Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

Plato looked at science (Mathematikos) which was concerned with spaciometry, the measurement of dynamic space, and which hoped to apprehend the construction of the universe by the gods, if they existed.and whether they cared or not if they existed. Men(thinkers and manipulators and measurers) existed      anthropoi they believed to the Theoi (dharma, Dyaus sanskrit) that lived above mount Olympus, in the visible theos which is the ouranos, the heavens, and greek men dared to challenge the Theoi the children of Dyaus/Zeus if they could be bothered to fight! One day greek men would sit among the Theoi, or pull them down trying!

Plato looked upon all this in mathematikos (science) and contrasted it with all other forms of knowledge (epistemologia) and declared to Lovers of Sophia, Philosophers that the greatest wisdom, the gift of Sophia was in Science, and that all epistomologia (knowledges) also gifts of Sophia should get off its backside and organise itself like science. He defended that opinion for the rest of his life .

Symmetry is involved in the yoke of BrahamaGupta but along with symmetry comes that which is not symmetry, and these stand side by side as a decomposition of spaciometry. It is this decomposition that describes or is described by the fundamental group structure of the void.

As a representation of this i posit the relationship between the sphere and the spherical vortices that constitute the vortex torus form, or as found in oceanic forms the shell vortices. These shell vortices have always been valued as precious objects and beads, but of course there is a time of human exploitation using these precious items,to further imperial aim!

The relationships between spherical space and spherical vorticular space is the relationship between equilibrium, and dynamic equilibrium and runaway disequilibrium leading to boom or bust, bang or crunch!

Brahmagupta advises on symmetrical yokes, but also on non symmetrical ones: n/0 and 0/n.

What 0/0 advises beyond all voids diffuse/evaporate into 1 void( or rather a supervoid which generates voids- in other words a nested fractal)  i have yet to discover.

[jehovajah]

If I was to say that shunaya meant dynamic space, what would you think? Nothingness? Hardly! You may think "wind", and indeed the Arab sifr does connote that. From this through various ethnological shenanigans we end up with zero. Confusion or what?

Now I mentioned dharma in conection with Zeus and I know this is a misconception, but it resonates so aim leaving it. I could not correctly recall the Sanskrit devah at the time.

Dharma is after all "the rules of perceiving" and that is very relevant to any science.

[jehovajah]

I have just seen a new banner logo : infinity is visible. Of course n/0 says that in a more cryptic way. So I guess 0/n means that infinity can be portioned and proportionate, where n/0 means visibility is due to disproportionate proportions,ie symmetry breaking. Finally 0/0 must mean "the possibilities are... Infinite"?

[jehovajah]

We sense, therefore we exist.
We perceive therefore our existence is distinguishable.
We distinguish sensors, therefore there is something to sense.
Because we sense and perceive and distinguish through and with sensors then an external internal sensory decomposition exists.
 The decomposition of sensory data into internal and external components will be distinguished as space, therefore space exists, and has attributable properties, the most fundamental being internal and external to us.

[jehovajah]

"  Origin and development of the concept of emptiness( wikipedia)

The theme of emptiness (śūnyatā) emerged from the Buddhist doctrines of the nonexistence of the self (Pāli: anatta, Sanskrit: anātman) and dependent arising (Pāli: paticcasamuppada, Sanskrit: pratītyasamutpāda). The Suñña Sutta,[6] part of the Pāli canon, relates that the monk Ānanda, Buddha's attendant asked,

 "It is said that the world is empty, the world is empty, lord. In what respect is it said that the world is empty?" The Buddha replied, "Insofar as it is empty of a self or of anything pertaining to a self: Thus it is said, Ānanda, that the world is empty."

He goes on to explain that what is meant by "the world" is the six sense media and their objects, and elsewhere says that to theorize about something beyond this realm of experience would put one to grief."

Now suppose that the word empty is translated "swollen" or "replete".

Bhudda therefore advises one to go beyond the senses to comprehend the world, In other words be wholly empirical, and therefore dynamical, livng and experiencing each fleeting moment, from moment to moment, And any conclusions you draw from that experience must fail and turn to dust with you,for they are as dynamic and fleeting as you are. Shunaya therefore has the curious attribute of being self reflexive, making sense in all its interpretations.

