Logic History Overview...

Logic History Overview...
Quantification Logic...

Saturday, August 27, 2011

Francis Bacon__The Father of Modern Scientific Induction...

A Priori Induction…
Every Word Is A Method…!!!
The A Priori Perception…
Gravity = EM-Hydro-Dynamic Balance…
The Three Fundamental Forces…!!!

Man, being the servant and interpreter of Nature, can do and understand so much and so much only as he has observed in fact or in thought of the course of nature. Beyond this he neither knows anything nor can do anything.

II

Neither the naked hand nor the understanding left to itself can effect much. It is by instruments and helps that the work is done, which are as much wanted for the understanding as for the hand. And as the instruments of the hand either give motion or guide it, so the instruments of the mind supply either suggestions for the understanding or cautions.

III

Human knowledge and human power meet in one; for where the cause is not known the effect cannot be produced. Nature to be commanded must be obeyed; and that which in contemplation is as the cause is in operation as the rule.

IV

Toward the effecting of works, all that man can do is to put together or put asunder natural bodies. The rest is done by nature working within.

V

The study of nature with a view to works is engaged in by the mechanic, the mathematician, the physician, the alchemist, and the magician; but by all (as things now are) with slight endeavor and scanty success.

VI

It would be an unsound fancy and self-contradictory to expect that things which have never yet been done can be done except by means which have never yet been tried.

VII

The productions of the mind and hand seem very numerous in books and manufactures. But all this variety lies in an exquisite subtlety and derivations from a few things already known, not in the number of axioms.

VIII

Moreover, the works already known are due to chance and experiment rather than to sciences; for the sciences we now possess are merely systems for the nice ordering and setting forth of things already invented, not methods of invention or directions for new works.

IX

The cause and root of nearly all evils in the sciences is this — that while we falsely admire and extol the powers of the human mind we neglect to seek for its true helps.

X

The subtlety of nature is greater many times over than the subtlety of the senses and understanding; so that all those specious meditations, speculations, and glosses in which men indulge are quite from the purpose, only there is no one by to observe it.

XI

As the sciences which we now have do not help us in finding out new works, so neither does the logic which we now have help us in finding out new sciences.

XII

The logic now in use serves rather to fix and give stability to the errors which have their foundation in commonly received notions than to help the search after truth. So it does more harm than good.

XIII

The syllogism is not applied to the first principles of sciences, and is applied in vain to intermediate axioms, being no match for the subtlety of nature. It commands assent therefore to the proposition, but does not take hold of the thing.

XIV

The syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and overhastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction.

XV

There is no soundness in our notions, whether logical or physical. Substance, Quality, Action, Passion, Essence itself, are not sound notions; much less are Heavy, Light, Dense, Rare, Moist, Dry, Generation, Corruption, Attraction, Repulsion, Element, Matter, Form, and the like; but all are fantastical and ill defined.

XVI

Our notions of less general species, as Man, Dog, Dove, and of the immediate perceptions of the sense, as Hot, Cold, Black, White, do not materially mislead us; yet even these are sometimes confused by the flux and alteration of matter and the mixing of one thing with another. All the others which men have hitherto adopted are but wanderings, not being abstracted and formed from things by proper methods.

XVII

Nor is there less of willfulness and wandering in the construction of axioms than in the formation of notions, not excepting even those very principles which are obtained by common induction; but much more in the axioms and lower propositions educed by the syllogism.

XVIII

The discoveries which have hitherto been made in the sciences are such as lie close to vulgar notions, scarcely beneath the surface. In order to penetrate into the inner and further recesses of nature, it is necessary that both notions and axioms be derived from things by a more sure and guarded way, and that a method of intellectual operation be introduced altogether better and more certain.

XIX

There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried.

XX

The understanding left to itself takes the same course (namely, the former) which it takes in accordance with logical order. For the mind longs to spring up to positions of higher generality, that it may find rest there, and so after a little while wearies of experiment. But this evil is increased by logic, because of the order and solemnity of its disputations.

XXI

The understanding left to itself, in a sober, patient, and grave mind, especially if it be not hindered by received doctrines, tries a little that other way, which is the right one, but with little progress, since the understanding, unless directed and assisted, is a thing unequal, and quite unfit to contend with the obscurity of things.

XXII

Both ways set out from the senses and particulars, and rest in the highest generalities; but the difference between them is infinite. For the one just glances at experiment and particulars in passing, the other dwells duly and orderly among them.

The one, again, begins at once by establishing certain abstract and useless generalities, the other rises by gradual steps to that which is prior and better known in the order of nature.

XXIII

There is a great difference between the Idols of the human mind and the Ideas of the divine. That is to say, between certain empty dogmas, and the true signatures and marks set upon the works of creation as they are found in nature.

XXIV

It cannot be that axioms established by argumentation should avail for the discovery of new works, since the subtlety of nature is greater many times over than the subtlety of argument. But axioms duly and orderly formed from particulars easily discover the way to new particulars, and thus render sciences active.

XXV

The axioms now in use, having been suggested by a scanty and manipular experience and a few particulars of most general occurrence, are made for the most part just large enough to fit and take these in; and therefore it is no wonder if they do not lead to new particulars. And if some opposite instance, not observed or not known before, chance to come in the way, the axiom is rescued and preserved by some frivolous distinction; whereas the truer course would be to correct the axiom itself.

XXVI

The conclusions of human reason as ordinarily applied in matters of nature, I call for the sake of distinction Anticipations of Nature (as a thing rash or premature). That reason which is elicited from facts by a just and methodical process, I call Interpretation of Nature.

XXVII

Anticipations are a ground sufficiently firm for consent, for even if men went mad all after the same fashion, they might agree one with another well enough.

XXVIII

For the winning of assent, indeed, anticipations are far more powerful than interpretations, because being collected from a few instances, and those for the most part of familiar occurrence, they straightway touch the understanding and fill the imagination; whereas interpretations, on the other hand, being gathered here and there from very various and widely dispersed facts, cannot suddenly strike the understanding; and therefore they must needs, in respect of the opinions of the time, seem harsh and out of tune, much as the mysteries of faith do.

XXIX

In sciences founded on opinions and dogmas, the use of anticipations and logic is good; for in them the object is to command assent to the proposition, not to master the thing.

