A fractal way of making sense of our experiences. Please contribute!

[jehovajah]

http://m.youtube.com/watch?v=OFzl37nX58A
At this stage Norman has established a coherent view on a representational form depicted on page by certain marks but thought of a responsive process of comparing, matching and verbally expressing: lego,analego, summetro, sumbolo in greek.
To call this Mathematics is deference to a subject name rather than to a description of the process . Mathematikos was a Qulification in scientific thinking really in astrological thinking.

The thought forms utilised by Noman are those from a broader subject called computational science in particular in application to computing mechanisms.
Processes of  listing or collecting are explored prior to a process of sequencing or ordering or succeeding.
The use of the concept of multiplication though tackled is finessed, because hardly anyone now comprehends factoring or fractioning a form by another!

Designing the factoring process defines what a particular style of multipliction could be.
Grasmann in my purview is the only philosopher who tackles this issue.
Here Norman defines multiplication as a kind of addition and this is the only real fundamental definition that cab apprehend the concepts of multiplication!

[jehovajah]

http://m.youtube.com/watch?v=7NefUTyUNuc
Here Nomn, indulgently, connects his thought forms to his Vexel idea.
The advantage is that he manages to embed his mathmatics in the over arching subject of computational science. That I concur with and find I have a long standing affinity for.

Of course this is a constructionist approach and some might wrongly claim intuition is excluded. Norman may harp on bout logical clarity etc, but the true process involves intuition, design, analogy, definition  And pragmatism .
Hermann Grassmann explains this in detail in the Ausdehnunglehre 1844.

[jehovajah]

http://m.youtube.com/Watch?v=bGJU9S3i8y8
Technical I know, but wht it is saying is that dropping the quantum theory element gives better results! So if quantum theory is the most accurate model we have thn it can be discarded in the short term nd certainly at the atosecond duration!
In fact the probability glasses we think it fashioble to wear obscures as much as it illuminates. It is a statistical process which works best on large data sets not so good on individual specific situation.
I worry that the mathematical measure that probability is is not understood as a global distortion of data. When renormalised to fit between values 0 and 1 actual data is distorted. Care must be taken to make the experimental outcome sit in the central part of the Bell distribution if classical results are desired!
Is the distorted outcome ever useful?
Pragmatically no,,but for experimental design it helps to minimise errors due to random effects of poor design or poor experimental control , or even  logical oversights . It mayt also indicate areas for additional research. ,
We are very human, we make mistakes, we measure incorrectly we design flawed processes , quantum theory takes that into account by using the probability distribution over the unit interval. . What is significant stands out, whether it be phenomenon or error!

[jehovajah]

Zeno's conundrum, the tortoise and the athlete teaches us something about our analytical and synthetical methods.
Firstly acknowledge anm8th sense, that of duration the analytical process is a process of cutting from above or over and over anew or again. The purpose to penetrate what lies within a form or structure, to reveal its constituents, the elements whose composition generates the analysed form, or the combinatorial arrangement of those elements that generate the same or new forms and structures.

Our sense of duration is often suspended, sometimes compounded into a false weight ( wait?) or eked out over massive accomplishments that seem to take no time at all! This unreliability in our sense encourages us to devise objective time keepers. the pendulum essentially..

So zenosnanalysis of the stages of motion suspend the sense of duration. In the meantime the analytical elements are increased in number without bound. .
This creates an issue for re synthesis! Suddenly the sense of duration kicks back in and the nalysed realises that it is not possible to finish the resynthesis!

The principle ofvexhaustionnremindsbthe analyser  he is part of the process!

[jehovajah]

We can also learn from Zeno that fractals are a natural requirement of analysis.
Firstly analysis is a scale free process! Thus it is up to the analyser to select an appropriate scale. . Fractals whether regular , irregular or dynamic set an upper and lower bound for analysis. The elements can be usefully identified by self similarity to the larger synthesised body. . In the case of dynamic bodies we can use the almost self similarity principle.

Ampere for example took a current in a circuit as his dynamic model of the elements of a rotational magnetic field around a current carrying wire in a circuit!

