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Author Topic: Fractal Foundations of mathematics: Axioms notions and the set FS as a model  (Read 126548 times)
Description: All ideas welcome.Needed to revise mathematical thinking and exploration
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jehovajah
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« Reply #450 on: June 05, 2013, 04:42:15 AM »

So i am nearly finished with the research into the background to that which we have given the name Geometry.
My conclusion is that we did not have a clear subject boundary called geometry until 1794 when A, M, Legendre published his now legendary(!) Text book Elementes de Geometrie.

You may like to debate that conclusion in this thread.
I cannot think of a better thread to do it in grin

Here's my blog link
http://jehovajah.wordpress.com/jehovajah/blog/2013/06/01/from-plato-to-legendre-geometry-in-aristotelian-crisis
I am still writing it however, so any good points raised might end up in it, suitably by lined of course wink
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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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« Reply #451 on: June 13, 2013, 04:50:54 PM »

I think the following is my best explanation of the fractal foundations of Mathematics!
http://jehovajah.wordpress.com/jehovajah/blog/2013/06/13/sequence-and-series-in-the-arithmoi

What do you think? Comments in this thread please. embarrass
Oh!! I just thought, I can now research the Babylonian and Chinese forms of the Arithmoi.....
« Last Edit: October 07, 2014, 06:27:10 AM by jehovajah » Logged

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« Reply #452 on: July 09, 2013, 12:23:05 PM »

One of the benefits of deep analytical research is the relief one feels when one stops banging ones head against a brick wall!

I have enjoyed some moments of synthesisi and quiet meditiation resulting in the following.

http://jehovajah.wordpress.com/2013/07/09/newtons-alchemy-a-quest-to-understand-the-active-principles-and-define-stickiness/

Read and enjoy.

If you have followed this thread you will know the themes touched upon in the post.

There is , of course, yet more!
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« Reply #453 on: July 13, 2013, 08:10:30 PM »

Of corse there is always more!
Quote
Book 5 is a Game changer!

The following videos explain the background to why!
[video=http://www.youtube.com/watch?v=UPlqJaUi5jE] [/video]
[video=http://www.youtube.com/watch?v=56Yy1odPLag][/video]

A quick overview by Norman of Book 5

[video=http://www.youtube.com/watch?v=YAdEfQsIGt8&list=PL6BB1525BCC3B1740&index=5][/video]

But Norman misses the connection between books 5 and 6, the Eudoxian theory of Proportions or rather Analogia! Eudoxus lays out the linea; basis for Analogous thinking,

Then books 7 to 10 recast the ideas of books 1 to 4 in terms of the Arithmoi, the general notion of mosaic grids or nets made from any of the ideas discussed in books 1 to 6!

Euclids Stoikeia thus represent n introductory course to Platos theory of Ideas/Forms in Books 1 to 6 finishing the first year of a 2 year course in introductory Pythagorean principles. The second year begins with Book7 and introduces the Gematria of the Pythagorean schools, the Numerology, the combinatorial philosophy of the Pythagoreans. This continues through to book 10.

Finally books 11 to 13 form a more philosophical discussion of the real dynamic world, and how the preceding principles in the 2 year course apply to the older spherical and circular schemes of proportions and spaciometry. Basing this discussion or dicourse on the Stereos, that is shadow casting forms, the projective geometrical I deas of Eudoxus in book 6 , come to the fore in the projection of the circle and the sphere.

Within the circle regular polygons wee constructed i book 4 , these were projections from the regular solids in book 13 onto the plane, the Epiphaneia, that is the light catching surface! The epipedos or Mosaic surface was explored in book 7, and what connects them all are the point and the Good line!

The notion of a good line as a constructed dual pointed line is fundamental to the connected structure of the Stoikeia. The gramme is officially defined as that greaness of magnitude that has no plates! this is literall a line drawn by a sharp implement, a cut into a surface!

Plates derives from the root plassos, which, briefly is the concept of plasticity as in clay, plastacene   or some substamce like putty. When squeezed this substance yields and spreads out . When worked its form transforms continually, so ones initial apprehension of its magnitude or greatness changes. We are lead into more and more descriptive notions of the distinctions appearing before our eyes! This behaviour is summed up in plates, which is consequently more than the simple translation of width, just as mekos is more than the simple translation of length.