XXX

Though all the wits of all the ages should meet together and combine and transmit their labors, yet will no great progress ever be made in science by means of anticipations; because radical errors in the first concoction of the mind are not to be cured by the excellence of functions and subsequent remedies.

XXXI

It is idle to expect any great advancement in science from the superinducing and engrafting of new things upon old. We must begin anew from the very foundations, unless we would revolve forever in a circle with mean and contemptible progress.

XXXII

The honor of the ancient authors, and indeed of all, remains untouched, since the comparison I challenge is not of wits or faculties, but of ways and methods, and the part I take upon myself is not that of a judge, but of a guide.

XXXIII

This must be plainly avowed: no judgment can be rightly formed either of my method or of the discoveries to which it leads, by means of anticipations (that is to say, of the reasoning which is now in use); since I cannot be called on to abide by the sentence of a tribunal which is itself on trial.

XXXIV

Even to deliver and explain what I bring forward is no easy matter, for things in themselves new will yet be apprehended with reference to what is old.

XXXV

It was said by Borgia of the expedition of the French into Italy, that they came with chalk in their hands to mark out their lodgings, not with arms to force their way in. I in like manner would have my doctrine enter quietly into the minds that are fit and capable of receiving it; for confutations cannot be employed when the difference is upon first principles and very notions, and even upon forms of demonstration.

XXXVI

One method of delivery alone remains to us which is simply this: we must lead men to the particulars themselves, and their series and order; while men on their side must force themselves for a while to lay their notions by and begin to familiarize themselves with facts.

XXXVII

The doctrine of those who have denied that certainty could be attained at all has some agreement with my way of proceeding at the first setting out; but they end in being infinitely separated and opposed. For the holders of that doctrine assert simply that nothing can be known. I also assert that not much can be known in nature by the way which is now in use. But then they go on to destroy the authority of the senses and understanding; whereas I proceed to devise and supply helps for the same.

XXXVIII

The idols and false notions which are now in possession of the human understanding, and have taken deep root therein, not only so beset men's minds that truth can hardly find entrance, but even after entrance is obtained, they will again in the very instauration of the sciences meet and trouble us, unless men being forewarned of the danger fortify themselves as far as may be against their assaults.

XXXIX

There are four classes of Idols which beset men's minds. To these for distinction's sake I have assigned names, calling the first class Idols of the Tribe; the second, Idols of the Cave; the third, Idols of the Market Place; the fourth, Idols of the Theater.

XL

The formation of ideas and axioms by true induction is no doubt the proper remedy to be applied for the keeping off and clearing away of idols. To point them out, however, is of great use; for the doctrine of Idols is to the interpretation of nature what the doctrine of the refutation of sophisms is to common logic.

Also:
In the late 19th century, Charles Sanders Peirce proposed a schema that would turn out to have considerable influence in the further development of scientific method generally. Peirce's work quickly accelerated the progress on several fronts. Firstly, speaking in broader context in "How to Make Our Ideas Clear" (1878),[66] Peirce outlined an objectively verifiable method to test the truth of putative knowledge on a way that goes beyond mere foundational alternatives, focusing upon both Deduction and Induction. He thus placed induction and deduction in a complementary rather than competitive context (the latter of which had been the primary trend at least since David Hume a century before). Secondly, and of more direct importance to scientific method, Peirce put forth the basic schema for hypothesis-testing that continues to prevail today. Extracting the theory of inquiry from its raw materials in classical logic, he refined it in parallel with the early development of symbolic logic to address the then-current problems in scientific reasoning. Peirce examined and articulated the three fundamental modes of reasoning that play a role in scientific inquiry today, the processes that are currently known as abductive, deductive, and inductive inference. Thirdly, he played a major role in the progress of symbolic logic itself — indeed this was his primary specialty.

Charles S. Peirce was also a pioneer in statistics. Peirce held that science achieves statistical probabilities, not certainties, and that chance, a veering from law, is very real. He assigned probability to an argument’s conclusion rather than to a proposition, event, etc., as such. Most of his statistical writings promote the frequency interpretation of probability (objective ratios of cases), and many of his writings express skepticism about (and criticize the use of) probability when such models are not based on objective randomization.[67] Though Peirce was largely a frequentist, his possible world semantics introduced the "propensity" theory of probability. Peirce (sometimes with Jastrow) investigated the probability judgments of experimental subjects, pioneering decision analysis.

Peirce was one of the founders of statistics. He formulated modern statistics in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883). With a repeated measures design, he introduced blinded, controlled randomized experiments (before Fisher). He invented an optimal design for experiments on gravity, in which he "corrected the means". He used logistic regression, correlation, and smoothing, and improved the treatment of outliers. He introduced terms "confidence" and "likelihood" (before Neyman and Fisher). (See the historical books of Stephen Stigler.) Many of Peirce's ideas were later popularized and developed by Ronald A. Fisher, Jerzy Neyman, Frank P. Ramsey, Bruno de Finetti, and Karl Popper.

Thursday, August 18, 2011

Economic Models' Math__Applies To Physics, As Well__& Vice-Versa...

(P.s. added Tim...)

http://en.wikipedia.org/wiki/Economic_model

"The economics/(physics) profession appears to have been unaware of the long build-up to the current worldwide financial/(mathematical physics) crisis and to have significantly underestimated its dimensions once it started to unfold. In our view, this lack of understanding is due to a misallocation of research efforts in economics/(physics). We trace the deeper roots of this failure to the profession’s focus on models that, by design, disregard key elements driving outcomes in real-world markets/(physics models). The economics/(physics) profession has failed in communicating the limitations, weaknesses, and even dangers of its preferred models to the public. This state of affairs makes clear the need for a major reorientation of focus in the research economists/(physicists) undertake, as well as for the establishment of an ethical/(scientific) code that would ask economists/(physicists) to understand and communicate the limitations and potential misuses of their models."

