discrete uniform distributions of N
Posted:
Dec 28, 2012 9:45 PM
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On Dec 28, 5:09 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <4a00241c-1c9f-4149-9c8e-73624b420...@s6g2000pby.googlegroups.com>,
> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > There are varieties of mathematicians, and they derive different
> > values from our study of the same principles. Some derive value from
> > the very closedness of things, that there are right and wrong answers
> > to our most fundamental questions. Some derive value from the very
> > openness of things, that there may be right and wrong answers to all
> > our questions. There are various motivations for its study and
> > application. Yet, it is largely so that given strong fundamental
> > principles, there are available general results, and mathematics is
> > for that those with totally different thinking, can share in truths,
> > given the principles of mathematics.
>
> Ross can convey less in more words than almost anyone else.
> --
Then, there's a consideration that there certainly is room for
progress, extra the standard, in the mathematics. Goedel proves this
using standard, modern mathematics: any finitely axiomatized theory
strong enough to represent natural arithmetic has true statements
about the objects of the theory that aren't provable, derivable, true
from the axioms of the theory: incompleteness. While that may be
so, it doesn't necessarily conclude that Presburger Arithmetic of
addition on integers is incomplete, as it is shown complete, and it
doesn't necessarily show that an axiomless system of natural
deduction, is incomplete. Yet, it surely gives the interested
mathematician the wherewithal to conclude that there are particular
true features of the objects of the domain of discourse, relevant to
the structure, that aren't those of our standard, modernly.
Then, there are as well considerations that traditional avenues of
inquiry, into the infinitesimal and infinite, have been closed off
from our selection of axioms with basically set theory's definition of
an inductive set or infinity, combined with the restriction of
comprehension of the axiom of regularity.
A. o. Infinity: defines a constant in the language of an infinite
inductive set
A. o. Regularity: restricts naive set-building from sets with
irregular transitive closures, a.k.a. Foundation
The other axioms reflect reasonably intuitive notions of the
composition of sets, as defined by their elements, that nothing
prevents elements from association.
Now, we know that there are various considerations these days, for
example of Aczel's reknowned anti-foundational, foundations, where the
anti-well-founded sets, as objects and members of the domain of
discourse, exist. As well, in NF and NFU, or as well NBG with
Classes, where NBG is ZFC with models, there are considerations of the
embracing of the domain of objects that would not be regular set-
theoretic sets: the class of ordinals, the class of sets, and other
structures that via their definition would not leave ZF containing
them, directly consistent. Obviously enough, this group-noun-game of
having classes for sets leaves what was the primary object, the set,
no-longer the primary object, just as type theories in sets, may find
that types are not the primary objects.
set theory <-> model theory <-> class theory
set theories <-> type theory (eg ramified, stratified) <-> category
theory <-> HOL
geometry
number theory
theory
With theory then and in number theory, a primary area of application
is that of real analysis, and continuum analysis, and here in as to
dividing the continuum, and having divided the continuum. Having
divided the continuum equally, is as to where the infinity of integers
is the continuum to the unit, as the unit is to continuum to the iota-
value, or from Newton: fluents and fluxions: this is a notion extra
the standard's theorems, but from the same principles.
So, there are traditional avenues of inquiry, into the infinite and
infinitesimal, as transcribed from antiquity, and revolutionaries like
Galileo, Wallis, Newton, Leibniz, Euler, Gauss, du Bois-Reymond,
Cantor, that are not within our modern mathematics, which as
curriculum is presented to as wide or wider an audience than ever
before. Then, where it is important for society to have mathematics
lead physics, we have seen in the course of the technological age that
where once mathematics lead physics and the capacity of experiment by
generations, technology has reached in as to where it is not the
capacity for construction of experiment, but the body of mathematics,
that lacks in the explanation of data. Our measurements reveal dark
matter, Avogadro's number grows, the farther we look the bigger it is,
the closer the smaller. We know there are truisms beyond the
classical, and the modern classical, the relativistic. There actually
are differences in things in the macro and micro scales from our
mesoscale, there actually is the anthropocentric, or simply in as to
our place in scale. Yet, there is still a reasonable expectation that
there is order in the Universe, and that then mathematics is and will
remain our best framework for physics, which otherwise would overthrow
not just dogma but science.
So, what are these truly revolutionary ideas in mathematics that will
truly reveal avenues for progression in our science? One might think
they would arise from the continued course in the foundations as we
know them of modern mathematics. Yet, in a hundred years, there isn't
a direct application resulting from that course. Measure theory is
built on the countable. The development of methods in physics is as
to the algebras, to reduce symbolic complexity and that is a right
course, and ever-more-complicated deformations of Euclidean geometry
to bend the methods to the results, and that is not clearly the right
course.
Then, the nature of the continuum is the matter of our discussion.