Mekos, plates and bathos are the three descriptive notions for a solid form. They do not mean length breadth and height of length width and depth as some have it. That is to say they refer to the conception from which we draw these 3 dimensions as instances. Mekos alone is sufficient to define length and girth, and our immediate and natural response to any form in the instant of us perceiving it. the other 2 arise as a consequnce of empirically interacting with the form. Thus plates , as explained above expresses this transforming girth or spread of a form as it is worked. Bathos arises from the experience of being in the form . It is easy to see that bath carries this idea into our everyday use very well. The scal e of the bath is hinted at by our terminology. Tus a bird bath, a swim bath are all baths of different sizes and ratios to one another. such ratios when precisely copied or scaled are called analogues. That is he forms are said to be in analogue to each other ,

Why avoid the word proportion? because the notion of analogue is far more powerful and useful. For example few realise that Logarithms are Analogues > By using the principles of Eudoxus in book 5 and 6, and even the basic terminology of Eudoxus Napier clarified the Analogue relationship between the sines and the arcs of the sectors of those sine ratios written as fractions , in long hand to at least ten places of digits. The 2 lengths napier refers to are the length of the sine on the perpendicular and the length of the arc on the circle. This length was obtained by rolling the circle on a flat plane without slipping.

Eudoxus defines proportionality using the good line. this good line is obtainable only by use of dual points marked off from the centres of 2 circles as ppoint of intersection of the perimeters. This is why they are called dual points because 2 circles are required to determine them. They cannot be place freehand, at least not unless you are as skilled as Michaelangelo! As Newton points out , the underpinning mechanics of this fact reveals that geometry is derived from and improves upon in recursive relation, the practices of mechanics!

This held clearly in mind, the length of the circumference of a circle was no real mechanical problem. The word kuklos means disc, and the disc is simply but carfully rolled on the pane to gramme or draw out its mark which remains fixed, not transformed by spread or depth. . One revolution enables the mechanic to represent the perimeter by a straight line segment , marked of in the epiphaneai. Eudoxus, and all greek pilosophy therefore deals with curvature pragmatically. They straighten it out by some mechanical means.

The issue of proportions that is analogies depends therefore on comparing the same things! thus the magnitude of a line, even though it is a straight line cannot be assumed!. The straight lie drawn by rolling a circle is a different magnitude to a straight line drawn against the side of a cube!. A straight line drawn with a mark cut across it is not the same as a straight line not segmented, because the cut means that the line is in fact a rectilinear form!(book 2)

however, what Eudoxus taught was hat by reducing all forms to points and lines, certain methods that were common to all forms could be deduced and studied (compared, analogised. In doing this, it becomes vital that the kind of things compared is noted. thus if we analyse circumferences as straight line representations, then any results strictly apply to perimeters of circles and not necessarily generally to any lineal magnitude.

This does not stop us from comparing differnt kinds of magnitudes, and in fact this is what Napier did in formulating his Logos: Arithmos, or logarithms. The ratios in the sines are compared with the ratios in the arc lengths that correspond. Thus the logarithms are an analogue system. However, because they are of different kinds we cannot use the equal sign or the dual concept. What we use is the Analogue concept of Eudoxus.

The reiteration of these principles from book 7 onwards is to teach the student that what applies to lines can be generalised through using the 2 gnomons, the parallelogrammic gnomon and the curved Gnomon in a circle used in the proof of Thales theorem. The parallel lines are crucial in preserving or transforming a shape. The rotation is crucial in preserving any shape but changing its orientation, and the projection from a point is crucial in scaling and transforming a shape.

With these 3 projections in or onto a plane : the circular, the parallel, and the perspective plus the analogous thinking of Eudoxus the dynamic reality around the student could be apprehended, sudied and utilised to make wise judgement as to the appropriate Kairos. After all this is what an Astrologer is expected to be able to do, to judge times and seasons for the opportune time to engage in any action. Thus Kairos is the fullest extent of the astrologers art, ot for rhetoric, but for pragmatic and wise livng in a dynamic universe.
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« Reply #454 on: July 17, 2013, 09:06:31 AM »

http://my.opera.com/jehovajah/blog/2013/07/17/newtonian-fluid-motive-as-spacematter

The principles of Logos Analogos already begin to bring a great result!
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« Reply #455 on: July 26, 2013, 11:46:26 AM »

I am going to write something on Harmonium Mensurarum, based on the lineal magnitudes, but requiring a prior " mechanics" notion, that is a motion of certain fundamental elements in a law abiding way.