Tim, I think it quite important to read David Hestenes' entire paper, to see where this paragraph links the maths of any and all model constructs together, and which ones are invalid and useless, verses, the truly valid and useful maths and models... Remember, at the most and absolute fundamental level, all these maths__being one of the fundamentals of all models__are all fully isomorphic, at limits__That's the real 'Conformality' of the entire 'House of Cards...' I think Hestenes has much better captured these facts with humor, than any serious post I could possibly make... The most important thing to recognize, is the fact that all these maths and models absolutely must be grounded in real World and Universal facts, parts, pieces and motions(the 9 degrees of freedom motions, i.e., vectors of real asymmetric wave actions, of the real fs-field...)__1st and foremost...!!!

http://geocalc.clas.asu.edu/pdf-preAdobe8/MathViruses.pdf

Imo Tim, you are going to have to understand the asymmetric/achiral control of the entire Universe's fundamental wave mechanics control mechanisms, in order to see how a Bose-Einstein Condensate truly forms and decays__over time__The true four dimensional aspects of FS-Matter, through its most fundamental wave-mechanics__Pure Wave-Mechanics as Vector-Phase-State-Spaces through Scalar Time Factors... Imo Tim, this takes a full understanding of the achiral actions of opposite handedness(the 3rd state), in a far deeper model of understanding chirality(left and right symmetric states), than you are presently looking at... I hate to keep chasing you back to this well worn point, but until you see what this model represents, I know it's not possible to see the Universe's most fundamental wave and motion mechanics(it's like a self-polarizing triadic orthogonality of the most fundamental hydrodynamic forces__the least understood of the least actions...) When you finally see what I'm truly saying here, the whole world of your personal Universal understanding will explode anew... Build the wire models Tim, and see the true Boogie-Man, in the Universe__It's far more fundamental than the analog-digital models, as they are thus far, interpreted almost entirely dyadically, and I'm constantly speaking about the triadicity of the fundamental polarizations__necessitated by the time scalar waves altered distance actions, in relation to the fundamental orthogonal polarizations, of the FS-Waves... This is more easily understood through 'Tensor Mechanics', as first started by Leibniz, and followed all the way through to Levi-Civita and the many more moderns, like Hestenes'(Geometrical Algebra...) Imo, math is absolutely necessary, but it must be the proper maths, and the even more proper use of the maths, in relation to real World and Universal facts...

A few years back, I named it: 'The Relational Geometry & Algebra of Responsible Intelligence…' It still holds...

A related link:
http://en.wikipedia.org/wiki/Nine-point_circle

The Achiral Non-Conformality, Within The Chiral Conformality...
The Achiral Non-Symmetries, Within The Chiral Symmetries...
The Non-Conservation of Time, Within The Conservation of Space…???

And Tim's Theorem:
http://en.wikipedia.org/wiki/Lester%27s_theorem
http://mathworld.wolfram.com/LesterCircle.html
http://en.wikipedia.org/wiki/Fermat_point
http://en.wikipedia.org/wiki/Circumcenter

And:
http://en.wikipedia.org/wiki/Binary_Golay_code
http://en.wikipedia.org/wiki/Ternary_Golay_code
http://en.wikipedia.org/wiki/Jakob_Steiner
http://en.wikipedia.org/wiki/Steiner_system
http://en.wikipedia.org/wiki/Steiner_triple_system
http://en.wikipedia.org/wiki/Steiner_tree_problem
http://en.wikipedia.org/wiki/Geometric_median
http://en.wikipedia.org/wiki/Ernst_Witt
http://en.wikipedia.org/wiki/Leech_lattice
http://en.wikipedia.org/wiki/%C3%89mile_L%C3%A9onard_Mathieu
http://en.wikipedia.org/wiki/Mathieu_group

P.s.
A few more of the notes I'm working with, Tim:

The Symmetry Illusion…

The Mechanics of Long Wave to Short Wave Photons__Radio to Gamma…
Peirce__The Teleological Logician…
Teleology__The 1st Beginnings of Knowledge Gives Ya The Ends of Knowledge__Primitive Logic…
Peirce__The Father of True Teleological Pragmatic Logic…
Only Teleological Intelligence__Exists…!!!
The teleological past, present and future of entropy__Gas shrinks upon losing all heat…
A living organism holds a continuum of ideas, to form real intelligence…
Peirce__The Teleological Genius…
Thought mechanics is outside-in__Not inside-out…
Archetype Thought Modes__Pure_Category_Model Logics…
Modal Teleological Epistemic Pragmatic Logic…
The False Vector/Scalar Models of Physics…
Sherrington's philosophy as a teacher can be seen in his response to the question of what was the real function of Oxford University in the world. Sherrington said:  "after some hundreds of years of experience we think that we have learned here in Oxford how to teach what is known. But now with the undeniable upsurge of scientific research, we cannot continue to rely on the mere fact that we have learned how to teach what is known. We must learn to teach the best attitude to what is not yet known. This also may take centuries to acquire but we cannot escape this new challenge, nor do we want to."

Peirce solved Cantor’s, Frege’s and Russell’s paradoxes, before they were ever created__Deep Relational Algebra__Foundations to Tensor Algebra…
‘The Axiom of Choice’ vs. ‘The Axiom of Determinacy…’
Both Geometry & Nature Necessitate ‘The Axiom of Choice’, or No Universe Could Ever Exist__Difference Necessitates ‘The Axiom of Choice/Action…’
‘The Axiom of Difference’ Mandates Choice, or Random Action, As Same…
‘The Axiom of Difference’ Makes Strong Determinism__Impossible…
The Universe Requires ‘The Axiom of Difference’ to Produce Different Quarks and Protons…
Factual Ideas & Goals force logic to actions, occurences and events__which change the world…
Iff granted absolute power, then & only then, can I emplace law to solve all the world’s problems…
Science = A Provable Concept…
Professor Hochberg’s book is best characterized by his own words from the preface: The book attempts to sketch, not work out in detail, an account of reference, meaning, truth and intentionality that stays within the “linguistic turn” characterizing twentieth century analytic philosophy. But it seeks to avoid following the contemporary variants of analytic philosophy that have turned from the analysis of things and facts to a preoccupation with and virtual worship of language and its use. The classical focus on ontology, combined with careful and precise formulations, that marked the writings of the early founders of the analytic tradition, has degenerated into the spinning of intricate verbal webs of analysis. The latter supposedly yield “theories of meaning” but more often signal the rebirth of idealism in the guises of “anti-realism” and “internal realism.” The focus on the world, as what words are about, is often lost as “analytic philosophers” concentrate on language itself—the world being “well lost,” in Nelson Goodman’s honest words… . We shall also note examples of a remarkable combination of arrogance towards and ignorance of the philosophical tradition that is displayed in some writings within the analytic tradition, including influential works.

