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

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## The Principle of Proportionality

This simple statement in Universal Hyperbolic geometry has the potential to revolutionise the Modern understanding oof Mathematics and reconnect iy with that of the Pythagorean school, principally as transmitted by Euclid.

Few realise that both Aristotle and Plato never qualified as Mathematikos. By this i mean they were not approved by the Pythagorean School as masters of the Pythagoreab system. It is doubtful if Euclid received that honour, but what he did do was more faithfully pass on their teachings as recorded in the Alexandrian libraries.

Of course Eudoxus it seems without much Question was a Mathematikos approved by the Pythagoreab school. Joining with Plato meant that the Athenian Academy was more faithful than the Aristotelian Lyceum. In fact Aristotle tefutes, fatuously many of the Pythagorean principkes on Arithmoi. We find then Aristotelian scholars at odds with Pythagorean ones on the nature of Arithmoi, and as a consequence on the interpretation of Gematria as Mechanical Geometry.

Both Newton and the Grassman's knew that geometry was derived from Mechanics , not Euclid's Stoikeia/Stoikeioon, which i might reemphasise was an introductory course to Plato's Theory of Form/Idea.

The remarkable fact about Greek pythagorean teachings is that they deal with the notion of an arbitrary magnitude. It is the patterning of this arbitrary magnitude which distinguishes the primitive concept into other primitives which in brief i will label point, drawn line, , light catching surface, and finally shadow casting form. These primitive magnitudes are developed into ratioed and proportioned forms, the logos and the Kairos. These forms are then utilised by the observer subjectively as Monads. The Subjective concept of Monas is then used to develop an extensive Algebra of mosaic forms called Arithmoi. The concept of Arithmos is simply that of a mosaic. To define it too rigorously removes its amazing power.

One other subjective thing we do is pronounce names ove the Arithmoi, distinguishing them by name, but also by other criteria. Thus the Logos which strictly are ratios comes to denote generally grammar and linguistics and eventuallu logic and study, while proportion and proportioning comes to generally describe reasoning and rational subjective thought or thinking.

The pronouncing of names is universally known as counting and or sequencing, and the names are known as counts or Numbers. Initially he counts were identical with the developing syllabries in the growing civilisations. as these developed and codified into Abjabs and alphabets so a distinction became possible between scrips for te important phonemes and scripts as graphemes in general.

In a parallel development simple marks became synonomous, in a one to one onto relationship with objects, and these relationships intertwined with the forms of the mosaics /Arithmoi. Thus pebbles for counting and pebbles for Mosaics are indelibly linked.

The modern concept of Number actually develops during the turmoil of the Renaissance period, Where the mixing of Wisdoms led to Algebraic and Rhetorical confusion. This confusion has persisted to this day, especially with the introduction of Rational numbers in place of Ratios and proportions.

Would we benefit from untangling this mess?

Surely

Firstly Euclid defines a Logos as a magnitude which is compared with another magnitude, both visibly greater or lesser respective to one another, or kinaesthetically greater or lesser respective to one another.

The lesser magnitude is then utilised to measure the greater. That is the lesser is the Metron for the pair!
Now there is one proviso, there must be an unlimited supply of both magnitudes! Both must be capable of exceeding each other.
Why!
Because if the lesser does not measure the greater exactly, more has to be added until it does, and also their has to be a lesser and greater, or an equality or duality. We need enough of both magnitudes to be able to come to a precise decision about multiples of the Metron, whichever is chosen as the lesser.

Thus we see a principle of exact multiples, that is exact division that is artios is the standard. This is the basis for rational numbers!

There is another proviso: that the things compared should be of the same kind. Those who use the SI units are familiar with this rule, but it also appears in Algebra as the rule that like things only can be aggregated. This leads to different collections of different kinds, and the practice of comparing different kinds by ratios. Thus we may discuss the ratio of cats to dogs, not realising that such a comparison has no common Metron. The use of numbers as such a Metron disguises this fallacy.

However, the development of determinants and matrices and basis vectors has been instructed on these types of ratios, so we have made use of this analogous use of ratio to good effect, in ways the Greeks may have found very confusing indeed!

The proviso of having sufficient of each magnitude so one can exceed the other is because the measuring process is bilateral thus the lesser magnitude may not be an exact divisor of the greater, requiring an adjustment to get a perfect result. This could mean having more of the grater magnitude so the lesser may fit exactly or more of the lesser magnitude so the greater magnitude in multiple form may fit the lesser magnitude in multiple form precisely.

Having established a Logos as a comparison of 2 magnitudes, it is as well to realise that this is an entirely subjective notion! Thus Logos refers to that mental cognition of 2 magnitudes being comparable and distinctively so. Logos as this cognition or precognition is the source of that inner judging and comparison associated with distinguishing magnitudes, and it is associated with calling out and naming distinction. Thus Logos, here called Ratio is much more than numerals written on paper or magnitudes conceived as being quantities in comparison and contrast. It is the very idea of magnitudes the very experience of them in distinction.

So now when two ratios share the same characteristic behaviour, corresponding parts compared in a similar and congruent manner, then that experience of duality, of similar behaviours of deja vu is called analogia. That is proportion is formally analogy.