It turns out that the result of allnexhaustive analysis is the seemeioon, often called a point. Whatever term we may use to describe it, it is the result of an exhaustive analysis. Once recognised or perceived, it is usual to find it everywhere one looks.

However the relatin between  any arbitrary 2 points is fixed or not fixed that is to say we perceive a relativity between any 2 arbitrary points, and this relativity we express as relative fixity or relative motion.

Of the relative motions, the primary one , the fundamental is that motion that describes a spherical surface. This spherical surface made by a point that is both in relative fixity and relative motion to another point is of course a relative surface.

The mutuality of these 2 arbitrary points allows us to define 2 mutual relative surfaces and from these to define mutual points or Isos points.,these Isos or dual points in fact define a circle of intersection. By constructing all mutual spheres from these initial 2 points which by the way are ultimately relative to an observer, and relatively sequenced by the observer, we may define the circular plane of dual or mutual points.

Within that plane we may pick in sequence any 2 arbitrary points and construct the 2 mutual circles for them within the plane and thus the mutual points of intersection that define a good or straight line, that is also a bisect or of the straight line which may be constructed throughn the 2 arbitrary points, and which also forms the diameters and semicircles and Ortho sectors of a circle constructed from the point where these 2 good lines cross.

It is in this way that I will continue to construct the Newyonian, or rather Apollonian reference frame by which we may describe al manner of  curves and motions. However, crucial to this exercise is the notion of relative fixity. Thus , when I describe a circle or sphere of any radius from a centre I am allowed to fix that radial displacement by any instrument used to construct a circle or sphere. Alternatively I am allowed to pick any point in the surface or circle and fix my instrument on the initial centre inorder to draw the relatively mutual spherical surface or circle. This is fixity, and defines rigidity or solidity, the importance of the notion of fixity lies in the application to measurement, which is an application of fractal construction based on Logos Analogos reasoning.
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« Reply #456 on: July 27, 2013, 11:50:38 AM »

<a href="http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1</a>

I return to this repeatedly, because it is such a beautiful demonstration of the fundamental principles of circle dynamics. I still can't get how elegant it is into my brain. But this much is clear, the centre of a circle or sphere mimics the motion of the perimeter  rotationally and translationally. This makes the circular disc and the sphere the perfect metrons, the foundation for all fractal metrication, measurement or quantification.

It also means that laws of Spacematter are best described in terms of the spherical dynamics  of Trochoids.
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« Reply #457 on: July 27, 2013, 12:38:20 PM »

The centres of the circles travel around a perfect circle and so the perimeters must also travel around a perfect circle, and yet i "feel" that the circles deviate from this orderliness! This is the reult of my relative apprehension. Thus i feel a force where there is none!, other than that required to turn the circles on perfect circles.

The trochoidal path therefore accentuates a proprioception that a variable force is being applied directionally and braces me accordingly. Now were i actually on the point that traces the trochoidal path, my reaction would be appropriate,

By this i learn, that as an observer, my brain computes or processes relative to where i think i am, or where i am best served, Either it processes and makes complete sense, or it contradicts some other sense to make nonsense of the whole. this nonsense is akin to confusion, and that is why i find it difficult to apprehend this elegant demonstration of "chaos" arising perceptibly from orderliness!
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« Reply #458 on: August 02, 2013, 07:06:45 AM »

Some background research into the Logos Analogos framework, which shows how Eudoxus , and the Pythagorean School in general coped with the fractal nature of the set FS, using the set notFS as the ground of their sensory experience.

While this is a standard specification , the rest of Book 5 and book 6 show how it may be applied to a whole range of nonstandard situations with a little imagination.

The kinds of magnitudes that it covers, actually define those it does not and helps to find a kind to which they do belong which then may be compared by some form of the methods!

http://jehovajah.wordpress.com/2013/08/02/eudoxus-on-logos/
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« Reply #459 on: August 16, 2013, 09:14:27 AM »

Some earlier insights relate to this post and can be found at
http://www.fractalforums.com/mathematics/foundations-of-mathematics-axioms-notions-and-the-universal-set-fs-as-a-model/msg25761/
Or on that page and those following.

