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 Author Topic: Fractal Foundations of mathematics: Axioms notions and the set FS as a model  (Read 109010 times) Description: All ideas welcome.Needed to revise mathematical thinking and exploration 0 Members and 1 Guest are viewing this topic.
jehovajah
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May a trochoid in the void bring you peace

 « Reply #480 on: January 22, 2015, 02:55:16 AM »

Euclids Stoikeua is a course in Pythagorean Philosophy.

In book 1 we are demanded to comply with certain pragmatic skills or not take the course!

Just as Platos academy reputedly has a lintel sign saying  " if you cannot do Surveying turn away!"
Apollonius probably modified the original set of demands to add the circle requirement.

So you need to be able to draw a good line between any 2 points;
You need to be able extend a good line indefinitely until it crosses another good line on the side where the two angles are less than 2 quarter chords/arcs;
You need to be able to draw any circle of any radius;
And you need to agree that all right angles are quarte arcs!

Propositions, definitions and methods of demonstration are gone through in part 1, including parallel lines, and how to compare line segments and move them into a standard structure for comparison and contrast..

Now in book 2 we are Introduced to the segmented line as a rule of proportions. The segmented line is a carpenters measure and so it has an arm that rotates from one end of the segment. Using this swing proportioning is introduced algebraically! Book 3 continues the rotation of a carpenters measure, but this time constructs circles on the segments.

Now the reader is able to apply skills to construct, compare snd contrast any form in the plane and any movement of the stars in the astrologers sky!

Book 5 and 6 now look at the formal comparisons of line segments and there proportions (analogos). This is how we learn to carry one idea as a ratio ( Logos) into another ( Analogos) and how that reasoning can be employed in many ways. Starting simply and moving on to increasingly complex thought patterns the student learns to appreciate the extensible nature of the segmented line segment.

Books 7 to 11 now introduce the manipulation of space by a solid Metron called Monas. The monads are divided out of foms like a mosaic. The factorisation of space by Division gave rise to the Arithmoi which begin book 7,

It is worth reading this book very carefully because right from the get go, an uncalled for mathmatical terminology is imposed on the Arithmoi by translators!. The Arithmos is a mosaic of a form and how the monads may be used. But they carry the thought patterns developed in books 5 to 6 into the regions of space and form. Each Arithmos therefore has its own unique " geometry" or rather collection of applicable and extendable thought patterns particularly regarding Logos( ratio ideas and measurement relationships) Nd analogos( analogous, and sympathetic ideas which exhibit a proportional link to the initial ideas) such a thought pattern of analogous ideas is not surprisingly called rational. It is this rational and ginormous thinking the course is attempting to develop, not the late 17th century hybrid called Mathematics!

Where then is the Aristotelian logic?

Pythagoreans and Aristotle disagree at a fundamentally qualitative level. Aristotle could not get his head around the two ness of 2! Whereas for Pythagoreans this is not a linguistic problem! Aristotle was afflicted with Aspbergers syndrome and obsessive Compulsive Disorders which drove him to taxonomise everything systematically. However he chose Language and Grammar as his fundamental taxonomic schema. Thus questions about language are ill suited to be taxonomised by his scheme: thus his mystification about 2

The Pythagoreans appreciated each form, format and structure whether in dynamic or static equilibrium. Thus the quality of 2 appears fractally in space everywhere and can be appreciated and identified. But then the quantity of 2 appears in every individual's consciousness subjectively, and concording to that individuals determination. Quantity is a subjective experience and also a utilitarian one.

Logos, the expression of a comparison is the only way one individual can convey the subjective experience of quantity to another: thus 2 cats : 1 cat is immediately understandable as a quantitative ratio, whereas 2 cats alone may refer to an unspecified group of human beings playing jazz! (for example!)

The subtle differences in the consequential dialectic( or linguistic logic) had to wait for Hegel to bring out in the west, although Indian, Chinese and Japanese astrologers were aware . The Islamic scholars however favoured Arustotles " logical" schemas and taxonomies, because they got access to the tutor of Alexander the Great! As a result our modern world reflects Aristotles views almost as if he were a Demigod of Nature. At the same time Hegel has had a continuing and subverting influence on this paradigm.