When reduced to numerals it is admittedly not as powerful a concept in appearance as it truly is!

Analogous thinking underlies all model making, and derives from comparison and contrast.

The machinery of obtaining a precise measures by counting Metrons serves to define the process of rationalising, and the repeats of that in an analogy serves to define proportioning.

But what of Kairos!

Tis is ;iterally the first time i have read book 5 in the greek and my surmising turns out to be based on the latinised translation of analogos! Thus proportionate and timely judgement which underpins Kairos does not directly derive from proportion as i thought.

Here is the reference to the resource on the topic followed by an excerpt
http://www.amazon.co.uk/books/dp/0791452344

Perhaps Isocrates' emphasis on kairos is best summarised n Panathenaicus - one of his most ambitious discourses, since undertaken and published when Isocrates was ninety-seven, In this treatise He sums up the the goals of his rhetorical paideia:

" Whom, then. do i call educated?,,,,,,,,,,First those who manage well the circumstances which they encounter day by day, and who possess a judgement which is accurate in meeting occasions as they arise and rarely misses the expedient courseof action; next, those who are decent annd honourable in their intercourse with all whom they assiociate, tolerating easily and good-naturedly what is unpleasant or offensive in others, and being themselves as agreeable and reasonable to their associates as it is possible to be; furthermore, those who hold their pleasure always under control and are not unduly overcome by their misfortunes,......fourthly, and most important of all, those who are not spoiled by success,,,,but hold their ground steadfastly as intelligent individuals(30-32)"

This pragmatic, personal, and socially conscious recapitulation of what it means to be "educated" encapsulates the principle of kairos in all its nuances: the importance of living by phronesis or "practical wisdom"...

Thus for me Book 5 has become a game changer when i read the Greek words directly. Analogos may well be a distinct root idea for Kairos

Kairos (καιρός) is an ancient Greek word meaning the right or opportune moment (the supreme moment). The ancient Greeks had two words for time, chronos and kairos. While the former refers to chronological or sequential time, the latter signifies a time between, a moment of indeterminate time in which something special happens. What the special something is depends on who is using the word. While chronos is quantitative, kairos has a qualitative nature. Kairos also means weather in both ancient and modern Greek. The plural, καιροί (kairoi (Ancient Gk.) or keri (Mod. Gk.)) means the times.

In the sense of proportioning time, allocating it for different purposes which are appropriate, measured and balanced, but it is clear that time and circumstance is the main magnitudes in comparison. Thus opportunity is compared with time and an appropriate match made.

[quote[ analogon, proportio, ratio ] see also analogies of experience , as – if , hypotyposis , presentation , reason , regulative idea , schematism A foundational term of philosophy, ‘analogy’ has a continuous if underestimated history since Pythagoras. A general theory of analogy was first developed by Eudoxus (?406-?355 bc) in response to the crisis of incommensurable ratios ( logoii ) encountered by the Pythagoreans. The overcoming of this early crisis of Greek reason ( logos ) is codified in the definitions of logos and analogos in book five of Euclid's Elements. These situate analogy in terms of the similarity between the ratios of different magnitudes, as in Definition 6: ‘Let magnitudes which have the same ratio be called analogical’ (Euclid, Vol. II, p. 114). Euclid makes a clear distinction between an analogy of terms and one of ratios, focusing his attention upon the latter. As a result, his account of analogy stresses the similarity in the relation between the antecedent and consequent terms of at least three ratios, and not any similarity between the terms themselves. The philosophical implications of analogical similarity were first realized by Aristotle, who shows in the Topics how it might be used to relate ‘things belonging to different genera, the formulae being “A:B = C:D” (e.g., as knowledge stands to the object of knowledge, so is sensation related ... log in or subscribe to read full text[/quote]
http://www.blackwellreference.com/public/tocnode?id=g9780631175353_chunk_g97806311753535_ss1-18#citation

## Pi and i Algebra

, , , ...

I said that i and pi are circle constants referencing the same ratio or proportion, but I is the formal name while pi is the formal name for the evaluation by a metron of the ratio.

The ratio is a constant that allies to many observable circumstances. Thus the ratio of the hemisphere to its circular base is the same as the ratio of demi hemisphere to its semi circular base, which is the same as a quadrant sphere to its circle quadrant base which is by this process of limits the same as the quarter arc to the radius. This constant ratio declares itself by the rotational relationship between each factor in the sequence. Thus by rotating the quarter arc on its base about the upright or right angled axle we obtain by degrees, or in stages each of the ratios . Seeing as rotation has made no material change except in the quantity of solid rotation, proportion demands that each rotational stage is proportionate to the "radian" of turn, and thus the ratio of the forms is equivalent to the ratio of the turns between them. But since we compare adjugates of the forms, that same ratio applies to each adjugate and thereby is discounted! Only , it is not forgot but set to the notion of identical, leading to the notion that the things or adjugates compared maintain a constant relation!!

The ease with which this notion is attained highlights the fundamental harmony , the Harmoniam Mensuram of Cotes, that underpins this STYLE of reasoning.

