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 Author Topic: Fractal Foundations of mathematics: Axioms notions and the set FS as a model  (Read 93720 times) Description: All ideas welcome.Needed to revise mathematical thinking and exploration 0 Members and 1 Guest are viewing this topic.
M Benesi
Fractal Schemer

Posts: 1075

 « Reply #495 on: October 27, 2015, 08:39:41 PM »

The point is under Newton and DeMoivres tutelage he had found how truly useful the unit diameter circle is.
Euler proceeded along similar lines but instead used the unit radius circle, making everything 4 times too big!

It is hardly conscionable that the Pythagoreans did not know that the true unity is the unit diameter circle, not the unit square..

The combinatorial properties of both hardly differ, but we may observe the fractal nature of space more clearly by using the circle Metron / Monas.

Nice.  The infinite sided polygon, which is also the one sided polygon, is... unity of the infinite and the unit.  Are you going to write a book from the material in this thread?

You see my new art?
 « Last Edit: October 30, 2015, 12:19:04 PM by jehovajah » Logged

jehovajah
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May a trochoid in the void bring you peace

 « Reply #496 on: October 30, 2015, 12:44:09 PM »

@M Benesi

Thank you for your contribution Matt.
No I have been rather distracted of late with health and family issues, but I will look your new art up, for sure!

I think the world will be a better place if I do not write a book!!! But I am happy for you or anyone else who wants to to use the material in consultation with me, to do so.

I cannot remember all the topics  I have pursued over the years, some are not conclusive I know, and some I never completed, but all allowed me to untangle myself from centuries of misinformation.

In addition, this has and still is an open thread. I am grateful to all those who supported my mutterings and meanderings and truly humbled by the awesome number of views.

My hope is that it has served to keep Fractalforums at the forefront of fractal sights and helped Christian to keep the site going along with all that the members are doing and continue to do.

As things go Fractals are fundamentally important to a topology of space and dynamic substances. That really is my conclusion .

The work I am doing on the Grassmanns is really to demonstrate that this conclusion was glimpsed by them in the early 1800's , before combinatorics was made mainstream by Erdos, Mandelbrot and others.including Turin.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #497 on: January 25, 2016, 10:01:51 AM »

The Ganitas
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #498 on: March 05, 2016, 10:30:18 AM »

As I have discussed in the Stretchy Things thread multiplication is an undefined process, and really we should refer to factorisation tables.

The archetypical factorisation process is Euclids highest common divisor method. As an aside book 7  of the Stoikeia is devoted to this process from the very first word to the last!

My reason for posting in haste is to record that this morning I realised that this process applies to th nfactorising of circular magnitudes and thus bears directly on the trisection of a gneral angle/arc !
I will detail this observation in another post .
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #499 on: March 07, 2016, 02:51:56 AM »

Important to assimilate these distinctions.
How these distinctions actually should be applied in constructing our descriptions .
Strange as it seems length area and volume are constructed products, and sometimes we cannot construct them!

So the thing we need to apprehend is the Monas, the Metron, and the Logos Anologos methodology
( Proportion). To replace these by numbers is to misunderstand Arithmos fundamentally!

Arithmoi are precise mosaics using a specific Monas whether that mosaic tessellated the spce or not was Not the main aim of the mosaic. Epipedoi, or speripedoi were also mosaics, but the elements were not necessarily a Monas , any forms could be aggregated together.
Arithmoi were canonical in that the constituents are homologous or homogenous. The first rule for the logos analogos methods is that the magnitudes are homogenous!!

The idea of a Metron is that a form becomes a Monas or a standard unit. This standard unit thus imposes homogeneity on any Katmetresee ( measuring by laying the Metron down ) of a larger form.

So an Arithmos is a mosaic of standard units, we can use any regular form as a Monas so that it can act as a Metron of larger forms

Given this factorisation of space by a given Monas used as a Metron , one can then derive proportions ( logos analogos) descriptions of relationships.

When it became fashionable to place the count of the mosaic in place of the actual mosaic, the idea of a quantity as free standing became formed and gradually led to that modern confusion called Number!