The Complete/Incomplete “Distinction…”
His paper “Universals”, which denies any fundamental distinction between universals and particulars, surmounts serious objections to a realist view of universals and, at the same time, solves several long-standing problems about them, dismissing other venerable enigmas as nonsense. Ramsey
 
Various reasons for making the distinction between universals and particulars – psychological, physical and logical – can be advanced. Ramsey argues that logic justifies no such ontological distinction. Allusion to the grammatical subject-predicate distinction will not do, since “Socrates is wise”, with the subject “Socrates” and the predicate “wise”, “asserts the same fact and expresses the same proposition” (p. 12) as “Wisdom is a characteristic of Socrates”, with subject “wisdom” and predicate “Socrates”.3

Moreover, there is, he argues, no essential difference between the (in)completeness of universals and that of particulars. “Wise” can, for example, be used to generate propositions not only of the atomic form “Socrates is wise”, but also of the molecular form “Neither Socrates nor Plato is wise”. But “Socrates” can also be used to generate propositions of both these forms: e.g. “Socrates is wise” and “Socrates is neither wise nor just”.

There is therefore a complete symmetry in this respect between individuals and basic properties (qualities). As Ramsey succinctly puts it, the whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality what is merely a characteristic of language (p.13).
The Particular/Universal Distinction ¬

One of his co-workers, Frank Ramsey, took Russell’s warning very much to heart. But Ramsey soon came to the conviction that philosophers had not only been misled by language to adopt and adhere to a subject-predicate logic and a denial of relations. Writing in his 1925 Mind paper “Universals” (hereafter U) Ramsey declared: “nearly all philosophers, including Mr Russell himself, have been misled by language in a far more far-reaching way than that; that the whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality, what is merely a characteristic of language.” (U: 13)
The Particular/Universal Distinction Illusion__The Language Illusion_At Limits…
“True” Is An Incomplete Symbol… Ramsey
Quantum Uncertainty & ‘The Axiom of Difference…’
Russell, Quine, Putnam, Chomsky__Purile Bastardizations of Peirce…
The Subjective Psychology & The Objective Philosophy…
Russell’s Phony Symbol Semiotics__A Logical Joke…
Language and Linguistic Non-Sense__A Running Series of Logical Jokes…
The Frege-Russell-Quine Psychological Era of Logic…
Russell’s False Symbol System__Confused and Conflated Semiotics…
The ‘One-Many’ ‘Universal-Particular’ Plays Out In Every Hand of Logic…
The ‘One-Many’ Central Logical Illusion… 
Russell’s Purile Tensor Logic…
The Mind Is A ‘One-Many Continuum’, Having To Interpret Both ‘Universals’ and ‘Particulars’__At Once__In Each Propositional Analysis…
The Central ‘One-Many Continuum’ Logic Problem Always Exists…
‘Formal Truth Formal Logic…’ Ramsey…
¬ () = X Is False…(let ¬ = the empty)
¬ {()} = X Is False…
1st Methodological Tensor Choice Modality…
The Continuum of Triadic Choice Modalities…
All Knowledge Is Combinatoric From Its Initial State, Except Psychological Non-Knowledge Systems…
Psychology__The Modern-Day Anti-Knowledge System__Nietzsche On…
Psychology can not be added, subtracted, multiplied or divided…
Psychology, being non-mathematizable, must remain non-scientific…
Hard Knowledge vs. Soft Non-Knowledge…
The Methodology of Discovery__Science…
Weak Belief vs. Strong Knowledge…(from Hume on)
The Combinatoric Compositionality of All Real Hard Knowledge Systems…
Economics is primarily historically combinatoric, except for the slight and short anomalies of intermittent negative psychological events…
Modal Knowledge vs. Modal Belief…
‘The Difference Axiom’ of Knowledge vs. Belief…
The Continuum/s Obviously Create/s Difference…
Determinism requires symmetry, whereas the Universe is asymmetric, thus the asymmetric actual eliminates hard determinism…
Francis Bacon’s Eliminative Induction…
Eliminative Combinatoric Induction Mechanics…
http://en.wikipedia.org/wiki/Multinomial_theorem
This my help with this Tim:
The Universal Isomorphic Algorithm__UIA = ∑∫∏v -> IC:M Iff / ≡ ∑’s •…(The universal isomorphic algorithm equals the sum of the integral product variables, implying the isomorphic center of mass, if and only if divided identical to the sum’s center…)
Proof
This proof of the multinomial theorem uses the binomial theorem and induction on m.

First, for m = 1, both sides equal x1n since there is only one term k1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m. Then

(x_1+x_2+\cdots+x_m+x_{m+1})^n = (x_1+x_2+\cdots+(x_m+x_{m+1}))^n
   = \sum_{k_1+k_2+\cdots+k_{m-1}+K=n}{n\choose k_1,k_2,\ldots,k_{m-1},K} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}(x_m+x_{m+1})^K

by the induction hypothesis. Applying the binomial theorem to the last factor,

 = \sum_{k_1+k_2+\cdots+k_{m-1}+K=n}{n\choose k_1,k_2,\ldots,k_{m-1},K} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}\sum_{k_m+k_{m+1}=K}{K\choose k_m,k_{m+1}}x_m^{k_m}x_{m+1}^{k_{m+1}}
 = \sum_{k_1+k_2+\cdots+k_{m-1}+k_m+k_{m+1}=n}{n\choose k_1,k_2,\ldots,k_{m-1},k_m,k_{m+1}} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}x_m^{k_m}x_{m+1}^{k_{m+1}}

which completes the induction. The last step follows because

{n\choose k_1,k_2,\ldots,k_{m-1},K}{K\choose k_m,k_{m+1}} = {n\choose k_1,k_2,\ldots,k_{m-1},k_m,k_{m+1}},

as can easily be seen by writing the three coefficients using factorials as follows:

 \frac{n!}{k_1! k_2! \cdots k_{m-1}!K!} \frac{K!}{k_m! k_{m+1}!}=\frac{n!}{k_1! k_2! \cdots k_{m+1}!}.

A related Pdf:
http://mathsci.kaist.ac.kr/~drake/pdf/towards-a-combinatorial-theory-of-multiple-orthogonal-polynomials.pdf

Best first source informations:
http://www.constitution.org/bacon/nov_org.htm
http://en.wikipedia.org/wiki/Pafnuty_Chebyshev
http://en.wikipedia.org/wiki/Andrey_Markov

And as Chebyshev's work relates analog to digital:
http://en.wikipedia.org/wiki/Chebyshev_filter
Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named in honor of Pafnuty Chebyshev because their mathematical characteristics are derived from Chebyshev polynomials.