Now this style is in the Greek known as rhetoric, and reasoning is derived from Latin" ratio" which in the Greek is called proportioning or Kairos, a distinctive approach to communication which Greek philosophers began to label by the term Logos. Thus this style of reasoning is heavily underpinned by the reasoners Sagacity, pragmatism and Sophia or wisdom. It should come as no surprise then that at the very foundation of rhetorical skills is a deep mystical apprehension of personal congruence with ones surroundings.

I have not shied away from considering this important foundation to all knowledge including the so called mathematical knowledges. As a consequence I am not surprised to find that true innovation and vibrancy in the so called mathematical wisdoms goes hand in hand with a deep esoteric mystical experience in apprehending any reality, so called.

Our languages and our cultures are not divorced from our experiences. To attempt to sanitise or hygienist, tidy up or excoriate these " irrational" notions is to cut away the flesh from the arterial life support. One might think one is harvesting fruit from the plant, except that fruit contains the germ of the originator! If this germ is continually suppressed, then anything made from its fruit will grow into some tangled , knotted plant, impenetrable and unsightly.mthis is the state of so called mathematics today. Those that tend this plant are as gnarled and difficult to comprehend as the garden they so lovingly nurture!

On the other hand, those who like birds flit through the matted mess, sucking up the thwarted seeds, and fly away to deposit them in freezer pastures may survive the original beautiful intent. Occasionally a woodsman, like Grassmann may come along intent on burning the bush to the ground! The trolls that guard this monstrosity however successfully fended him off! Maybe their senses have grown dull with inbred thought, for Mandelbrot has managed to infect the bush with the most deadly virus. When it has done its work Grassmann's original strain may more easily be grafted onto the remaining stock.

Grassmann's strain is not pure, but it is hardy, and it may tolerate the purging by the Mandelbrot virus and grow to produce wholesome ripe fruit.

What can be said of pi, can also by analogy be said of i, but I is a pure label for a ratio, whereas pi is an impure label for the result of a metric process. Algebra has airways been about assigning pure labels to observables whether they are forms or ideas or processes. These pure labels have no metric content, and that is there freedom. But they must label there subject or else they become meaningless.
When one labels the subject in focus, one creates a set of adjugate labels that relate to each other. These adjugates are relational, but not metricated, without a metric, the pattern of labels may be found to apply to many circumstances, that is a wide range of subject foci. This is recognised as analogy, metaphor, simile or similarity. Thus describing a situation with these metric free labels may so organise ones apprehension of the structure of reality, that one is swayed away from continual vigilance and empirical observation. In particular, the introduction of a metron may ruin this sense of orderliness! Thus some who do not wish to be changing as Shunya continually changes may resist this continual life saving and life giving obligation to be empirical, to test the labels in the metricated subject focus, and so to clip the wings of vaulting flights of fancy and fantasy!

To some this may be onerous and an unwelcome experience, but to those who engage it it is a purging experience, pruning the plants in ones garden, the setFs, so that by this evolutionary process plants fit for purpose grow in the mind, bearing with them the DNA of all!

That one may survive whatever the prevailing environment is the true prize of this process. And to appreciate that one must look beyond the individual, constructed subjective self to the colonic environment from which it springs, and beyond that to the dynamics of the ceaselessly changing Shunya !

## Relativity

I opine that the notion that is encapsulated in the word relate is the most fundamental notion that can be adduced!

Both sequence and form exhibit it as apriori, and like communication we cannot but relate subjectively, in subjective processing. Thus, even though we may declare that 2 objects or sequences are unrelated, it will be a statement that is a matter of opinion and not an incontestable fact. We may always find some way to relate every form and every sequence.

I will explore the notion of "relate"

The motion of light is not straight. It is very fast, but it is not straight. Or to put it another way: if our idea of straightness is based on light, then it is "bent"!

Etymology of the English word relate
the English word relate
derived from the French word relater
derived from the Medieval Latin word relatio (laying of matter before Senate, such motion; referring back case to magistrate; narration, relating of events, recital; reference to standard; retorting on accuser; giving oath in reply)
derived from the Late Latin word relatus (narration, telling of events; utterance in reply)
derived from the Medieval Latin word referre (bring, carry back, again; give, pay back, render; it matters, makes a difference, is of importance; report , bring back news; record)
derived from the Latin word ferre (to carry; to bear; bring, bear; tell)
derived from the Proto-Indo-European root *bher-
using the Latin prefix re-
derived from the Latin word relatum
derived from the Medieval Latin word referre (bring, carry back, again; give, pay back, render; it matters, makes a difference, is of importance; report , bring back news; record)
derived from the Latin word ferre (to carry; to bear; bring, bear; tell)
derived from the Proto-Indo-European root *bher-
using the Latin prefix re-
Date
The earliest known usage of relate in English dates from the 16th century.
Derivations in English
corelate, related, relating
Usage
Word found in Modern English