This however had lead to the Pythagorean school being undermined and eventually the Stoikeia being obsoleted! However at the time of Hegel Euclid's Stoikea was still very iconic and influential, and worthy of salvation. Only one main family emerges as actually doing something to Save Euclid the Grassmanns.

It was commonly felt by the time of Gauss that LeGendres version of Euclids Stoikeia was seriously flawed and in need of revision from the ground roots up. However it was not stated as an attack on Euclid, but on Geometry , and specifically the old school geometers. However Russel Ascerbically attacked the Geometrical board of scholars using Euclid as a cloaking device!

Hermann Grassmann and the Grassmanns in general felt it was not geometry but Philosophy that was at fault, not clearly distinguishing between formal and Real expertises.because of this Confusion going all the way up to Gauss, geometry had been given a raw deal, Euclid had been wrongly maligned, LE Gendre had screwed around with Euclid to such sn extent thst the false ideas nd goals of a prior generation had become the misleading and ultimately fatuous goal of his: to resolve the 5th Postulate( demand) as if it were a proposition!

I have placed it in the extension of a good line demand to highlight its pragmatic nature!

By reseating Geometry on the real expertises Grassmann opened up a new era of thought patterns to take us on into the 21st century.

I hope to continue translating Hermanns works here in due course.
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 « Last Edit: January 22, 2015, 04:24:06 PM by jehovajah » Logged

May a trochoid of ¥h¶h 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 #481 on: January 22, 2015, 04:26:46 PM »

But if some budding translators want to have a go and start adding the first chapter of the 1844 Ausdehnungslehre Please, Please have a go and post it here!

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May a trochoid of ¥h¶h 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 #482 on: March 04, 2015, 11:45:11 AM »

Ok, So I cannot restrain myself .

I am working through the first chapter of the first section of the Ausdehnngslhre 1844. But rather thn publish it here I will start a new thread.

This thread is very long and represents a considerable reading task. The Ausdehnungslehre 1844 certainly should be published as a fractal foundation of Mathematics element, but it is unfair to inflict the reader with such a task to see how it all fits together wonderfully.

Those who have read this thread over the years may know that it represents a journal of discovery, and recovery. As such it is eclectic and meandering . However the Ausdehnunglehre, though born of the same deep enquiry, and practically investigating the same topic using similar tools, despite its imperfections, is a  more systematic and purposeful treatment or exposition. Indeed it is a true course in a method developed by Hemann Grassmann from studying the great Natural philosophers and Astrologers of his day. Read this and think like the Greats! Or rather Study this and become a self activating genius like LaGrange, LaPlace, Euler, Hegel, Steiner, DesCartes and many others.

So see you over in the New thread
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May a trochoid of ¥h¶h 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 #483 on: April 04, 2015, 11:31:47 AM »

I coined the name Manipumé, Hermann Grassmann in fact the Grassmanns coined the name Formenlehre for the revision of that subject formerly called Mathematics!

Here is Normans vision .

I am persuaded by The Grassmanns that their style has more to offer quicker than any others, and I know it is richer and well permeated throughout the theoretical cadre, so a revolution in that direction would be achievable with less " bloodshed" and defensible vested interests!
Unterwerfen, Wirrung und Wiederholing!
Vivre la Revolution!
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May a trochoid of ¥h¶h 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 #484 on: April 21, 2015, 08:02:53 AM »

As usual this is an accessible introduction to the problems of modern mathmatics nd physics.

The attack by Berkely on the Fluxions of Newton is often cited as incisive, but in fact the defense of the Fluxions is found by reading the Astrological principles and Sir Roger Cotes exposition. Norman picked out but one of many methods Newton expounded upon to demonstrate his Fluxions. The method is not dubious and certainly not definitive of a Fluxions.

The binomial expansion as an infinite process is subject to the principle of exhaustion. However Newton does not(!) eliminate by setting to zero! Neither does he subscribe perse to infinitesimals , hi approach eliminates the error by subtraction!

Thus his series expansion, like the Taylor expansion enables him to take various approximations as assays of the models for the dynamic situation,.