Tuesday, August 16, 2011

The Mathematical Saboteurs...

(Tim, this is funny, and oh so true...)
MATHEMATICAL VIRUSES*
David Hestenes

ABSTRACT.

The discovery of Mathematical Viruses is announced here for the first
 time. Such viruses are a serious threat to the general mental health of the mathematical
community. Several viruses inimical to the unity of mathematics are identified, and their
deleterious characteristics are described. A strong dose of geometric algebra and calculus
is the best medicine for both prevention and cure.
 
1. INTRODUCTION

Computer viruses have been prominent in the news lately. The increasingly widespread and
frequent communication among computers has facilitated the spread of computer viruses
to the point where viruses are seriously regarded as a threat to national security in the
United States.
The computer virus (CV) owes its name and perhaps its genesis to the biological virus
(BV). Like a BV, a CV cannot function by itself, but when attached to a host it replicates
repeatedly until it impairs the functioning of the host, sometimes to the point of disabling
the host altogether. Moreover, it is infectious, spreading from one host to another on
contact. The host of a CV is a computer program, to which the CV is attached as a
subroutine, replicating and spreading whenever the host program is run. Similarly, a BV
is a fragment of DNA containing instructions for self-replication which are activated when
the BV is in a living cell. Fortunately, antiviral agents can be developed to cure infected
computers as well as biological organisms.
My purpose here is to call your attention to another kind of virus | one which can
infect the mind | the mind of anyone doing mathematics, from young student to professional
mathematician. As I believe this is the first published paper to explicitly identify
such viruses, I take the liberty of naming and describing them as follows: A mathematical
virus (MV) is a preconception about the structure, function or method of mathematics
which impairs one's ability to do mathematics. Just as a CV is program which impairs the
operating system of a computer, an MV is an idea which impairs the conceptualization of
mathematics in the mind. Indeed, as one definition of \virus," Webster gives \something
that corrupts the mind or soul." Since the identification and classificalion of MVs has only
just begun, it would be premature to attempt a more precise definition. The better course
is to examine some specific viruses to form a firm empirical base for further study.
My first example is an easily recognized MV which is extremely virulent and as common
as the common cold. I call it the
coordinate virus, designating it by MV/C to denote genesis
and type, and characterizing it as follows:
MV/C: Coordinates are essential to calculations.
Physicists and engineers are especially susceptible to this virus, because most of their
textbooks are infected, and infected teachers pass it on to their students. Mathematicians
as a group are less susceptible, because many of them have been innoculated with a good
course in abstract algebra, though, as we shall see, some resistant strains have survived in
certain mathematical subspecialties.
The diagnosis and treatment of MV infections is still in its infancy, and it is especially
delicate when the infected host is not aware of his illness, as is usually the case. Diagnosis
of infection with the coordinate virus MV/C is comparatively easy, owing to the presence
of a well-defined syndrome of symptoms which I call
coordinitis: Typically, the infected
subject fails to distinguish the abstract vector concept from its matrix representation and
consequently has great difficulty conceiving and manipulating invariant functional relations
among vectors without expressing them in terms of coordinates; he is likely to regard the
real and complex numbers as more fundamental, or even \more real" than vectors. To
cite a specific example of such symptoms, one textbook by a distinguished physicist asserts
that \the vector calculus is like a folding ruler, before you can use it you have to unfold it"
(by which he meant, decompose vector formulas into components). In my own experience
of more than two decades teaching physics graduate students, I have observed that most
of them suffer from coordinitis and as many as 25% may be permanently crippled by the
disease.
Some may think that the coordinate virus is harmless or benign. After all, it is not
fatal. The afflicted can still limp along in their mathematical thinking. However, they are
condemned to a world of prosaic mathematical applications. They will never be able to
scale the Olympic heights to inhale the pristine air of abstract mathematics. Let there
be no mistake about the nature of the coordinate virus. There is nothing wrong with
using coordinates when they are appropriate. It is the insidious idea that coordinates
are somehow more fundamental or concrete than other mathematical objects that limit
conceptual capacity.
Though I would like to lay claim to the important discovery that mathematical viruses
exist, honesty compels me to admit that mathematicians must have known about them
all along. For as soon as a mathematician is introduced to the MV concept he begins
to notice viruses everywhere, and he is equally adept at naming them. Invariably, at
the mere utterance of the words \Bourbaki virus," a knowing smile breaks across the
mathematician's face, as if he is sure that we are both privy to some private indecency. I
will not attempt to describe the Bourbaki virus, for I doubt that the field of MV diagnostics
is sufficiently mature for the task. It is an important task, nonetheless, in view of evidence
that mathematics is still suffering from the ravages of a \Bourbaki epidemic."
While it may be suspected that many MVs are at large, in any attempts to detect and
neutralize them we must be alert to the dangers of misdiagnosis. One man's purported MV
may be another man's inspiration. Medical history is riddled with mistakes in diagnosis
and treatment. To avoid similar mistakes we must carefully establish suitable diagnostic
criteria; we cannot rely on mere hunches or opinions. The main burden of this paper is to
set down some specific criteria for the general mental health of mathematics and use these
criteria to identify several MVs by their deleterious effects.

The Way It Is, Tim...

David Orlin Hestenes, Ph.D. (born 1933) is a physicist. For more than 30 years, he was employed in the Department of Physics and Astronomy of Arizona State University (ASU), where he retired with the rank of Research Professor and is now emeritus.

Hestenes has worked in mathematical and theoretical physics, [1] geometric algebra, neural networks, and cognitive research in science education. He is the prime mover behind the contemporary resurgence of interest in geometric algebras and in other offshoots of Clifford algebras, as ways of formalizing theoretical physics.

From 1976 to 1979, he was an Editorial Advisory Board Member (formerly called Associate Editor) of the American Journal of Physics. He is currently on the editorial board of the journal Foundations of Physics.

In 2002, the American Association of Physics Teachers awarded him its Oersted Medal for his notable contributions to the teaching of physics. He has been a Principal Investigator for NSF grants seeking to model instruction at both the high school and university levels.

http://en.wikipedia.org/wiki/David_Hestenes

Tuesday, August 9, 2011

The Conformal Mechanics of A Cyclic Universe...