These elements ave been variously interpreted ad atomic structures with electrons circulating, to electrons themselves in some way rotating. , Biot and Savot just elected to use rectangular magnetic force arrows!
 . Consequently their model is accurate enough but limited to the circumstance they depicted"

Rayleigh in fluid mechanics proposed a scale factor determined by the ratio of momentum to viscosity effects,  the Rayleigh number guides application of the same principles at different scales. .thus synthesis and analysis of any fluid dynàmic investigations nd explanations.

[jehovajah]

http://m.youtube.com/watch?v=0BUR9mKvmi4
Norman clarifies his discussion of a theory of numbers and other mathmatical conventions.

The difficulties are related to scale. The issues go away when we renormalise or redefine the scale unit.
This is essentially a psychological application of fractal philosophy. We state that the arithmetic definitions illmapply to units. That leaves a unit undefined and therefore free to be set by the perpetrator!

In addition we use nalogues to justify or gain insight in applying this schema to almost self similar situations.

So our limitations naturally invoke this cyclical application of a methodology . This is innate within the development of fractals , the recursion or iteration of a defined process .

The distinctive structures ( topological) generated by such iterated processes are what Benoit called fractal. Some rigorous definitions initially limited the powerful applicability of this term, but I for one accept a looser set of defining properties.

We my not be able to prove Mathematically things which we intuitively recognise as fractal but that is no great loss co,pared to the ability to construct these topological patterns either as a resynthesis or as an experiential outcome .

Normns technical problem arises because he bases his foundation on numbers however captured. But his technical difficulty is pragmatically avoided by rescaling. .

The Pythgoreans based their philosophical treatment on topological forms . From foms we define counting and measuring , and these counting/ measuring practices give rise to mosaics called Arithmoi . It is one aspect ofvarithmoi that has come to be called numbers. By forgetting these connections moden Mathematicians including Norman, try to found their ideas on logic and a Minimal symbolic label or marker.

Grassmann particularly Justus attempted to found mathematics on logic and natural topological forms ( not the supposed ideal forms of the Greek philosophers) . However, natural forms are not regular , but irregularity is not without structure, Benoit by coining the term fractal sought to portray the roughness of natural forms but not as random roughness . In fact the complexity of randomness as a notion is redefined in the light of fractal topology.

[jehovajah]

http://m.youtube.com/watch?v=rCDRCGjmaO8

The Axioms of the Stoikeia do not exist! Axiom derives from the notion of an axle . These notions became identified as central tenets of any subject, that is the central dogma! . When classicists invented and used the term it was not to develop a Euclidean/ Pythgorean Teaching scheme or curriculum.  Instead it was more in keeping with a Platonic or Aristotelian discoursive approach where the central or axiomatic topics of a philosophical debate or discussion is where the continuing discussion starts anew,
Thus axioms originally required prior knowledge , learning or experience in a subject.
So what are the 5 in the Stoikeia?
They are Demands ( aitema) placed on the student to enter the course!
The word postulate means to beg or request with supplications! In polite society this is equivalent to demanding!
A list of items is a list of things required!

We therefore must not downplay the abolute embarrassment Mathematicians feel,about spending centuries trying to "prove" the 5th postulate!!!
It was and is a simple requirement on students or urveyors!
To do the course you need to be able to provide a straight edge that can join two points with a straight line! .,we take such rulers for granted, but if you do not have one how to you make one? It is not as easy as you might think!
You need to be able to extend a straight line, you need to be able draw a circle of a given segment size. Given a stiff segmnt and a fixed point a circle can easily be drawn, but to draw any circle you need to be able to provide this type of implement.

The 5th requirement is actually a requirement to draw straight lines that intersect however far they have to go to intersect! Surveyors need to be able to measure long distances and use straight lines that intersect . You can't use a curved line!

However spherical geometry assumes curved lines that intersect, that is a different more advanced philosophical discussion tackled in ancient times. 

[jehovajah]

Cantors demonstration of the continuum being a different infinity to the natural numbers infinity collapses to nothing when you realise that it is impossible to list all the real numbers so called and impossible to list all the natural numbers.
But we can structure the natural numbers but we can't structure the continuum? Well yrs we can structure the continuum by adopting a fractal structure . It might seem that structure must be larger than the natural number infinity, but of course we can not prove that , only assert it.