Later a Persian cleverly showed that ratios could be written as fractions, and that a limited set of ruled governed these fractions . It took a while for the furore to die down, but eventually the powerful use of fractional measurements on rulers led to the measuring line concept. In particular the Indian 8, 9 or 10 cyclical systems, Vedic, Bahai, and Brahmin systems, under Wallis were powerfully integrated into the measuring line conception. This organised structure was later used By Dedekind et al. To derive the modern Number line system. In the mean time numbers had takn on a cardinal, or lordly position, and that was distinguished from their ordinal use, as well as their dimensional multiple use.

So to the circle and to the diagonal of the unit square.
A fairy tale is widely told about the Pythagoreans being thrown into consternation by irrational quantities.
Firstly these were known as incommensurable , and that meant that the divisor Algorithm could never be finished! The simple solution was to define these incommensurables as new units, that is "protoi "! Thus protoi Arithmoi are all about these incommensurable magnitudes! Prime numbers are how the Pythagoreans studied incommensurable or irrational proportions( that is through the application of Euclids algorithm).
The fundamental unit Monas was thus the unit diameter sphere!
All of the results for a square can be obtained using a unit diameter circle.

It is clear that the square tessellate the plane, but the circle leaves gaps. This was not a problem, but a clear indication of the incommensurability of space even in the plane. However one could choose a Metron to suit the required outcome .

Thus in attempting to describe the proportion the iterative / inductive method of Euclids algorithm is the classicl fractal process upon which all fractal processes fundamentally rely

When this algorithm is applied to the diameter versus the perimeter of a circle of unit diameter it was known to be incommensurable. That did not prevent greater proportions being sought! Today we still attempt to develop even greater proportions!
Was it ever hoped that the proportion would be found? No. The reason is homogeneity. A circle is not homogenous with a straight line!

Careful astrologers up to Newton never expected a curved line to be straightened . The inhomogeneity of th comparison is why no proportion will ever be finalised.

However, for pragmatic Mechanics and gear design proportions were approximted by truncation. For those who deified the watchmaker as a type of divine Mechanic, the implied perfection covered over the inherent error introduced by approximation. Within hose errors lie th chaos theory which now accounts for many unexpected behaviours in the divine clockwork!

So the trisecting of an arc was inherently an incommensurable outcome, because 3 is the third prime or proto Arithmos!

Nevertheless a multiple of any prime is never an issue and this is the way to approximate to the trisection of n arc by using proportion and the ultimate proportion form: the circle / sphere.
 « Last Edit: March 08, 2016, 08:58:25 AM by jehovajah » Logged

May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #500 on: March 08, 2016, 09:39:06 AM »

Firstly, the perimeter of a circle is a magnitude. It is not a straight line magnitude. The question is how can one factorise that magnitude?
Using a compass or divider one can mark off homogenous arcs. These can then be utilised in an Euclidean algorithmic process to factorise the perimeter.
Many other methods can be devised to distinguish these arcs, but the classical one used chords or secants to bound a segment of the circle. This made for easier comparison by physical movement of the segment.

The mechanical/ physical movement in measurement is always downplayed, discounted, disparaged . Neusis was given a bad reputation among certain Academicians. Engineers and Tekne orMechanics relied upon these physicl motions to generate the factorisation of magnitudes. Euclids Algorithm is a mechanical, physical process.

Therefore Spaciometry is a dynamic subject a study of dynamics in space ! The circle is the chief rotational dynamic, but not the general one !

We all accept that the radius cuts the perimeter into 6 equal arcs! In point of fact we cannot prove that. We accept it and indeed define it to be so. It is an Ennoia a commonly accepted judgement that experts all agree on . These experts are mechanicsl engineers, not " mathematicians" !

Mrchanicsl engineers can also demonstrate the proportionality of the magnitudes of different diametered circles. They can demonstrate how the gear ratio works pragmatically according to the diameter. Thus they can demonstrate how a circle with one third the diameter of another will rotate 3 times along the perimeter of the larger circle. Thus mechanically any general arc can be segmented mechanically and " precisely" that is, with due care to avoid slippage .

It is these mechanical relationships that underpin the values in the trigonometric tables. The development of infinite series definitions for trig values is based on interpolation of mechanically established differences. The difference expressions are a testament to a dynamic application of the Euclidean factorisation method . It is these methods that were renamed the Calculus , and differential equations are no more or less than glorified difference expressions.

Solving a differential expression requires specifying a desired out come to the interpolation process. Then the differences can be mechanically derived by motion of gear wheels!