Etymology of the English word relative
the English word relative
derived from the Old French word relatif
derived from the Latin word relativus (relative; referring; having reference)
derived from the Late Latin word relatus (narration, telling of events; utterance in reply)
derived from the Medieval Latin word referre (bring, carry back, again; give, pay back, render; it matters, makes a difference, is of importance; report , bring back news; record)
derived from the Latin word ferre (to carry; to bear; bring, bear; tell)
derived from the Proto-Indo-European root *bher-
using the Latin prefix re-
derived from the Latin word relatum
derived from the Medieval Latin word referre (bring, carry back, again; give, pay back, render; it matters, makes a difference, is of importance; report , bring back news; record)
derived from the Latin word ferre (to carry; to bear; bring, bear; tell)
derived from the Proto-Indo-European root *bher-
using the Latin prefix re-
Date
The earliest known usage of relative in English dates from the 16th century.
Derivations in English
corelative, relatively, relativeness, relativism, relativist, relativity, relativize
Cognates
Dutch relatief, French relatif, German relativ, Italian relativo, Norwegian relativ, Spanish relativo, Swedish relativ
Usage
Word found in Modern English

Mathematicians and physicists and Philosophers seized upon this word in an existential frenzy, and gave it a new referent: a comparative process in which all were judged by each others standards. No one had supremacy or right to impose there standards on others as the coercive force of authority, but all were of equal authority and all had freedom of speech to voice their opinion, including their opinion of others and their actions.

This was not a democratic process, in which the mob or the people decide who has stirred them most, and thus they decide, nor is it a debate in which the aim is to win the approval of the audience as victor of the field of discourse. It was rather an intellectual and academic freedom to propose and suppose what honest opinion or devil's advocate opinion a proponent held, regardless of moral or political or even social censure. In open society these censures were and are endemic, but within certain clubs, societies and guilds each member was allowed to relae his opinion and position,

Thus relativity and relationships are bound together by these freedoms that are not public, but which nevertheless are expressed between parties, when closely observed.

The notion of dependency is also crucial to relativity, the sharing of matters which require advice or opinion or a decision from others.

Other notions take it back in History but forward into the future"Harmonias, Summetria, sunthemata and crucially, sumbola

Now i have exposited the strong subjective motives for the words, using it as an anchor we can look at how the nonverbal behaviour of the notion is utilised to connote ,by analogy, spatial and ultimately temporal dispositions.

In order to relate one must position oneself. Thus every preposition of space becomes a utility to describe this positioning or disposition while relating. Relating thus has a strong subjective motive,and thus emotional tie and a concomitant strong positional requirement. Thus the word itself has a powerful synaesthesia, which promotes its use as a term or denotation of spatial and temporal disposition. Add to that the compounding effect of family spatial bonds and temporal bonds and the absolute value of the notion is grounded. Thus "relate" in all its forms has a clear utility and flexibility of meaning which allows it to signify both spatial, temporal,subjective and objective, lingual and familial dispositions; ie "relationships"!

Thus we may begin to see the physical bond or tie as indicative of a binding relationship, regardless of the subjective feelings of the participants bound by the tie. Similarly a Boundary is significant as a tie that binds those within in it in a relationship. The first relationship is denoted as intra, the second as extra or external.

Relationships thus typified, strongly support internal and external coherency as indicative of bonded or boundary relationships(sunthemata)

When an object possesses an attribute of harmony it is because the external or internal relationships produce a subjective response that suggests fitting('ar) and oneness(monias). Such notions are direct derivatives of this bonding and binding and bounding perception of relationship.

There are levels of harmony: surface, where everything looks great on the outside, but internally may be complex, chaotic and disordered; ans internal, where everything looks great on the inside and may or may not look great on the outside. A particular value is something which has both external and internal harmony attributed to it. The relationships in these harmonious situations vary in complexity,

The attribute of summetria represents a subjective apprehension of an object as itself. It is a reflexive notion of relationships, and we develop these types of notions because we do not use referents external to the subject matter, that is we do not analogise.

When i study any formi compare. If i compare the form with itself or elements of itself i use the notion of summetria. The powerful result of doing this is not tautology, which of itself is fundamental, but identity. The ability to identify and have a notion of identity is crucial to so many systems, Thus self reflection and self referencing and selfishness is crucial to identity, and to boundary issues. The bonds, the bindings and the boundaries in summetria all relate to one nother, lead to and from on another and confine to one another. The notions of group or set or collection even sequence are not possible without summetria, the ability to define and denote belongingness, what is part of something and what is not.

Sumbola is that process that allows us to compare complex systems by means of a representative form or mark or subjective notion. By doing this i combine the summetria the sunthemata of a form into some representative form and then look for the relationships between the notions. From this i derive the above distinctions again, in a fractal pattern which is self similar, almost, and repeats at increasing scales of complexity of forms/symbols in relationships. One additional emergent notion from this comparison is analogy.

Analogy is where i compare a form with some other form that seems particularly close in its sunthemata and summetria of relationships. The Harmonias of the comparators may extend between the comparators, and thus a greater sense of harmonias is achieved. Disharmony may be evident from the comparison, and thus a bound is set on the analogy . These experiences form the bedrock foundations of aesthetics.

Thus through a lifetime of interacting with space and engaging in these comparative processes and refining the subjective notion of relationships i obtain to a certain aesthetic wisdom that may guide my behaviours at all levels and my perceptions a all leves, and my expectations and anticipations of the future.

Wholeness in the idea of relate is based on the notion of summetria, with only inductive empirical evidence. It is thus a subjective decision to accept the designation of whole for anything. Thus unity has no basis and is entirely a subjecive choice as to where that designation might apply. However, once we have established a referent for uniy certain relationships apply without variance! These invariant relationships are the subject of philosophical enquiry and discovery.