Today's post is
http://my.opera.com/jehovajah/blog/2013/08/15/eulers-i

Which I reproduce below.
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« Reply #460 on: August 16, 2013, 09:18:00 AM »

The eye of Euler was a unique one. It is true that he lost the use of his orbs around the age of 60, but he continued to produce mathematical researches  until his death. He was able to continue because his expanded sphere of consciousness made all things revolve around a common central insight: the Sphere.

Before him sir Roger Cotes, under the tutelage, correspondence and general influence of Newton arrived at the same harmonious conclusions about all measurements. The sphere was the standard!

This may come as a surprise to many mathematicians and physicists, who being badly tutored and oftentimes misdirected in their studies, are unaware of the profound underpinning Pythagorean scholarship of their crafts. Indeed, so dark is the light of their teachers that they fein to scoff at what they know nothing or very little of!

Nevertheless, greater minds having toiled all their lives in study and application like Euler, have confirmed the soundness and veracity of the Pythagorean scholarship. In so doing, they condemn the error introduced through Aristotle, who not finishing his studies at Plato's Academy ventured on his own opinion and by his own reasoning to establish a complete categorisation of knowledge by which he made the sphere into a cube, losing, if it were possible the very nature of curvature by measurement!

At Plato's Academy, Eudoxus was usefully employed in bringing Pythagorean credentials of Mathmatikos to all who were enamoured of the adventure. One of these was Euclid of Alexander, who returning to his home city founded a Platonic Academy in Egypt, which promulgated the Pythagorean scholarship. As did the Athen's Academy. They were rivaled by the Aristotelian Lyceum, which having imperial backing flourished for a hole, but foundered with the fortunes of the Alexandrian line , eventually being scooped up by Arabian patronage.

The Athenian platonic school and others suffered fates as terminal as the Lyceum, but the underlying Pythagorean model revived and revives. Itself continually in many guises. Th body of Knowledge survives today because of its boldness to follow the communal brotherhood model as opposed to the Mithraistic, Egyptian mystery religious models.

 Such models were much in favour by Gnostic, Zoroastrian and many other Cultic assemblies, who obscured their financial and greed motive by a promise of secret wisdom and special access to divinity. Much of what they taught is found in the religions of today. But contrast that with the empirical and pragmatic philosophy of the Pythagorean schools, who studying the stars planets and all around them declared all things divine and worthy of respect, study and modest temperance, balance and moderation.

The talk of secret rites and inner sanctums etc reflect more the Cultic behaviours of those around them than that of the Pythagoreans, who from the outset were a public teaching order, given special patronage to continue their researches in return for public dissemination of their findings. This of course attracted great hostility from rival groups who did not have this patronage or social standing, and frequently resulted in persecutions, infiltrations , political agitations , agent provocateurs. The response was to form a tight knit community who shared the Ideals of Pythagoras and who thus established security models which were claimed to be rites of passage by detractors.

That giving their knowledge a public airing was ultimately the correct model, is born out by how influential the Pythagorean school of thought is in the public domain. This did not prevent secret cults or conspiracies from going about their business, but it did make the public aware that this was how the world worked.

The Pythagorean school was a model for monastic life in the west, but it derives it's style of teaching, koans and self knowledge from a tradition that goes back to the Akkadian and Sumerian Magi, as well as the Egyptian scribal and priestly schools.

The Pythagoreans  used a fundamental pattern on the floor and walls of their" temples" to the Musai. These came to be known as mosaics. But these mosaics, often abstract but not always, we're a pragmatic description of what may be known and how it may be known. This is the topic of Epistemology in philosophy. The fundamental point is that we have to record everything , by every means and on every surface. The more permanent the recording medium the longer that record will remain.

But this record is useless without trained interpreters and scholars. So the complete package is a school of scholars nd trainers whose task it is to record phenomenon, and to maintain and interpret the same. The role is to tend to and grow a vast library and educated elite who could pragmatically, both academically , poetically and Artisanly make use of this growing wisdom for the benefit of the whole of society, not just a select few , who could pay through the nose.