Hi Tim, first I wanted to mention I just finished reading Penrose's new book: 'Cycles of Time, An Extroadinary New View of The Universe...' Roger's described the Universe very similar as I've been describing it for the last 30 years__Hawking Decay cycle, to recycle, in order for the 2nd law to make sense... If I didn't know better, I'd think he'd been reading some of my web-posted material. Here's a review link to his book: http://bagotbooks.wordpress.com/2010/09/16/cycles-of-time-roger-penrose-at-the-royal-institution/
http://arxiv.org/ftp/arxiv/papers/1011/1011.3706.pdf

Tim, your post, along with that old one of yours in my last post, pretty much fills in a lot of what I intended the symbols to represent. The thing is as I mentioned__I've really only recently developed all the idea links, so I haven't really had time to access it fully yet__but I do think we can add up all the system mechanics, the way I did with economics money mechanics systems, and truly draw new ideas, facts and mechanics with it... As is also mentioned in Penrose's book, there are still areas of fundamental physics and maths, where we simply can't yet do the complexities__but I do know this algorithm's capacity may open up some of these areas__though, I do still know some of them are impossible, as many of my incommensurability posts have mentioned, before. Still, if we attempt new additions of systems, systems' variables, and systems' mechanics__and place them into one composite positive and negative model/concept__we can derive real new information, as that's exactly how I used the isomorphic logic to that algorithm, to solve my economic headaches__back in the `80's. At the time I tried inverse/reverse/perverse applying the economic algorithm__E = 1/5X__back onto the physics problems, I was then working on, but it didn't work. I wasn't privy to the knowledge of using functions to do math back then, but my writing of that formula algorithm, in the last post__was a direct interpretation of logic functions into mathematics, or algebra/calculus__which I've really only learned over the last six or seven years of studying Peirce, and all the mathematicians and logicians he studied. That's really how I recently applied it to physics, where I couldn't years ago__as I had no deep logic functions knowledge then__just logic and math, geometry, algebra, trig., calculus etc. I just stumbled upon the E = 1/5X formulation of foreingn exchange calculations with trade to GDP's accidentally, then, by simply comparing the concepts I was working directly with. What a difference 25 years, self-education makes__but I do really have to slow down the research, and realize what exactly's in this ol' skull o' mine...

Anyway, let me get to your post, by way of this paragraph:

I see all this clearly. Even if I'm off a little on some aspects, I know the logic is sound, but considering what I'm saying here and if your representation is addressing all of this, then at it's current composition, is it only addressing the relationship of such things, or does it allow such a means of perhaps extracting quantitative values of those things being conserved?
It's really both, Tim, as far as I can see... I think it's a good method of raising the bar on the goals we seek__to some universal understanding level, for our or any others' minds to meet at a common point__and at the same time, while only preliminarily viewing and thinking about the algorithm's potential__I do think we can truly extract new, quantitative information and values with it. I haven't really even tried using it yet, but in quick thinking about it, I'd suggest preliminarily making a list of all the knowns and unknowns you may like to process toward new information__then generally place the information content into the positions within the algorithm, and just generally think out the logic__much the way you did in the paragraphs above this paragraph in your post...

Penrose also offered some new algorithms in his new book, that operate in similar fashion, and an engineer in the campground here, just gave me an old 1899 Maine Maritime Academy math book, our navy academy here, and there's some very interesting old algoritims, especially the 6th root of numbers, for doing water to steam calculations. Penrose's are more in the similar vein as my new algorithm, but that old 6th root of numbers, I've already extended to the 9th root, to cover the 512k bits of computer storage registers, etc. 2 is the 9 root of 512. I'd just not thought of all the applications of higher power roots, even where powers of base 10 are used__even though I've used them, but I mean as to the many opportunities to explore with them, in so many other areas__just thinking...

Anyway, it's still all so new to me, I've still gotta have time to process how I may wish to use the algorithm, or how I may simplify it__but what it's designed for is to simplify complexity__already existing__so I don't yet know if I want to reduce it's power representations. Just play with it, by adding groups of concepts and variable concepts, finding the center of conservation values for all the variable and invariable concepts, by possibly building extensional concepts necessary to find answers by forcing results to their most central values__just a quick suggestion, and don't really know if it'll even be of any value__till I really access it deeper. All I can tell you right now, is that the central numbers between all number differences within concepts, has a lot to do with a lotta' mathematical solutions__as that's what I'm always using more than anything else, and Peirce's 100 thousand pages of information contained an extreme amount of the same__especially his mechanical engineering and statistics. It's just my brain ain't functioning right now, Tim, as I just finished burning through Penrose's complete book today__220 pages...

Play with the ideas, and ask me some questions later, Tim, when me brain ain't so full of others' ideas... I'm taking a break from researching, over the next 3 to 5 days, to get back to my own self-functioning. Gotta' help my son move all my and his stored belongings to his house, as a lot of the stored stuff is his, anyway... That'll give me the break I need...

The Universal Isomorphic Algorithm__UIA = ∑∫∏v -> IC:M Iff / ≡ ∑’s •…(The universal isomorphic algorithm equals the sum of the integral product variables, implying the isomorphic center of mass, if and only if divided identical to the sum’s center…)

Does it allow further applications, or merely bring others to this level of understanding the relationship of such things? Or am I a little off the mark?
No, you are right on the mark__and it allows application attachments and extensions__that I do know__as I've extended my economic mechanics extensively, since first developing the logic, this formula is founded on. Tim, the formula is so powerful, it can even process legal values and morality__as I've often used the logic in public meetings, and people have asked how I came to know those conclusions__and I'd always explain the logic to em, but it usually just goes over most people's heads. It seems they can't understand how math and logic can interpret morality and law, etc.__when it's simply processing to__the center of fairness, which is the same as to the center of mass, or even thought's mass, if you can believe that... Gee, I think that's mathematical__at least it is in my book... There's just an awsome power in processing to the center of all or any differences__which is basically, all that algorithm does... It's just designed for handling massive amounts of information...

It's just 'A Mathematical Diamond Rule...'

P.s.
Btw Tim, this logic idea was first discussed on TQ with Ayia-Oba back in Feb. `07, and I reposted that post, as the first post, on my 'East Meets West' Thread, back in Sept. `09...