The "scatter pattern" of sensory intensities in space are processed and sequenced in the raww form up to the highly structured related form which i perceive as my current world. By building my world on these invariant relationships i aam able to maintain identity in a changing land and sound -scape(old German skapaz) that is a set of sensory relationships. If however one of these accepted invariants changes, either in value or significance or position or dominance, or even out of the processing loop, then i experience profound change. Such sets of invariants are sometimes termed paradigms.

Thus relate and relativity have a deep and abiding influential power on my subjective processing of sensory information, and the structure of relationships forms who i profess to be as and according o how i Accept them.

Acceptance is the final choice power i have over everything, and the structure of my experiential continuum is the structure of those relationships i accept and how i relate to them.

## Perfection is..

Perfection is an ideal state of mind that rationality has conceived , a set of principles concocted by reason alone and deliberately divorced from reality and emptied of conflicting empirical data.In that sense it is truly autistic, and an unachievable goal. Such goals may take us out to the extreme edge of probability, and test the notion that anything is possible, of infinite possibility; or they may dash our hopes against the rocks of impossibility and the foaming seas of despair, drowning us in such overwhelming triviality, sucking out of us the very air of creativity. Kairos then is the only wise way forward, proportionality in everything. Whatever your faith or Fate, kairos is wisdom.

R=cos(ø/2) 0<ø<π
R'=cos(π+ø/2) 0<ø<π
Lai Zhide vector in polar form (R,R'} Where thre brsckets (} mean a sequenced set S{sub}F[/sub} on generalised reference frames. In this case the reference frame is two vectors(axes) diametrically opposed or contra to each other.

Lhai Zhide revolutionised the Taichi and the Taijitu, "modernising" them to reflect indian Astrological insight. The Tai jitu (Yin Yang) in particular became a powerful measuring tool which preceded and exceeded the utility of the protractor, in that it measured not only angle and radius but also time of year.

The scientific use of the Taijitu has been obscured by its astrological associations. I do not demean astrology,AS UNTIL VERY RECENTLY ALL ASTRONOMERS WERE ASTROLOGERS! Modern astrology is but one small part of what astronomers used to do to "earn a crust". After all who was going to pay someone for gazing at stars?

In any case, the Lai Zhide Taijitu is a fundamental cosmological tool, that simplifies may calculation set ups, and illustrates the power of the circle boundary and interior in measuring our reality, especially in terms of vectors.

in rotating vector form

r=cos(ø/2)e
and in path form z= cos(Ω/2)z + Ω where initial z is vector x and the initial path Ω.

All of these require the iterator ø or Ω to be enacted to describe or apprehend the form. Thus iteration is fundamental to description of dynamic forms where the form has an essential change in the compass multivector network characterised by rotation. Even a straight line has rotation inherent in its compass multivector description.

Iteration is the secret power of "mathematical", that is philosophical dynamical analysis. This iteration is disguised by terms like compound, repetition, recursion, even iteration itself. As a consequence it seems a difficult action to pin down, or describe, and many hae "played" on this basic indistinct notion.

Indistinctness will not do for computers. Iteration has a hard and clear definition in the "for" loop. Because it is in computer science does not mean "mathematicians" should ignore the definition(s).

For All z; For All Ω; z=cos(Ω/2)z + Ω; until Ω==2π

## Within Shunya "i" relate,compare,contrast,describe,iterate and experience the outcomes

In his seminal work on Couples a science of pure time Sir William Rowan Hamilton ably demonstrates the power of the Eudoxian trichotomy, and wit it establishes the fundamentals of Euclidean gematria, as far as he understands it. His intention was not to develop spatial gematria but a temporal one which was nevertheless , per force, analogous. It is a pure vector analogue as well as a "number line" analogue. but this is simply because he chose to emphasise those analogies. His demonstration is of far greater applicability.

The basis of the analogy is the sequential stement above, which is of high probability to be the same for most animates- the "i".

Relating is in fact a spaciometric activity between the subjective and objective experience of sequenced attentional focus, with intentional relational or relativistic interaction with the shunya field. Thus i may objectify a region by location. This is a semiotic act. I may then objectify again region by location, and ascertain that there are in fact differences in location relative to "me", and infer differences relative to each "other", and thus settle upon the notion that i have regions in my experience that differ by location, multiple region which i might describe as 2 in this instance.

Re iterating with this new description enables me to approach a settles set of descriptors. Essentially the relae process has provided me with more than just a semiotic location, but with a language description of a compaison and contrast process i have automatically engaged in.

If i do not focus attention on an external "object", my attention turns inwarde and follows exactly the samr courses of action.

Thus i follow this process on one or multiple attentional "targets".

The comparison and contrast aspects become increasingly more complex and convoluted and interesting. We often hear that it becomes confusing, but in fact w are well able o unconsciously handle the task, if we understand that the process involves generating new language, not utilising old language . It is the attempt to use old language "correctly", that hampers my creativity.

Thus we come to Eudoxus who in his heory of proportions looked at this very issue. If i exclude the subjective region from the discussion some say that the discussion becomes objective. However this is a tutology. The point of view i adopt is a s subjective as anything else, but it does help to focus the attention on the "objects" and to highlight the comparison and contrast topic. In this case one object only seems to invite description. This of course is a subjective description of an "object".