This kind of Utopian ideal has a long history, but is never better expressed than in the mythical life of Pythagoras, and the literary copy redacted into the Jesus myths.

The mosaics are fundamental to the notion of logos and Analogos, on which all Metron theory and thus measurement is based, and from which the notion of a Fractal is mot expertly drawn by Mandelbrot.

Because Mosaics are in the early days mostly abstract patterns, the grammai that form these patterns and the seemeia are ultimately of fundamental significance. In a mosaic, as opposed to a wall painting which has an accurate representational feel, the discrete combination of elements is emphasised, and then overlooked to gather the emergent information.

The artistic movement of pointillism expresses this fundamental analysis of vision in the late 19 th century. However the Pythagoreans went beyond visual representation. They researched all senses and concluded hat all knowledge is constructible in the way a mosic is constructed.

Pythagoras is reputedly given an insight into the allure of "Musai" inspired sounds or Music. Again the mosaic pattern of strings reveals itself in the Logos on strings and the Analogos between strings on an instrument? String theorists today are pursuing this kind of mosaic description of reality, unaware that it has already been done by Eudoxus!

The sphere was absolutely foundational to the Pythagoreans, and thus to Plato. But what was not adduced by scholars of the renaissance was how completely it was studied nd understood. The Arrogant Renaissance Movement took the old knowledges and first promulgated them, and then claimed to advance them! A more sober assessment is that certain Renaissance men made their fortunes by making uch claims hike burying their predecessors works which often were written in Greek or Latin, nd accessible to the few learned men and women  of an Aristocratic family.

Ah, but it was ever thus!

The sphere and it's counterpart the disk were well studied and laid out in mosaics on the floor of the Academies of Plato and the Moussaion or temples or Monastic dwellings of the Pythagoreans.

The one Metron that was used for the circular disc was the diameter. Among the many logoi represented in the Mosaics was the logos for the semi circular perimeter to the diameter. This was pragmatically laid out by turning a wooden wheel or disc or shield on a lathe. Then by carefully halving it, and checking that both Semi circle were exact copies of each other by placing one on top of the other.

Then the semicircle was copied into the mosaic by a careful artisan. From this much was learned about the nature of the circle, it's centre , it radii etc. but the diameter was a fundamental, good or right line! Today we say straight, but when analysing the nicest concepts we must be careful not to put our conceptions onto them.

This fundamental good line was investigated. In fact, how "good" it was was defined and redefined by many different constructions over time. Euclid mentions that it is defined in his time by dual seemeia . That means a compass like instrument is used to determine dual points from 2 arbitrary initial centres. The diameter arises from where thes dual points transecting the circle perimeter into x2 equal circumferences or arcs.

Today we even define the circumference incorrectly!

Next by carefully rolling the semi circular dist in a straight line a segment is marked of equal to the circumference of the semicircle, that is the length of half the perimeter. It is an arc length, but more precisely sn arc magnitude. The reason why circumference is defined is to distinguish an arc magnitude from a diameter magnitude!

They needed to be distinguished as different magnitudes because they were both represented by a lineal magnitude!

The ratio or logos of a semi circular circumference to its diameter was defined by Eulet. It is defined as i

This is the same ratio as the quarter arc to the radius, this too is i

The ratio of the whole perimeter to the diameter is defined as Pi. This was also given by Euler. Finally the ratio of the altitude of a point to the radius Was defined as the sine, but the ratio of the change in arc length to that radius was 1 : 1 -  1/x where x was very large. This was the basis of the Napieran logarithms. These were logarithms of the then extant and detailed sine tables, this ratio being ver close to sin (Pi/2 - I/x).

Both Cotes and Ruler recognised in Napiers method the ratio or Logos 1: 1 + 1/x where x was very large, was a significant value for the circle. This tended to the limit that Euler denoted by e. both of these were expressed in terms of the binomial theorem to obtain these results, and thus the basis of ny logarithm is the limit to which such a ratio tends as x grows very large under a binomial expansion of the form
( 1 - 1/ x)^x or (1 + 1/x)^x.