Above and Below The Surface.....The Inner Relationship Of Temporal and Spatial Values

added p.s. and p.s.s. below after post.......

That's not a headache to me ole buddy. I think I follow you for the most part as with the old post you provided being one of the more clear moments I tried to express the asymmetry actually conserving a more fundamental symmetry. I can understand the representation and translate the symbols, but the main thing you'll have to help me with is understanding how to apply it and what type of information it will extract. For instance, does it have the ability to form quantitative calculations because the way I understand it so far is through the isomorphic relationships of the various systems it is intended to represent.

The universal center of mass is the barycenter product of the various galactic centers of mass, which are barycenters of their constituent black holes, solar systems etc. which further break down all the way to the atomic levels. The coupling forces of each of the constituent systems concentrate their mass values to form point center representations whereby their composite mass acts at the next larger scale to further concentrate it's mass value. If the universe was to cycle to a contraction aspect, the universal composite barycenter would be the dynamic point to which all systems condensed towards. This is part of some of the aspects that I interpret as having to be considered from your representation. Also needing to consider the actual transitional phases through time and space working from these center masses whereby all of the quantitative dimensions eg energy, momentum, velocity, frequency, wavelength, etc, of the various systems which establish the characteristics of such systems are conserved in an isomorphic manner whereby their very distribution determines the identification of the system being observed. The key to understanding such is by way of realizing that seemingly unrelated and non-interchangable properties are actually interchangeable when considering the transition through time and space, and not just space or just time as most dimensions of a system and the equations thereof attempt to define. For instance, Einstein established that energy and mass are interchangable at the constant value of cc, and Planck used his value to equate energy to frequency, yet such values as velocity are also interchangable with various other internal system values if one considers not just the ever present state of a system which most of science's focus is on, but also the ever present cost to remain in such a state. I would argue with Newton here, an object in motion might stay in motion, but such motion is far from free. Velocity is the quantitative value of a system maintaining a stable state for any duration. To relate to Pierce here, remaining in the present moment in any one state is a required value of the sum of all other system values. Rearrange or morph the values or Va of a system and a different system is formed with some of it's quantities surfacing in the everpresent realm whose values are readily available to measure, while other quantities are conserved to those temporally hidden values which only show up as an aspect of a rate of change or velocity contributor.

I guess what I'm getting at is that some values of a system can be considered as intrinsically linked to spatial aspects, while others can be considered as linked to temporal aspects. Thus being as the spatial dimensions are more available to us, those values upon it's surface are better understood, while those hidden by temporal aspects below the surface have allowed the relational aspect of all values to remain hidden from our perception until we consider that moving through time isn't any more free of effort than moving through space or distance. Yet, how we move through both simultaneously is resolved by how we distribute the internal motion or Va of a system. And as suggested before, some changes of a system are to the extent that the system perhaps hasn't enough time to morph itself to remain functioning as a group, thus not meeting a parameter whereby quantities of substance are moved as a result whereby balance is maintained by imbalanced phenomena. Which if veiwed from the reversible aspect, sheds light on not only decay, but also construction.

I see all this clearly. Even if I'm off a little on some aspects, I know the logic is sound, but considering what I'm saying here and if your representation is addressing all of this, then at it's current composition, is it only addressing the relationship of such things, or does it allow such a means of perhaps extracting quantitative values of those things being conserved? Does it allow further applications, or merely bring others to this level of understanding the relationship of such things? Or am I a little off the mark?

P.S. The best way I can make an analogy is to consider that we can imagine as some already do, the present as being refreshed at a constant interval. It doesn't matter if it's steady or exact, because there is no external reference, thus just as we can't tell how long we've been a sleep, if one interval is different than another, there's no way of knowing, but all points are refreshed simultaneously. Now if we have a photon, the distance it must be moved every time the system flashes/refreshes is much greater than that of a macro object. However, the time or bandwidth it takes to constantly reconstruct a massive system is much greater than the photon. Thus, the difference of the distance being moved every cycle of the photon is equal and innerchangeable to the greater complexity of the massive system, whereby unrelated values are found to conserve to each other by way of their relationship to space and time and the simple cost of being preserved in the everpresent present moment.

P.S.S. Once we zoom inside the massive objects, their subatomic structures actually cover the same distance per hypothetical refreshing as does the one bit photon, such distance is merely confined to a localized region ie angular motion rather than extended through space thus accomplishing the accuracies of the linear distance variance discussed above.

Monday, August 8, 2011

Ok__Let me give you a geo-head-ache...

The Universal Isomorphic Algorithm__UIA = ∑∫∏v -> IC:M Iff / ≡ ∑’s •…(The universal isomorphic algorithm equals the sum of the integral product variables, implying the isomorphic center of mass, if and only if divided identical to the sum’s center…)

"If one knows the logic’s inference path, one can do the math…" me

How's that for 'smart-ars'...? It's a geo-formula, and it's fully universal__and this is also what I interpreted your newest mirror analogy into__but it's actually from an older economic concept I developed, back in the mid-`80's__by grouping all 231 global nations into one concept, within the borders of a fictitious model of America__as I was having trouble wrapping my logical conceptual mind around all the foreign exchange mechanics of the world, outside America's borders, and its known mechanical influences and controls upon America, and America's lesser controls upon all these other nations... The point is Tim, I hadn't realized I'd already stumbled upon a 'Universal Algorithm' I'd been looking for, for years, until you posted the other day about the mirror analogy__and of course mirror analogies triggered my mind into its memory analogies of your mirror analogies' relationships__as I see the mind thoroughly inter-connected in all its inference processing mechanics__clear back into our earliest memory states, and intelligence concepts of__Simply trained memory states, to me anyway...