Two objects invite description relation and comparison and contrast and finally language distinctions by iterating through the experience with different outcome products.

So why does this not happen with one object? Well it does, as a subjective description which we have just specifically excluded! We cannot avoid these types of processing no matter how much we want to be objective.

Eudoxus now takes the comparison and relation through another iteration, by moving the 2 objects together spatially and laying one down on the other. This process is called katmetreese. It is a fundamental "action " of FORM comparison, and spawns the whole of empirical Science.

By analogy i can now describe the subjective, "mental" comparison of objects as a type of form comparison done using the sensory processing networks within my biological frame. For this to happen it is necessary to have a "memory". a fundamental consequence of a sequencing space that imposes sequence, as discussed in an earlier post. These memory object comparisons are the basis of "logos", that is symbola or symbolic representation, and a manifestation of kairos that is direct proportion.

When i have come to a settled description and language about an object, after iteration, i have then to live my life interacting with these subjective objective representations. These provide me with additional comparisons and relations particularly the combinatorial ones. Thus the sunthemata and the summetria become subjectively significant from empirical experience and my inherent operating system, the logos kairos sumbola sunthemata summetria reponse is revealed .

This OS is in fact the basis of my initial reactions to the shunya field, which i have characterised by the title.

Now Eudoxus method applies to more than 2 Objects. However to apply it we have to recognise the objects in a higher level form or summetria. We tend to call these a group in modern mathematics, but fail to really explain what is being done. Euclid in fact in his teaching material sets out these comparisons in a readily apprehended form. These comparisons are set out as geometrical constructions. Thus if i hve 3 things to compare, i can compare them 2 at a time and draw a conclusion that way, but i can also compare them as a single unit, monad, a triangle of things and draw conclusions that way.

Thus i first have to subjectively describe the new unit consisting of 3 objects. This means that not only do i describe each individual object, but also each pair, and all their relations as part of a description of this 3 object unity. Having done that for 2 such 3 object unities i am now in a position to compare at the levl of the 3 objects, or at the level of the pairs of objects, or at the level of the individual objects.

It is clear that s more and more things re compared together the complexity of the comparison, potentially , seems to increase. However, utilitarian objectives make short work of what relative comparisons need to be made, and unconsciously many of the comparisons are made in processes we have come to call aesthetic.

At ach level there is an overarching simplicity covering an underlying complexity and an iterative convolution. Because these are sufficiently complex relations we experience greater degrees of freedom of choice as to how we may proceed, and process, and these degrees of freedom relate directly to the complex synaesheias we experience in our sensory mesh processing systems.

Thus when it comes to 3 dimensions the distinctions are qualitatively different from the Epihaneia that Euclid introduces the class to , and so in his later books he begins to explore how the gematria is derived in 3 dimensions. Of course, at this stage the set of comparisons, combinatorially can be very rich and interesting, and thus provides a means to capture the complexities experienced in "Nature", and a true theurgy toward any demands a god may place on a mere man!

Whether it be Gods or the State or the President, the theurgical nature of science is set by external parameters to which some of us respond with our whole lives!

http://www.enotes.com/topic/Sequence_theory
http://blog.aaronwalser.com/2009/12/sequence-theory/

http://math.stackexchange.com/questions/20277/graph-theory-proving-that-a-degree-sequence-is-graphical-havel-hakami

http://www.gametheory.net/mike/applets/Random/

http://www.purebits.com/mlsteo.html

http://cslipublications.stanford.edu/site/1575862174.shtml

Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison

David Sankoff and Joseph Kruskal

Time Warps, String Edits and Macromolecules is a young classic in computational science, scientific analysis from a computational perspective. The computational perspective is that of sequence processing, in particular the problem of recognizing related sequences. The book is the first, and still best compilation of papers explaining how to measure distance between sequences, and how to compute that measure effectively. This is called string distance, Levenshtein distance, or edit distance. The book contains lucid explanations of the basic techniques; well-annotated examples of applications; mathematical analysis of its computational (algorithmic) complexity; and extensive discussion of the variants needed for weighted measures, timed sequences (songs), applications to continuous data, comparison of multiple sequences and extensions to tree-structures. In molecular biology the sequences compared are the macromolecules DNA and RNA. Sequence distance allows the recognition of homologies (correspondences) between related molecules. One may interpret the distance between molecular sequences in terms of the mutations necessary for one molecule to evolve into another. A further application explores methods of predicting the secondary structure (chemical bonding) of RNA sequences. In speech recognition speech input must be compared to stored patterns to find the most likely interpretation (e.g., syllable). Because speech varies in tempo, part of the comparison allows for temporal variation, and is known as "time-warping". In dialectology Levenshtein distance allows analysis of the learned variation in pronunication, its cultural component. Levenshtein distance introduces a metric which allows more sophisticated analysis than traditional dialectology's focus on classes of alternative pronunciations. A similar application is the study of bird song, where degrees of distance in song are seen to correspond to the divergence of bird populations. A final application area is software, where Levenshtein distance is employed to located differing parts of different versions of computer files, and to perform error correction.