However this structure is nothing new. We have extensive magnitudes and intensive magnitudes and fractals are on the interplay between intensive and extensive magnitudes.
We know qualitatively intensity is different to extensivity.

I hear the conceit of potential infinity and actual infinity, but that is semantics. People want to keep the number infinity concept rather than the endless process.

http://m.youtube.com/watch?v=5rDxNjouzdg

[jehovajah]

http://m.youtube.com/watch?v=ywSQRggFjVg
These methods as explained are flawless! They indicate how thoroughly Newton apprehended the algorithm of Euclid in Stoikeia 7.
The setting out  and the algebra are 2 modern intrusions to try to " clarify" the rhetorical explanation usually given.
Of course in introducing these notations students tended to forget the process for continuing fractions or power series etc. thus they believed the = sign , not given in the original, meant actual equality! Thus we get into the Berkley misunderstanding of these Newtonian quantities, and the modern concern about convergence of series.

The long division arithmetical example is in fact a lay out established around the time of Brahmagupta, and promoted by him, but his layout relies on the power series base 10! Thus the polynomial reduction calculates the coefficients for this serie! Convergence is not an issue!

The second example highlight how to do a power series to a different base  and again convergence is not an issue.

If one uses the negative index for the powers less than one then the coefficients are relative to those.. Note how in calculating 1/8 the coefficients of the powers of 10 are calculated as a whole number by using not the next decimal place value but 10 times the next decimal at each stage.

One very important practice not explained here and again misunderstood when infinitesimals are mentioned the quantitative size of the magnitude and of the procedural product of a multiplication process with that magnitude are kept in mind!! So the product of any magnitude less than 1 in size with any other magnitude gives an output that is less in size than the product with 1 !!

Hence when reducing by Euclids algorithmic method the divisor is continually reduced in relative size until a whole count results . This is true in polynomial " division" as in ordinary arithmetic with " numbers". The practice of writing in power series is to acknowledge the structure on which numeral systems are based . The Hindu Arabic is a base 10 polynomiàl series, but of course other bases may be used and frequently are.

In addition the reduction from larger divisor/ factor to smaller factor does not have to proceed from left to right . In the case of 1/1-x where x is bigger than one the reduction can be performed the other way as the size is clearly larger than the item being factored. Alternatively one can proceed with negative powers ( a modern alternative) and negative size. In Newtons time these representations would not be understood unti Wallis established a consistent Algebraic notation for fractional and negative logarithms/ exponents / powers.

Today's emphasis on convergence etc is due to this fact , symbols completely mystify students who can not then relate the notation to actual quantities in their experience . Early Astrologers were taught better than we often are and so were not explicit about many obvious things.

[jehovajah]

http://m.youtube.com/watch?v=i5TFXyI4UMM

I am grateful for this video, which is why I posted the previous post. Newton never created a flawed system or method, he just lived in a time when others would steal or corrupt another's ideas fir financial or social or political gain. His academic papers written in Latin were sometimes written in code for that reason .

Here we see how the fulmination of Betkely to this day clouds the apprehension of Newtons method!

Suffice to say Berkely was well refuted by Cotes aad others at the time,mbut in a time of clerics, clerics rule! And Berkleys ferment had his intended effect regardless of its effect on Newtons reputation: religious philosophers came to heel while those who were atheistic we're forced into a semantic cubbyhole from which they to this day have not emerged!

The ideas of infinite / endless is well defined in philosophy, but in religion it is by design mysterious! Thus to some clerics it is blasphemous to even talk about infinity without acknowledging God or Jesus! But the Greeks did just that , because it is a semantic label for a human experience not a divine one! To any clear thinking Astrologer just as we can count without end, so we can factor without end. But Lucretius and Democritus insisted that there Must be an end to cutting a material object , even if we can continue to break it into parts in our minds! Thus the word atom was given its material significance as the material that can not be divided further!

To accept this stance is to immediately distinguish between matter and spirit. This basic asserted dichotomy underpins the argument between materialists and those who accept an immaterial reality. Clerics of course favoured the latter, but as Berkeley observed many philosophers were abandoning the tenets of the church for a materialist point of view.
Logically there is no irrefutability or either position, and this is what Berkeley was driving home. No one cn deny the other by the tenets of the dichotomous stances, both demand an equal faith response!