Brahama gupta is said to have drawn a perpetual motion machine, a wheel based design, therefore a yoke of symmetry, BG thereby advises not perpetual motion of the wheel but perpetual moion of the void, a dynamic void, for it is not a wheel but a symbol of the void he drew. `in this way BG extends his operator beyond scalar representation in arithmetic to geometrical representation in spaciometry. No doubt he applies it to the sphere.

The perpetual rotation of the yoked decompositions we are more familiar with in the sin and cosine version and the Euler Cotes formula

 where represent the constant rotational dynamism.

This of course would not effect the perfect circle or sphere as the addition of its elements are zero!

As in the decomposition into + and minus there is a decomposition into clockwise and anticlockwise, and these are the effective constituent elements of a rotational machine. These i think would necessarily be spiral, and components would be trochoids of spirals. Both cannot exist in he same machine if it is to work, and therefore work has to be done to establish this state of affairs. Therefore we cannot get more out of a machine than we put in, unless we break the symmetry yoke in the system. In oing so the system will explode.

Newton observed, but few have fully understood that all forces we investigate  are equilibrium forces, and arise only in respect to matters of equilibrium. However there are disequilibrium states, but whether they are nested in a larger equilibrium system etc is only a guess at this momemt.

[jehovajah]

The decomposition of the void is the natural subject of the philosophy of the I ching. For this subject chinese philosophers have devised maps or flow chart like diagrams called Taijitu.

Lai Zhida it is said introduced the taijitu into the yiology of the I ching in 15th century. In doing so he was introducing a "modern" scientific revision of the Yi Ching. 5 centuries earlier Zhou Dunyi introduced a version called  the Taijitu Shuo which applied it more generally,and particularly to human activity in all its forms.

Zhu XI particularly established this philosophical discipline.

According to the chinese Fu Xi is the originator of the Yi Ching, and we see here the spoked circle, the twisted vortex and antivortex and the cone out of which Nuwa and Fu Xi emerge, holding symbols of the heavenly powers.

The development of the Yi Ching continues through philosophers and astrologers to apply a divination method to more and more contexts.

Although dense and rhetorically complex the Yi Ching is basically simple. 64 Hexagrams have over time been established as the reference frame. This reference frame is a human measuring instrument of possiblities from the void, and as such represent a decomposition of the void into these 64 distinct groupings.

Thus the void is full of all potential but only 64 are studied by the Yi Ching.

The fundamental decomposition of the void in China  has always been accepted as yoked opposites, in this case Yin and Yang. It is my consideration that this decomposition influence BrahmaGupta, and in fact the Chinese Yi Ching was a curiosity of his in the 6th century AD around the time of chinese indian cultural exchange.

Using this fundamental yoked pair as a guide to each line of the hexagram, some activity is used to "draw" decompositions from the void, to Realise a supepositional state! However, it was thought that one realisation was not sufficient only giving one aspect of the superpositional state, so over time 6 lines were developed, and thus called the hexagram. As stated 64 hexagrams were developed as sufficient and necessary.

For each of the 6 lines in a hexagram the yin yang decomposition would have either a static or dynamic resolution! so the yoked pairs were recognised as being equilibria that are static and/or dynamic, and decomposed into anticlockwise and clockwise dynamic equilibria and original and reflected static equilibria: thus 4 states.

In terms of the Yi Ching this gives 64*6*4 =1536 different state readings and 16*6*64 exegetical interpretations for a pair of readings and 4^3*6 *64 for any triple of readings etc..

This was felt sufficiently complex for human affairs!

Zhou xi and Zhou Dunyi developed aspects of this traditon to give a sufficiently robust description of probable outcomes to Government decisons, so in that sense it was used as a kind of game theory.

The Yi ching has clear application to quantum chromo dynamics and other quantum descriptions.

In regard to Brahmagupta, i see that considering these things lead to his advice on decomposing the void, and brought from chinese influence the notion of shunaya into indian philosophy in 6th century, through a long cultural contact starting in 4th century?.

[jehovajah]

Of magic squares, magic circles, the Yi Ching and the Yin Yang as decompositions of the void into yoked unities, the concept of which has given birth to Taijitu of the fundamental process of the universe.

Of how these elements have been algorithmically combined by the Yi ching and BrahmaGupta into the basis of Science from Quantun Chromodynamics to the vast reaches of the known universe expounded in Relativity theory.

TBC.

[jehovajah]

Of cradles of civilisation and the mutual development of asian science and near eastern science and the western offshoot through greece. A cultural twinship against the aryan invasian theory and for the out of india theory

Civilisation and human presence not the same.