What is the continuum? We know that our tools of real analysis,
founded on countable additivity in measure theory, give us results
matching those of geometry and experience in the meso-scale. As well
we know there are true features of these objects of discourse not yet
resolved in our modern theory. Then, for what may be directions for
progress in continuum analysis, there is obviously the broad vista of
refinements and developments in the standard, but also there's a
reasonable consideration that the alternative avenues don't end where
we've left them, nor do they necessarily lead off to the weeds or
where the standard could not maintain its track.
So, there is found this simple construction with its plainly
reasonable features, and as well, surprising features, that the
integral of the function, EF, is one, besides that in the fundamental
results of modern mathematics, _that it isn't shown uncountable by the
cohort of results otherwise establishing uncountability of the
reals_. Then, where its features are well-modeled in what are
standard mathematics, as modeled by real functions, in real analysis,
and as having its existence inferred from the existence of a
reasonable notion of uniformity or regularity of the naturals, in ZFC
and number theory, then this gives the reasonable justification that
those abandoned directions are not without course, and not without a
reconciliation, that our edifice of modern mathematics precludes their
existence.
Then, with the other interwined, if not inseparable theoretical
studies, there are notions for set theory, and theory, that an
axiomless system of natural deduction provides a foundation for
results as applied. As well, in the course of the investigation of
these matters, simple axioms of a spiral-space-filling curve of a
natural continuum founds a geometry: from points and space before
points and lines. Then, the demands of the conscientious
mathematician of at once adhering to rigor, and our established
consistency in results, and as well acknowledging and even inviting
those results as would supercede what as modern mathematics precedes
progress, has that those results encompassing what came before, extend
the sphere of knowledge, and find room for developments that may truly
be innovative in discovery, beyond the refinement and specialization
of methods, to their completion, and placement of axiomatics in the
axiomatized.
And, EF is part of that.
So, the notion of a uniform distribution over the naturals, and the
very notion of a constant rate of change over the naturals, seems to
be built into the simple monotone progression of the naturals. One
might find that counting the integers leads to the first being counted
more, and conversely that for any there are more to be counted and
that the more there are, the more there are. While an inductive set
may truly be primitive in our theory, as well in frameworks where it
represents the meter of change, a copy or structural reflection of it,
would be high above. The numerical continuum, first as infinitely
many individua, then as integers, then as a basis for functions from
integers, then for example as reals, is a way to approach that the
numbers aren't defined, instead derived. So, the notion of a uniform
distribution follows from first principles.
Then, the structure of a uniform distribution of naturals, here for
continuous distributions and discrete distributions U_c and U_d, or
U^bar and U^dots, would first see that they are distributions. As
functions defined on the naturals: EF = U_c and 1/omega = U_d. Then
it's quite remarkable that EF, as p.d.f., is its own CDF. The
structure is for our theory of probability. Now, it is well known
that probability doesn't necessarily admit such a function, as
standard real analysis doesn't necessarily admit such a function,
though of course it's readily modeled by real functions, and other
functions with known utility are so constructed (re Dirac's unit
impulse function). Then while non-standard (and not to collide with
the statistical sense of standard as having unit variance), while non-
standard, these functions are at least modeled by the standard. And,
where a corresponding framework for structure of the continuum as non-
standard (here as super-Archimedean ordered field and as well ring of
iota-values) establishes the basis (and in the sense of the vector
basis) of the unit as the range of this function, then as well where
probability is founded on measure theory and real analysis, the
structure of a uniform distribution of the naturals is as well at
least partially evident.
Then the use of a uniform distribution of the naturals, has that it
would correspond with notions of density in the integers of number
theory, where cardinality is mute to it, and about how then
conditional and joint probabilities would be definable in terms of
this simple function, correlating expectations from number theory and
density of integers with statistical expectations. Then as well,
where there are surprising features of the functions that are each of
a continuous and a discrete distribution of the elements of the
support space, these might lead to applications and then, a use of a
uniform distribution of the naturals.
So, with a rationale and a justification for the development of these
ideas in probability, and the corresponding framework of continuum
analysis expanded with renewed investigation of avenues once blocked
for a need to reconcile with our standard, modern mathematics, then,
that is of general interest to many. And, with the fact that re-
exploration and discovery of the good mathematics that great thinkers
see in the true nature of the continuum, may well be a most fruitful
avenue for discovery of novel mathematical structures for physics,
where transfinite cardinals haven't panned out in applications and are
shoehorned into countable additivity for useful measure theory, then
the rationale and justification for the notion, structure, and use of
the uniform probability distribution, and its corresponding
mathematical framework and foundations, is a course for better
understanding of the continuum.
And EF: on the line, in the line, the line: starts that: the
continuum of the line.
Regards,
Ross Finlayson
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