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.