It is therefore simple to see that
Pi = 2*i

What is obscure is why  i is defined as _/ -1

Euler defined it as the Sqrt of -1 because no one else before him had bothered to define the negative square area. Although Bombelli had utilised the definition, no one understood that a line or rectilineal form above the diameter had to be given a different sign to one below it . Bombelli did, but did not apparently, to my current knowledge specifically identify squares or quadrature in this way. However, he gave the exact rules for using these types of magnitudes, which he apparently claims was a daring gamble on his intuition. I believe Bombelli came to the same conclusion I did after meditation, that if you are going to define negative magnitudes for accounting then you must also define negative magnitudes for geometrical accounting.

Euler specifically defines the quadrature of the unit circle for positive and negative magnitudes of squares. What his contemporaries failed to pick up on was that the Sqrt of -1 is -1 AND +1 while the sqrt of +1 is -1 OR +1.

Both Euler and Bombelli used the definition of quadrature in Stoikeioon book 3 to define the new fanged negative quantities. Nobody else really cared enough to tie down this detail, and so confusion reigned for nearly 5 centuries!, resulting in the hybrid we call number thoughtlessly or at best poetically.
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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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« Reply #461 on: October 07, 2013, 09:00:16 AM »

Crap!

Its not often i sit down to write tat word about Mathematics, but the more i research into it the more i realise we have inherited a crap version of Astrology, a crap version of Arithmetic and a crap version of Phusis or Physics. In addition, a whole branch of mathematics called Algebra could be freely translated from the Arabic as Tortuous  crap!

Pretty strong ,huh?

Well Newton did not think so. He refused to be drawn into Wallis's grand scheme for an Algebra, preferring to name what he did Analysis( De Analysae), and in fact he went as far as to distinguish "number" as some analytical  model concept, which was useful to demonstrate consistency but not useful to describe dynamical systems, mechanical an extensive magnitudes.

His Principia is a landmark work in the field of Naturak Philoaophy, in which he sets out some Astrological Principles, principles for Astrologers! He knew Mathematikos to be a qualification in the Pythagorean Schools equivalent to a Doctor of Philosophy and a Fellow of the Pythagorean Master scholars! And the work of the Pythagoreans was an extraordinary devotion to Natural Philosophy of the universe around them, using the Astrological tools For measurement.

The tools for measurement are both subjective and objective, and require record keeping. The simplest tool was the artists hand or fingers , and painting tools and media. Thus we find star maps painted on walls ceilings and in caves going back into the mists of time . We also find carvings. The nature of these images and sculptures is to draw a picture of the heavens in some familiar animal form!

Some pictures however capture more terrifying things that occurred in the heavens, and theese we are only just deciphering because of Plasma Physics!

The next great tool was a piece of string with a heavy weight. This line used as a plumb was essential for sighting, and the analogue to it was a pole fixed in a plumb position. The light and shadows cast by these arangements of poles naturally led to marks of positions on the ground. These marks were the position at which shadow poles  cast their tips. plumb lines in their framescast their patterns, and from which sightings could be taken.

These marks were incorporated and inspirational of anoter art form: the Mosaic.. The Mosaic is the most fundamental development in Astrology. It turned astrology into its cmmensurate Geometry. It made astrologers make records on the ground, configure and meditate on the ground and ratio and rationalise on the Ground. This was the essence of Geometry, Also mistranslated as Gematria, but this mistranslation carrying the core significance of Geomeereesei, Katameetreesei.

Apart from epipedos and speripedos, there is no word for Mosaic besides the Pythagorean concept Arithmos.

The method of the Pythagorean scholars was Analysis and Synthesis. The Euclidean Stoikeia represnt the Synthesis of their prior analytical researches, and it is still only an introductory course to Pythagorean/Platonic/Socratic Philosophy of Eideion!  That is Ideas / Forms..

The courses that followed this course or were discoursed alongside this course were more subtle, complex and demanding of rational, that is ratio compared with ratio, or in the greek Logos Analogos, THINKING. A deep meditation was involved, and the focus of that mantemata was the artiststs paintings, drawings, mosaics, sketches, skesis, schemata and symbolic use of these natural forms of expression and copying.

Without the fundamantal grid pattern of a mosaic, nothing could be objectively measured, no consensus could be reached in these discourses, no invariant formulae observed.

Thus the understanding of the arithmos, the analysis of it is what reduces to what are called grammai  and seemeia, extensions and indicators, or equivalently Strecken und Punkt, or in englis segmented lines and reference points .

this was the highest astrological art, the Gematria of the Pythagoreans, the shunyasutras of the Indians who were influenced by Hellenistic concepts. This was turned into "gibberish" by the arabic scholars!