12/1/99

ISBN (Paperback): 1575862174

Subject: Computer Science; Sequences; Pattern Perception

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1176350045

## Multiple and the hidden notions of Symmetry and Summetria

One particular proof of Pythagoras Theorem based on the plane circle highlights the close link between a multiple form (pollapleisios) and similar forms, and also the distinction between summetria and symmetry.

There are also similar notions to distinguish by duals such as external/internal and objective/subjective.

Summetria is a very much broader notion than symmetry,in that summetria is perhaps best associated with group identity. Group identity is that curious notion that a group has an identifying attribute which may be external or internal objective or subjective.

I may attempt to distinguish a group from a collection,but soon run in to the essential tautology in such fundametal notions,and the "circularity of the distinguishing definitions. In any case the perceived differences arise not in the object but in the subjective machinations of cultural language differences. Thus the same notion is hijacked from a different language and culture in order to proppose some "finer" distinction, the result being confusion. The author is simply advancing his or her particular iew under the guise of an essential distinction which may or may not exist, and which in many cases may have a perfectly acceptable cultural alternative word, voiding the necessity to use loan words from another language or culture.

So yes i am guilty of this "spicing" things up when perhaps there is a shorter Anglo Saxon idenity that may be used. However one is who one is, and ones subjective juices flow as they flow, and i do not proose to change that any time soon.

However a collection is distinguished by activity, and this in itself allows a group or collection identity to be distinguished: we may distinguish by the activity and the actors for example. If the collection is housed in one place we may distinguish by the location or the boundary of the location.

Now intern-ally the objects in the collection or group may have a common origin, or a similar provenance or a congruent utility, and thus have a group characteristic that identifies.

The process of identifying is thereby illustrated to be absolutely subjective, and i may therefore form subjectively a collection identity "at will". Summetria refers to this ability to form collections at will, or by predilection or subjective processing of any sort.

The notion of multiple is therefore based on this subjective freedom to form groups by some subjective process which are either wholly subjective and therefore inscrutable by others or some degree of objective, and thus open to scrutiny and tending toward transparency.

The Pythagorean Philosophy deals with this tautology about summetria under the notion of the duality of monads and henads, and in fact ses up a standard model that is to be fractally applied, not taken as the "one and only". Thus analogy is invited at all scales from the standard monad henad duality model. Euclid, as a Platonic Pythaorean analogises it as monas and arithmos.

Euclid does a little more,by pointing out one particular form of summetria that he wants to develop. I must again stress that this is one form of many, and Euclid in doing this was not attempting to preclude others, but he was attempting to apply kairos and harmonias in the design and construction of his teaching materials. These are described as aesthetic principles, and highlight and underscore the basic design sentiment that the greeks and other philosophical societies laboured under.

These design sentiments have been much explored and many new designers ,having understood them, now free themselves to evolve other design ,aesthetic values. Some examples may be organic or environmental design ethics, promoting natural form and arrangement, a Feng Shui.

Summetria then is a fundamental powerful design modality, that affects every decision we make regarding appropriate or inappropriate, right or wrong placement activity,aggregation activity and construction and layout activity,and presentation activity. Thus even "neatness" and "order" and "systematic " and "logical" and "rational" are descriptors of the influence of summetria.

Euclid chose in his teaching materials to identify symmetry. Symmetry is distinguished by the notion of an objective/external metron. A metron is anything that is used to measure by repeated application of the metron in a "laying together to cover" action . This is a combination of 2 or more actions distinguished in the greek sugkeimia and katametria. On the whole arithmoi tend to be forms with some aspect of symmetry, that is ability to be measured by a common measure.

The idea of symmetry is further identified in terms of commensurables, in Euclid. Thus Euclid sets out the essential notions of symmetry by these notions without excluding other possibilities.

Meros and Pollapleisios are hooked on to this notion of symmetrical and commensurable arithmoi, and thereby the idea of a "rational" scale of things is developed, entirel within the notion of symmetrical multiples. Hamilton's [paer on "Couples or conjugate functions" is a masterpiece in deriving these symmetrical relations in a Euclidean form.

In a Eudoxian manner Euclid distinguishes the whole number or integer notions of accurate and approximate and how they play out as multiples. Now in doing so he introduces another design measure and that is the gnomon. The gnomon foms multiples by "-axis", that is by "folding" into the form made by the gnomon the constituent arithmoi. The axis then refers to the particular vector arrangement of the sides of the gnomon. Thus where the sides of the gnomon point are distinguished as distinct vecors ,and these vectors as lines of magnitudes are called axes(cf axle,axiom)

Symmetry is also further distinguished by Euclid in terms of "iso", which means balanced equally. This is a kinaesthetic visual synaethesia that describes a symmetry based on the sameness between the kinaeshetic and visual signals. It therefore is useful to describe smoothness and evenenss and identicality and congruency.The notion includes comparing in 2 ways. Thus a line is good if it is measured the same("iso") way after flipping it through π radians. This notion applies equally to "straight" lines and circular arcs.

The notion of symmetry and meros lead to the idea of symmetry up to scale. This is usually defined as similarity, but this i not a rigorous definition in some instances..