However, as logic was intended when designed by the greek schools of Rhetoric, one may present an argument to persuade by fair means or foul, and ridicule, ad Hominem, and many other logical devices were outlined and taught in the schools of rhetorical logic.

Berkeley picked on Newtons infinitesimal notion because he knew his audience did not understand it! He was careful not to claim Newton said, for libel was a serious issue in his day, but cleverly he points out that an apparent contradiction lies at the heart of Newtons method.
Then he asks in an attempt to ridicule : what are fluents,Fluxions and these vanishing quantities?

Now in Europe infinitesimals were considered as a joke ! But this again wasvduevo trying o call them numbers!
Everyone thinks they know what a number is!! Clerics were of the opinion they were ( whatever they are) divinely revealed by God and set out sensibly in the bible and in Nature. Thus infinitesimals were undoubtedly the  work of Error or at worst of the devil.

I can not emphasise just how tiring this kind of thinking is! For a classicist to read the Ancient Greek clear exposition of arithmos and how to philosophise quantity with them must have been a huge relief.

So infinitesimals for Newton, like infinity were semantic word labels for endless processes! Once you stop such a process it is by force no longer infinite!!! Thus to use a symbol is not to make it an Arithmos!
 2 other things highlight Newtons explanations . Fluents or fluids were time dependent, and his symbol is gor a small moment in time. While an extensive unity may be objectified, time is n intuitive srns that I highly subjective. The sense of duration requires such LanguGe as nascent, evanescent, vanishing etc. a clock, or pendulum do not replace our time sense, they merely provide a scaffolding in which to erect a onsidtent and conventional measure.

The second was a concept of dynamic variation. Even classical philosophers considered forms in static poses or stages of pause. Zeno sets aside time intervals in his famous argument to focus on the clear stages for comparison , thus fluidity is not incorporated in classical Mechanics.

It is still difficult to get this idea across even today! Fortunately we cn use the high speed camera metaphor to capture Newtons vision of fluents and fluxions. We can now visually track a fluid action and see the intensity of a Fluxions by comparing 2 sequential frames in such a movie.

Newton did not divide either. He factored. When you factor you do not Divide you look for the factor that gives a certain quantity, that is you look orthe Quotient!.

Often the concept of multiplication is used to explain quotients, but it is in fact the concept of counting factors of a given form that we need to inculcate. Our understanding of factors is codified in the binomial theorem but essentially the mosaics of old or a pile of bricks demonstrates factors and their equivalent topologies.

The rule not to divide by nothing is therefore patently nonsensical, but when you set up such algorithmic presentations of the factoring process and systematic synthesis or Combinatorial schemes these rules of thumb often pop up.

.let us now consider newtons fancy: suppose we have an infinitesimal moment in a fluid situation. This is like taking a snapshot at a trillion frames a second . The infinitesimal momnt is between frame 1 and 2 ! But that is at a trillion frames a second. What if we increase the frame rate to 10 trillion ? Then frames 1 and 2 now show a different relationship. If the quantities were not fluent there would be no change.

The symbol for this time step can not be fixed and it can not be evaluated because once you do you are no longer fluid! We know how high spped cameras Freeze fluid motion. The concept of freezing fluid motion is entirely novel to Newton. Leibniz still clung to the notion of a differential and that in an extension. Newton realised a differential in time might freeze a fluent and reveal its Fluxions or freeze frame changes.

So what about a term which contains an infinitesimal as a factor? In this case Newton new they were pragmatically useless. They can not be measured but if they could they would make the product so small as to be vanishingly small! Newton does not drop them, he just works with approximations from that point on
There is no need to go to a limit because a fluent is dynamic, continuous and clearly flows before nd after any chosen time frame. The Fluxion for a constant flow should be the same no matter when measured . The tiny piece he ignored in approximating is vanishingly small and not measurable in practice, especially if working at the limit of ones ability to measure.

The approximations of Newtons methods have served a pragmatic technology well.

[jehovajah]

http://m.youtube.com/watch? V=eOPpZ7eOAtA
The Sandkrit philosophy of the Upanishads heavily influenced Greek thought and then the Prussian and European intellectuals.
This clear explanation of Brahman also explicated the central role of Shunya that is Everything .
Shunya is the substratum from which all forms derive.