Anyway, the point is more that a 'Universal Formula' was even discoverable at all, and that such a formula was sitting in me lil' ol' brain since the mid-`80's, as a logical concept, I'd never realized could be written as a symbolic mathematical algorithm, though possibly a little bit complex algebra/calculus. Don't take the formula as fully formed just yet, as I simply wrote it as I followed the logical inference path, from your mirror analogies, to my older economic group integration analogies. The key in the formula is the 'Isomorphic Variables' of group unifications of concepts, whether they be our 'Structured Matter Concepts' of macro physics, our 'Unstructured Matter Concepts' of FS-Fields to any smaller infinitesimal fields, or my 'Foreign Exchange Integration Concepts'__the main factor is the realization of a 'Variable Mechanics' being able to be mathematized at all__at least one of this magnitude of 'Universal Import...' Tim, as far as I can see, this formula has the capacity to take Dave's Va = Vr + Vu formula, and fill in all the 'Necessary Variables' Mechanics Maths' to actually solve for all the force and mass unknowns__we are certainly looking for, imo... Also, by going back over many of your other analogies and graphics__I can probably reduce the algorithm's present complexity, even more. I've just simply not thought about algorithmically mathematizing all this material, as we've proceded__as my brain only kicks in fully, when it's gathered enough information, to satisfy my insatiable thirst__Just the way I am... :-)

You see, I've actually been looking for this algorithm for my economics proofs for some thirty years, without ever realizing it sat directly within the logic I'd already discovered years ago__but just didn't have the translation knowledge back then, to turn 1st order logic into 1st order math__which I've since learned preliminarily how to do, at the smallest and largest levels__at least as to its general operations' rules, which I think is all we need to start with... Any missing pieces we can work out along the way, but I only discovered this a few days ago, after you posted the mirrors' analogy, to trigger my memory to my own older analogies__so I've really been working on the economic end of the proofs, which of course involve so much of the mental gymnastics of recent posts, as economics is so psychologically dependent__if you hadn't noticed of recent, over the U.S. Debt downgrade, etc. Investors are all nervous nut-cases__and economists can't work totally in hard logics and maths, and imo, neither can physicists__though they may think they can__as the real world of others' actions is always going to upset the best functioning apple carts. Anyway, the formula allows for all these 'variables', even the psychological ones, to be factored into the total mechanics of this new 'Universal Algorithm...'

It's funny such a formula of such humongous economic complexity, could actually interpret into a first order arithmetic and logic, so simple, in its overall performance__a child could see its effectiveness, though you probably ain't following me yet__I think you will, as I later procede...

(Here's some of my relational notes about the above... This stuff's all preliminary Tim, as it's too new to completely work out yet, but I know it works, as the economic logic, I've been using for years, works__and that's of course what the algorithm is founded on__but, being Universal__it functions for physics, as well... Btw, I first discovered my first economic algorithm, E = 1/5X, while searching into physics' maths__thus solving for this highly complex integration of money mechanics, which this algorithmic logic and math turns into__and so the circle goes, round and round... Tim, I gotta' be free, in my thinking, in order for my mind to be creative, even though it may be difficult for you to follow__sometimes... It all fits into one 'puzzle-picture...')

"Where's the demarcation line, between important and unimportant quantum infinitesimals...?"

Mathematical Logic__The Highest Best Ordered Logic…!!!
The Mathematical Ordering of Fully Functioning Inference Logic…
FOP’L(first order proof logic)__Why the laws of thought are absolutely necessary…
The naked mind must respect the laws of thought, or truth, math and logic are impossible…
Mathematics dictionary…
Background Logic & Foreground Memory…
Time Conservation__Events Preserve Progressions…
Conservation of Logic__Only Logic Conserves Truth…
Logic Is Conservation of Truth Mechanics…
Logic To Prove Necessary Action…
Combinatoric Conjunctive Logic…
Integral Group Compactifications…
Imaginary Time Reversal Logic…
Self-Intelligence Calculation Space…
Working Memory Concept Space…
Isomorphic Groupoid Arithmetic/Algebra…
Time & Economic Split-Ends…
Intellectual Privilege & Priority of Objective Intellect, Over Subjective Feelings…
The Absolute Necessity of Our Knowledge of The External World…
The Unification of Science & Philosophy…
The ‘Conservation of Time’ integrations of math progressions, over events…
Combinatoric Conjunctive Time Logic…
The Combinatoric Time Matrix…
The Time-Distance Distinction…???
The Time Dimension of Distance__The Bandwidth Necessity Expands As Values Approach c, And When The Bus Compacts To Limit__or Structured Systems Broaden Out As They Reach Field Velocities, Then Decay Back Into Fundamental FS-Field, By Decompaction Necessity…
Explanation & Explication Can Not Do, Without Logic & Arithmetic…
QM’s Theoretical Measurement Scale…!!!
Logic Is Mathematics…
No logic exists without formal logic…
Concept Inference Mechanics Avoids The Pseudo-Linguistic Problems…
Truth__The world is the way it is, as it is at this moment, or at any other moment__as it can be no other way. The world exists as it exists…
The Concepts-Meanings Distinction…
The Isomorphics of All Small Fields Add-Up, To The Large Field Total__The Addition to Continuum Necessity…
The Logically Scientific Background of All Knowledge…
Logic Is The Default State of Path Necessity…
Information Quantification & Validity...
The Fundamental EM-Hydrodynamic Scalar Wave, of State Changes At Limit__i.e., When Low Density Fermions Decay To Photons…
If it ain’t scientific, it ain’t provably true…

UIA = ∑∫∏v -> IC:M Iff / ≡ ∑’s •…(Formula sums to the center of all sums, while differentiating all its time fixed, and variable parts__Seems to be a universal algorithm of possibly all universal formulas…?)

__...

Chew on that a bit...

Later Tim

P.s.
Just looking back over our older posts Tim, and I found this post of your's, which also relates to my post and formula: May help add clarity:

There seem to be many relationships which satisfy the instantaneous current state of a system Lloyd, but this tells us little about how a system got where it is, and the steps taken to get there with further state changes and such which would be the unified calculus type relationship of change over time by way of distributing composite 'e' through its many constituent values of v m a f λ, etc, as Va is conserved. When all values have to be satisfied in some conserved manner, we can better understand the effects of rad decay and the atomic clock relationship of QM to RM. This is the asymmetry conserving a deeper unified symmetry which I've mentioned. This is how Dave referenced Va I think. When each system achieves absolute motion, it must satisfy the conservation relationships of the many internal values. An autonomous system is reduced to its specific distribution of Va=e whereby each system has the same ingredients, but the ratios by which they are mixed together determine the characteristics of the system being observed. It's conservation in every direction thus homogenous mechanics. Consider what would happen to the constituent electrons and such if we were to slow or speed up a large composite body if they must further internally conserve the Va of their systems. Studying the composite conservation from Va of the larger body internally into the individualized systems as with our past discussions of Jupiter's effects and such opens up many avenues of exploration if we can work these mechanics out properly.