There is an amazing Harmonic analyser that does Fourier analysis by gear wheels! .

The mrchanicsl basis to so called mathematics must beer emphasised in the new iteration of the Thought pattern Doctrine set out by the Grassmanns
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #501 on: March 16, 2016, 04:07:16 AM »

Many constructions de emphasise the circles dismissing them as construction lines. However the circles are the fundamental process! They are the ultimate proportioner. The Logos Analogos methodology utilised the circle diameter, chord and sectors to proportion regular polygons .many of these methods are.  Refined from the basic Euclidean algorithm . Basic trial and error methods are haphazard, unless organised by the highest common divisor method.
The right triangle in the semi circle makes an ideal scale recorder! Thus as those circle ratios are discovered the associated chords are arranged straddling a common diameter. These becme the sine lengths and the cosine lengths for given regular polygons .

In addition, arcs delineated by chords could also be compared, and the notion of an arc magnitude developed in analogy with chord magnitude.arc magnitude proportions could be compared . Comparison between arcs in different diameter circles were compared by gear ratios .

Many relationships were painstakingly uncovered, discovered and recorded by segments on a line! Sectors , triangles, parallel lines became common expertise used to effect these comparisons .

These expert understandings, used in conjunction with Euclids algorithm leads to many regular polygons and chords and arcs.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #502 on: April 02, 2016, 09:39:41 AM »

The application of the Euclidean divisor algorithm to factoising the circle into commensurable arc lengths is possibly easier to visualise, particularly if you have drawn circles and factorised them into 6 arcs. This by the way is  common judgement established as a definition! We have no way of precisely confirming this measurement so we define it. It has served us well .

To disect a circular arc by an other arbitrary circular arc on the same Circle : Mark off a Reference Point on the perimeter/circumference . Mark off the 2 arcs by this reference point .

Taking the smaller disect the larger until there is either an exact match or a remainder.
Mark off the remainder on the citcle from the Reference Point.

Repeat until an exact match is obtained or it becomes boring or too small to continue!
In general we expect incommensurability as did the Pythgoreans. Where commenurability was discovered these were celebrated as Monas, and the mosaic patterns ( fractal patterns) they formed were called Arithmoi!

Embedded within this praxis is the additional notion of " square rooting' which comes from the notion of transforming one form Ito an equivalent square. This made the suare the ideal tessellating Monas but not the fundamental one. The circle is always the fundamental ideal Monas despite not tessellatingthe plane.

Of course I speak here regarding the plane, which is itself an ideal surface!
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #503 on: April 09, 2016, 01:26:35 PM »

I have referred to this disconcerting fact that mathematicians have foisted a lot of no sense on themselves by accepting Cantirs proof of real numbers as an uncountable set.
In my experience my Maths tutors were very upset when I would not accept it as proven and expressed the illogicality of it. They spent a great deal of time trying to get me to accept it draughting in a visiting Irish reader to convince me by isomorphic projection theory!
I was never convinced but simply conceded to allow me to get on with the rest of my studies!!
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #504 on: May 15, 2016, 12:15:49 PM »

The sets FS and. NotFS were defined in the Cantor sense, but not in the Von Neumann sense even though this is clearly fractal in the self referential and iterative inductive sense.
Bertrand Russel and AN Whitehead unsuccessfully attempted to establish mathematics in their Principia, so why Von Neumanns work is seen as differing I do not yet know. The construction is the same.
Both Whitehead and zrussel were heavily influenced by the Prussian RenaissNce in which Hegel played a major revolutionary part, but specifically they were admirers of the Grassmann project!

Bearing in mind Gauss's Protege Rheimann it is a not widely publicised fact that the Grassmanns were more influential than he!
So why did Russel fail? Because unlike Hermann he refused to consign mathematics to the dustbin of history! Instead , like NormN they tried to refund Mathematics, a task that ultimately must fail due to logical inconsistencies!
The Doctrine of Thought Patterns is one suggested name Hermann came up with for the whole field. Today we might consider computer science as being the general term for it! Certainly movements like Wolfram display the evolutionary consequences of the Grassmann approach.
Can Mathematics be saved? I think not, and it is already dead !
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #505 on: January 02, 2017, 12:27:18 PM »

Norman nais it!
I travelled through the 500 odd posts in this thread to come to these concrete realisations .