The pragmatism of Confucianism created a stable void in personal thought which the Buddha and Buddhism filled appropriately. The taijitu expressed this fusion of inner striving and outer order and harmony. Through the bridge of Buddhism the chinese had access to Hinduism and it s influence on Indian thought and practice. in the fusion with buddhism.

"Samkhya
Main article: Samkhya

Samkhya is the oldest of the orthodox philosophical systems in Hinduism. Samkhya is a strongly dualistic philosophy that postulates everything in reality stems from purusha (Sanskrit: पुरुष, self, atma or soul) and prakriti (matter, creative agency or energy). There are many living souls (Jeevatmas) and they possess consciousness. Prakriti consists of three dispositions known as qualities (gunas): activity (rajas), inactivity (tamas) and steadiness (sattva) which arises when the two other gunas are held in equilibrium. Because of the intertwined relationship between the soul and these dispositions, an imbalance in disposition causes the world to evolve. Liberation of the soul happens when it realizes that it is above and beyond these three dispositions. Samkhya denies the existence of God.[2] Western dualism deals with the distinction between the mind and the body,[3] whereas in Samkhya it is between the soul and matter.[4] The concept of the atma (soul) is different from the concept of the mind. Soul is absolute reality that is all-pervasive, eternal, indivisible, attributeless, pure consciousness. It is non-matter and is beyond intellect. Originally, Samkhya was not theistic, but in confluence with Yoga it developed a theistic variant.  "

The gunas correspond to motions and distinguish 3 sorts of motion: acceleration(rajas), static equilibrium(tamas), dynamic equilibrium(sattva). Though dualist the chinese notion of yin and yang (yoked pairs) is not present in indian's oldest philosophy as cited here, therefore i venture that yoked pairs is a unique to chinese philosophical idea that influenced later Indian thinking in the person of Brahmagupta.

[jehovajah]

Of chinese science/astrology and quantum chromo dynamics.


Hinduism is like saying Britishism or Americanism: it refers to a geopolitical regional entity not a faith. Buddhism again refers to a belief system based on "buddha's" rules of perception, and as such is a container of any cultural ideas of those who apply the rules of perception, the Dharma. Indian philosophical thoughts in all there varieties were thus brought into contact with chinese philosophical pragmatism or Confucianism.

Confucianism again is a carefully constructed container like a "dharma" for behaviour not perception, thus Buddhism and confucianism actually complement and attract each other.

Through this connective tube of buddhism and confucianism indian philosophies flowed into china and chinese pragmatic philosophies flowed into india. Thus Brahmagupta came to examine chinese science and astronomy and the Yi Ching. AT about the same time Lai Zhide examined indian science astronomy and astrology. Lai Zhide revised the Yi ching to put it on a more astronomical, astrological basis, that is a more scientific basis and redesigned the Taijitu accordingly. Zhou Dunyi revered the traditional taijitu in his later combination of the buddhist, confucianist taoist  streams into neo confucianism.

Brama gupta in studying idea from the Yi Ching was lead to ad to the 3 motions of Shunaya the notion of the yin and the yang, a balance that was yoked and could perform the 3 motions in a yoked form.

Fro this Brahmagupta derived the rules of yoked motion. These rules are and always have been a meditation and a sudhanta,

From india through the arab empire it spread throughout the world and impacted on everything. Through it the world has derived the so called "complex numbers", but which Bombelli called "adjugate" to numbers, that is yoked, and he recognised the pairs that he called "conjugate": the yoked pairs.

Through the Bombelli operator the rest is history because we end up at Quantum Chromo Dynamics through a wonderful and curious route.

However as i have pointed out there never really was a problem with the square root of -1, rather it was a problem with misconceptions of Brahmagupta's sudhanta.

For me the √-1 is simply +AND- 1.

[jehovajah]

Shunaya means the void which i instinctively know is magnitude. However magnitude is also instinctively a measure but what measure?

I have to explore the concept of measuring, the activity i engage in voluntarily and involuntarily and identifiably as measuring. To do this i have to start with an axiomatic model of measuring.

[jehovajah]

Indian concept of oneness.

The Indian concept of shunaya has a deep meditative role in mathematics for indians , but the concept of 1 Eka  has a transitional role, that is it transitions from the individual to the whole as a universe,

Thus the concept of unity as a meditative  search into the meaning of the universe is entirely greek!  So the indians studied shunaya, the greeks unity monos.