Strictly speaking, they were referring to the tortuous tautological rationg and comparison that was involved in the texts they had to hand. Al Jibr refers to this twisting and turning almost as if in mental agony. In fact, once you got used to it it is quite addictive! But to about 96% of the general educated public it was and is mental torture and extreme cruelty!

The problem with algebra is it allows crap to creep in with only a handful of people able to challenge that utter nonsense! and that is what has happened , and why i started my blog with those immortal letters!

Google "jehovajah Grassmann Wildberger " for more details.
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« Reply #462 on: October 20, 2013, 11:36:24 AM »

Well, if you can't believe I am still at it, neither can I!

http://my.opera.com/jehovajah/blog/

At least I have bern able to put Sacred Gometry in ome perspective.. The mathematics of Eectomagnetism, Fluid Mechanics and Euler circles is still on my mind.

Quaternions,Clifgord Algebras, so called number systems are revealing themselves to be fomal geometries or rather spaciometries. Strecken at last have a distinct meaning that is more immediate than vectors, and the whole game of Mathematics is revealed as a lost orphan needing a tick but nurturing mother like computer science to curb its wayward ways!

Computer science is now the Queen on the block! Processing chips can do so much more than jut run formulas, thru can help in formulating, is playing and interacting dynamically with models of our space.
As a mathematician ( whatever that may be nowadays) why would you want to be ith out our computert programming buddies?
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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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« Reply #463 on: October 24, 2013, 03:34:57 AM »

When I was 12 I decided to devote myself to mathematics! I did not even know what it was.

I can remember concluding that mathematics was some foreign word, probably Italian, meaning methods, because all I seemed to be learning was one method after another: one trick followed by another quick method to be memorised and rrpeated back parrot fashion.

It was not until 2009 , after I had been to uni, after I had taught math for 10 years, after I had given up on it, that I finally looked up the words etymology.


It was not Italian it was Greek. It was associated with Thinking scientifically.The PIE root suggested handling as well as thinking as a human , using hands to explore.

About 2 years after that I found it was a Pythagorean qualification! A bit like a doctorate in Philosophy in the topic of Astrology.

About a year after that I found out that the Pythagoreans built mosaic designs in their monastic temples.in their temples they observed the stars and planets and sought the inspiration of the Muses!

Thus it turned out that those professing to be Mathematikos were in fact professional Astrologers. Their lifetime devotion was meant to be to the Musai, from whom they sought wisdom, inspiration and divine knowledge. The mosaics were the fundamental tools of their divination, and recorded in their floor patterns the secrets of the Stars and Planets, and the Harmony of the spheres.

In these places of contemplation, music, poetry and dance recorded inspirations and revived their souls .

They called these mosaic floor patterns Arithmos, and there were many Arithmoi. Without them there wasno geometry , no counting , no music. Such was the saying of the Pythagoreans.
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« Reply #464 on: October 24, 2013, 01:03:58 PM »

When I first started this topic, it was after years of Autistic bewilderment! Like all who have pronounced autistic traits I wondered at the difference between my experience of reality and the experience I was told I should be having. If I was not told it directly, then it was communicated indirectly in the Language of my enculturation .

It is this combination of environment and intrusive language programming that helped and hindered the formation of my own self identity in time.

There was much to learn, and I had an appetite for learning. But in the early stages I was very uncritical of what I learned. Learning itself was a powerful drug dependency.

At long last, the vicissitudes of life took their toll on my naive approach. It was not that I felt naive, which I eventually did, it was more that what I had learned did not match the content of what I had experienced for myself. My vast internal landscape of learning and knowing was useless in practical life .

Fortunately my parents taught me practical skills which served to keep me mentally intact, until I went to university. That was when I should have completely gone to pieces and had a nervous breakdown! However at the time I had developed an intense interest in religious matters which bought along with it a faith, a praxis, a theosophy and also a more general philosophy, a support network and a diversion from abstract learning to learning practical social skills in support of the religious requirement to socialise!

However, I did manage to squeeze in a bit of language learning in Greek, Hebrew Aramaic, Latin , which my school French, and Germn lessons had prepared me for somewhat.