The fundamental. Binomial series governs his whole development of Fluxions.
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May a trochoid of ¥h¶h 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 #485 on: April 21, 2015, 08:45:12 AM »

The fundamental elements of Logos Anslogos, which is dealt with from the beginning of Book5 onwards in the Stoikeia.

http://youtu.be/_c5v14ZIUO0
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May a trochoid of ¥h¶h 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 #486 on: May 11, 2015, 09:15:57 AM »

Am loving this guy at the moment!

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May a trochoid of ¥h¶h 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 #487 on: May 11, 2015, 09:38:07 AM »

Was ïst fichtig?

Und such

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May a trochoid of ¥h¶h 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 #488 on: June 16, 2015, 03:10:30 AM »

Norman describes the nexus of measurements tha occurs with the unit circle.

The core circle idea is different to Eulers circle which is based at the origin with radius 1.

However, Normans rant about dividing a circular arc is based on a mythical concept: the " good" or straight line as the simplest line.

There is no simplest line, but the most natural line will be a trochoidal one !
The 1600 to the 1800 s saw a remarkable decline in the understanding of the Pythagorean philosophy as Platonism was extracted and exposited to fit certainly modern ideas and interpretations.

Newton and Jakob Steiner we're arguably the last reat synthetic geometers, although DesArgues projective approach resurrected many of  Apollonius teachings.

What obscures all modern apprehension is the Number concept.

Replace the number concept by the Pythagorean Combinatorial concept and ones mind becomes free.

The protractor idea does not remove approximation. No ruler removes approximation, no electromagnetic radiation removes approximation. But proportion enables us to count, to order and arrange in space haphazardly or systematically. And it is this combinatorial synthesis of prior analysed elements that underpins all mathematical thinking.

We can proportion the circle, what we can not do is homogenise the circle to a straight line!

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May a trochoid of ¥h¶h 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 #489 on: June 16, 2015, 12:25:22 PM »

Relevant video

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May a trochoid of ¥h¶h 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 #490 on: June 24, 2015, 07:56:02 AM »

The idea of a protractor is fundamental to astrology .

Here Norman explains the Ptolemaic protractor as it appears in the garment of ratiosl trigonometry and more generally in the geometry and trigonometry associated to conic sections.

The general concept of rotation can be distinguished by the establishment of a circular arc measure.
Here points on the core circle can be computed by trial and error to produces a regular sector division of the core circle.

Finally the presentation on relativity is very geomtrical.
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May a trochoid of ¥h¶h 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 #491 on: June 24, 2015, 01:24:13 PM »

Sir Roger Cotes discovered the power of the unit circle in harmonising all systems of measurement .

He was in consultation with Newton DeMoivre and Halley at the time , but died before he could explain himself and the curious logarithmic expression or nowadays we would say " function" he had found.

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.
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May a trochoid of ¥h¶h 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 #492 on: July 14, 2015, 06:09:05 AM »

This video introduces a fundamental 17th century notion that underlies the Mechanics of Newtons Principia. The notion of curvature! Especially as the process of considering larc segments that are tangent to a given curve starting with a straight line segment!

Arc segments were fundamental to Christian Huygens thinking, but in fact go back to the Pythagorean school of thought. Thus the notion of an inverse as somehow capturing curvature , that is reciprocal relationships encodes circular and spherical surface relationships.

The significance of this, as a general overview , is that we can go between formulae for straight line segments and those for curved line segments by replacing factors symbolising these straight line segments by their inverse.

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May a trochoid of ¥h¶h 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 #493 on: July 15, 2015, 10:03:00 PM »

This confusing documentary perpetuates many myths that are no longer feasible in the conceptualising of mathematics . Indeed mathematics is a word that is fast becoming obsolete and irrelevant.
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May a trochoid of ¥h¶h 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 #494 on: July 28, 2015, 08:25:25 AM »

Norman tackles the knotty problem of Set theory .

http://youtu.be/dyCRPT6iFBg

Note the German definition.

Normans approach ignores the philosophical and theological background to 18th and early 19th century "Mathematics" . Norman does not ignore the background ignorantly! He deliberately excludes these founding ideas and ideologies as " not clear enough", and wants to redefine mathematics . I want to let the term become obsolete!

The Grassmanns were of the opinion I hold.
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May a trochoid of ¥h¶h 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!
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