The earlier greek philosophers meditated on the world, the universe and came up with "unit" as the basis , The indians came up with shunaya

Man is the measure of all things is 0/n.

"NUMBER VS. MAGNITUDE
In the mathematics of early Greece, there was a strong distinction between discrete and continuous measurement.
Number refers to a discrete collection of atom-like units.
Magnitude refers to something that is continuous and that can be infinitely subdivided.
Rational numbers can be expressed as decimals that repeat to infinity.

Early Greek mathematicians divided mathematics into the study of number, or multitude, and the study of geometry, or magnitude. The multitude concept presented numbers as collections of discrete units, rather like indivisible atoms. Magnitudes, on the other hand, are continuous and infinitely divisible. Because length is a magnitude, a line segment can be divided as many times as one likes. The Pythagoreans believed that magnitudes could always be measured using whole numbers, which would imply that lengths are not infinitely divisible. Other schools, such as the followers of Parmenides, known as the Eleatics, believed in the infinite divisibility of magnitudes.

Parmenides taught that true "being" is unity, static, and unchangeable. This is similar to the idea that "all is one," which implies that concepts such as multiplicity and motion are illusions. If everything is part of the same thing, then there are no "multiple" things and, consequently, no motion, which is the change in position of one thing relative to another. Pythagoreans believed in multitude and motion perhaps because these concepts are intuitive, part of collective common experience. A consequence of the Pythagorean notion of multiplicity is that magnitudes should be commensurable. To the Pythagoreans, the idea that between any two quantities in nature there exists a common unit of measure, a common denominator, may have been comforting. It perhaps suggested that the rational mind can always find a solid basis for comparison, and does not have to rely on guesswork to say definite things about reality.

It would be easy to dismiss the Eleatic view, if it were not for the arguments of one of Parmenides' most famous pupils, Zeno. As we shall see, Zeno argued against the Pythagorean notions of multiplicity and motion, using infinity to show contradictions in this view. Prior to Zeno, however, problems with the Pythagorean viewpoint arose from within their own ranks in the form of an independent thinker by the name of Hipassus of Metapontum. Hipassus showed that magnitudes are not always commensurable, an idea that upset his peers to such a degree that, as the legend goes, he was drowned for his heresy. In the next section, we shall examine the idea and consequences of incommensurable magnitudes.:"

a lexicon

mona- and mono- unit/single. page 331 and page 613. Entos is more unity by inclusion.But i have at last found the common counting namers in greek and en is "one" even if it is written α (alpha). En of course also means in, within, so the idea of wholeness being inclusive is apparent.

Before i leave this i point out that greek thought was based on units, that could be established at any scale, so the pythagorean notion of being able to define a unit that would measure any magnitude is intuitively correct. So what Hipassus and Theodorus showed is that irrespective of what unity you choose there will always be magnitudes that cannot be measured. Both i would say suffered the same alleged fate. Eudoxus covered over the crack by positing a theory of proportion, which satisfied all that proportioning and reasoning could always be done and so it did not matter as these magnitudes themselves could be used as unities. Therefore from the time of eudoxus it has always been known that arithmoi are scalars and that all proportions and ratios are essentially holding information about scale, All manipulation is manipulation of scales and rules found true for scalarswould universally apply. Thus there was no need to actually see an atom as the scaled up versions exemplify their behaviour.

Atoms accordingly would have different sizes and these sizes would form fundanental unities like the proto arithmoi. What these atomic magnitudes were was the focus of their continued research, to no avail. until the unity scalars could be arrange conveniently and themselves ino a scale of scalars, the continuous fractions in use t the time was tedious to the greek mind, however it was beloved by the indians like a rhythmical song.

[jehovajah]

Of Cantor and shunaya.

Cantor in his exploration of sets came across the infinite sets, and on exploring them came face to face with shunaya. It so unsettled him that he thought he had sinned against god and was going mad! after recovering he left us with the notion that there are different kinds of infinity. This is what Brahmagupta meant when he wrote n/0 the countable fractions of infinty!

Cantor through modern number concepts added the notion of uncountable infinity, but NOT different magnitudes of infinity. There is only one "magnitude" to "infinity" or shunaya and that is shunaya.