This was a period of intense religious self examination, tempered with searches into the latest psychological and neurological research. It was a time of reevaluation and learning to socialise and communicate.

By a certain time I was ejected from the fellowship I had been nurtured in, because I was too independent in thought. This was traumatic, but trauma I was used too as part of the religious training I received. Religious people are taught to expect, deal with and cope with trials said to originate in gods will.

It was during this time that I embarked on a self evaluation of what I was and what I believed, and how I justified it. It was then that my mathematical training was put to use in setting up axiomatic systems to explore outcomes. This is when I gradually realised the venn diagram concept of the set FS and it's complement NotFS.

It was years later in 2009 tht I actually started to write these axioms out. The intention was to revise the whole of mathematics which had become an irritation to me, as others seemed to delight in descending it ino pitch black obscurity!

I set out hopefully. I had a view that if we got the sequence of the theoretical dependencies right we would get the whole subject on a more congruent footing. I knew I could not prove consistency in the subject, but at least I hoped I could restore clarity.

My initial concept was that numbers start with addition, and repeated addition gives you multiplication. I felt that algrbra was to deal with variable measurements, but still numbers, and geometry was the application of this measuring scheme to construct things. The concept of so called axiomatic based logical proof was a mess that needed clarification. Who needed it anyway?

I also studied physics and chemistry and saw the direct application of mathematics, and the indirect application of proportions. Nevertheless the link between mathematics and the other subjects was obscure to me. I viewed mathematics as entire to itself. It gave me great delight for a while. But as I got older and wiser I valued its abstract numbers less.

At university I managed to get involved with some of the early mainframe computers. The language of Algol60 did give me a nervous breakdown, on top of everything else. But by the second year I was healed and enured to the mental strain it engendered, principally because it was more autistic than I was!

I learned then the power of programming codes. I later delved into the electronic signalling of various gates and the Boolean truth logic used to get a handle on what flowing circuitry was expected to do. I recognised it in purely mathematical calculus terms. I was therefore amazed when Sinclair brought out the ZX Spectrum and it could animate little objects on a TV screen. I learned BBC basic, then eventually Qbasic. I went on to take some introductory courses in c, c++, object oriented programming in c, HTML and java. And I learned a bit of visual basic from the computer at work to do some programming of spreadsheetsm databases and word document pages.

I knew enough to realise that mathematics had been supersceded by computer science and programming in particular!

The set FS was in response to reacquainting with Mandelbrots Fractal Geometry. At my first reading I was bamboozled. I had very little computing experience, no computer graphics experience and no access to a computer even an Atari, which I later found out my dad had purchased out of interest, but not told me.

By the time I was reintroduced to Fractal Geometry it was almost a paradigm shift in mathematical thinking in regard to chaos theory and dynamical systems.

It seemed to me immediately that all my internal knowledge and that written in books formed a set I called Fractl space or FS for short. I was unclear about the boundary for FS and to avoid the Russellian Paradoxes I decided to leave it undefined. But I could bound it by NotFS which I did.

With this vague set of ideas I decided to build an axiomatic system that would iterate to a better approximation of reality as I went along.

This thread was the outcome of that effort, with my allied blog site.

Over the years a few have interacted with me but none have sought to collaborate. Perhaps it was fate, but in any case I have not been able to put the subject own since then, as you perhaps have gathered.

Today I am heavily into Newton Euclid, and Grassmann. I have had to iterate beyond my initial concepts as I learned their historical roles and introductions. But in Newton, Euclid and Grassmann I appear to have tapped into a living vein of ideas that have flowed between them.

Ausdehnungslehre is translated as Extensive Theory, but it is better understood as Dynamicl Measure Theory, and probably evn closer Fluid Dynmical Measure theory. I read little snippets of Grassmanns thinking and I am launched into incredible extensive thoughts about space! In that regard his work is a method of analysis and synthesis that organises the set FS and extends its boundaries into not FS in many surprisingly fundamental ways.

Do not prevent yourself from reading his 1844 classic, nor Newton's Works, nor Euclids. The more you can read them in the original language the more you will be rewarded. Hopefully you will learn how to gradually break free from all self imposed limitations and develop a richer experience of reality.
« Last Edit: November 09, 2013, 01:53:12 AM 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!
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