Date: Dec 28, 2012 9:45 PM Author: ross.finlayson@gmail.com Subject: Re: Continuous and discrete uniform distributions of N 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