BG used an integer n when he expressed countable fractions of infinity. If we use a real number r then r/0 represents uncountable fractions of infinity. These scalars are intuitively (mod 0) and form a (mod 0) clock arithmetic.

[jehovajah]

OF gravity and electromagnetohydrodynamics and the nuclear forces and yoked pairs.

I found an interesting child's play centre from Holland which is just right for the budding nuclear physiscist.

Equilibrium is the source of all force vectors, and yoked pairs describe the first level of euilibrium action. I suspect there are prime ntuples of yokes!

[jehovajah]

For a while i will think about my response to shunaya.

Instinctively i sense its magnitude (the Logos Response) its omnipresence (internal to external continuum) its transparency (optical attributes like opacity) its relativity(to me) and its penetrability, its enfolding enclosing lightness that does not smother but wisps of sensations tremble over me through its unrelenting touch.

I have a sensation: i have a thought.

I have an urge to measure, to experience the magnitude and to apprehend it, I have an instinct to organise my sensations: to make sense.

I have an urge to respond, to react and to act and to enact and to re-enact, to rehearse and review what i can recall of my sensations and to remember them to re-engage with them and to manipulate them as tools to remind me , to relate me  and to distinguish me from the void, from shunaya.

My fundamental measuring tool is me.

My Logos Response provides me with regions and boundaries and ratios and proportions. My response is to react and call out, to name each sensation by a response and to remember each response as a name and by this utterly instinctive iterative process to develop an internal language that models my experiential continuum.

And with my language tool i iteratively respond in convoutions to the void developing a mesh of networked ontological connections
Modeling as in a mirror the biological mesh of which i am slowly becoming aware, and ultimately the QCD mesh of motions of which i am gradually becoming aware.

With my language tool i measure and compare and distinguish and record through my biological mesh network and communicate within myself and without. I becomes we and me becomes us.

We come to know.



Of the myriad of proportions that we sense we pick as we may any as our unit for any particular urge that we have.
And so to measure shunaya:

To measure shunaya is a mystical activity, a daily meditation, an intense need that we have to experience and apprehend shunaya. What tools we use become the basis of our science, our iterative convoluted exploration into knowing shunaya.

What tools we use what tools we construct provide no greater knowledge of shunaya, but rather provide us with a different aspect of the workings of our modeling of shunaya. Shunaya is only apprehendable by us as ourselves.

Shunaya is the great mirror in which we see dimly reflections of our own selves.

There are many things in our world, and we learn hat if we pick the right unit we may measure them all. By this experience we learn the value of scale and the operations of scalars, and the peration of scalars is to aggregate and dis-aggregate: we must measure!

The art of measuring is to enjoy the iteration, and so our art is an expression of our iterative urge to measure.

We aggregate to measure and we number: pronunce the names of each aggregate stage in an eternal hymn; we make a count.
We aggregate in bundles to add rhythms to our chant, and our bundles multiply, and so we measure to the sum.
We aggregate in bundles of bundles and so we manipulate adding pattern and complex rhythms to our iteration,factorising our operations and so we sum, and dance and sing and make marks in the sand.

We drag a marker on some surface and this becomes another unit which we manipulate and aggregate and notice how they spin and move and clump together, and cover a surface and so we geometrise, all the while singing and dancing and drawing and building, putting unit blocks together, and learning through our science, through our art, through our play, our singing and dancing  the wonders of shunaya, the applicability of our model of measuring shunaya, our place in shunaya, and what we are relative to shunaya.

The songs and rhythms of aggregation and their counterparts dis-aggregation are the songs of scalars, and they apply universally, but the patterns of aggregation, the rhythms of aggregation are endless and iterative and convoluted and beauiful to us. That one particular rhythm, the indian p-adic rhythm should be so beloved is only a stage in our exploration of the measuring of Shunaya.

We have transcendental rhythms now, rhythms of π and e and yoked pair rhythms of i, and radial rhythms and the rhythms of trochoids, and the rhythms of permutations and combinations, and all of these are the rhythms of our response to our interation with shunaya through unity.

We know. And now we know that we know.....

[jehovajah]

It is hard to think of a spiral reference frame when you are schooled in cartesian and polar. Yet few of us have seen the full panoply of the polar reference frame!

However this to me is a great model of such a spiral reference frame.


http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4775

I have moved deeper since those early days of searching and now look for a trochoidal reference frame. Look closer and you will see...

The yoked pairs and the trochoids that form them.