I can see from this notion of meros that symmetry includes within its broader definition the notion of multiple, and thes multiple symmetrical parts may constitute a symmetrical greater(meizonos) whole. Thus a similar figure may be constituted from minor(elasson) similar parts arranged within the frame of the major(meizonos) simmilar figure.Thus in this way symmetry may be preserved even at different scales.

The preserving of symmetry is not a requirement of Phusis, it is an entirely human conceit. However, it has proved very useful, but when applied rigorously it fails. Most objects n Nature are assymetric, but some are almost self similar, which underlies the notion of fractal geometry.I can explain this assymmetry very simply: The circle form on which Euclidean symmetry is based is a special form of spiral.

The Euclidean Stoikeioon are not a fundamental geometry. It is hard to assert that such a thing exists except in the minds of some early nineteenth century synthetic geometers and mathematicians. Euclid's course is a course in theurgical dialectics based on the Pythagorean philosophy as interpreted by Plato. Thus the plane geometry is not the foundational source of notions of symmetry, space is. The human interaction with space is pricipally £D, and no matter how much we imagine otherwise we have no real sympathy for 2d or flatland or anything remotely to do with living in a 2 dimensional surface.

WE always approximate to flat, even or surface by a logical trick of removing the dimension of an object we may call depth. Similarly ,like Euclid we remove a magnitude dimension called breadth to approximate length. Thus our notions derive from 3d space and are modified by relational restraint to apply to a lower "order" "form". Hence Euclids notions of 2 dimensional symmetry should not really define symmetry. If anything 3d symmetry should define symmetry. However when 3d is used we have additional attributes to factor in, not the least being perspective and parallax.

Summetria ,then is the more general group/collection identity notion frm which we derive 3d,2d and 1d symmetry,by relational restraints. In all dimensions symmetry includes multiple forms, major and mainor parts, proportions and ratios,kairos and logos.
The biggest symmetry used in mathematics is the equality symmetry, and this is also the basisi of the tautology at the foundation of definition.

## The Logos Kairos Sumbola Sunthemata Summetria Theurgigical Response

The Logos Kairos Sumbola Sunthemata Summetria Theurgigical Response arises out of the visual . auditory gustatory kinaesthetic (proprioceptive) neural network interaction with the Shunya field. This neural network is maintained and enabled by the cellular microbiological interactions with the shunya field, maintaining a regional distinction between the biological field effect of the shunya field and the environmental field effect.

The Logos Kairos Sumbola Sunthemata Summetria Theurgigical Response is the main component of the subjective conscious process within the biological frame of the neural network, with the subjective unconscious processes arising within the biological structure as an evolutionarily determined outcome of shunya field interactions.

On an environmental objective subjective description, the biological framework arises from the interaction between, and exists within a macro structure consisting of, similarly formed biological units. The macro structure is evolutionarily determined and environmentally bound within the shunya field.

On a Shunya field perspective, the environmental boundaries for each subjectively distinct regional product within the field form a fractal distribution of the essential subfield structures of the multi polar shunya field.

This is a first draught of axiom 1 and it is a derive Pythagorean metaphysics.

One of the important corollaries of this axiom is the formation of the neurological "we" and the corresponding subjective "I". Both these subjective constructs arise out of the interaction of the biological frame interaction with the social and environmental interactions within the shunya field.

There is no thing, wich in and of itself is evil, but rather, overmuch makes it so. For things progress from bad to worse and then beyond to evil, declaring in this comparison the truth of the former proposition

It is tautologically true that what may be said of space is said in attribution. Thus the properties of space are those i attribute to space, and consequently the laws of space can be n mor than the laws of that which i have attributed to space. And then, even this ensemble i attribute to space!

Said in another way, i attribute to space my model of space, informed by my engagement and interaction with space and its populace. Such a model is by all means derived and deduced from my interactions and intra dependence on, in and with space, and conducted through induction to a high state of provenance and attribution. In so far as the induction agrees with my experience of space i base my life and interaction upon this model, often in sublime ignorance of what the model lacks.

`in this light, idraw wicked attenti0n to the obvious and extant situation that no action in or of space occurs without or unless space is geometrically deformed. I would prefer spaciometrically deformed, but in this case that terminology hides the simple observation. Thus i do not think it proper to draw any distinction beteen physics and geometry, seeing as the former employs the latter to great effect in describing, measuring and predicting the behaviours of spatial objects in their relative motions and positions, providing the agent has the means to aggregate th constituent forms into a subjectively comparable form.

The proviso arises from the active agency within all our models required to determine iterative outcomes, and to compute the surface products of the same.Where the subjective processing is not possible, little merit is gained by any action, and little sensibility is obtained from apprehending the observed behaviours.

However, the fractal paradigm, based on principles of scale, self similarity, iterative process, followed by careful colour and surface assignment provides a overarching environment to solve these difficult ,descriptive situations.

At the heart of it all, our attribution works for us or against us according as we accept the consequences of our own acts of attribution. Moderation, terefoe leads to a moderate view of all things, without excess or excessive formulations.

Kairos, my friends. Let all things be done in kairos.

## The Seventh Book

, , , ...

The first thing that i notice in book & is that the Monas and the Arithmoi are defined relative to each other.

The second is that the arithmoi are aggregated and measured, not counted.

The significance is that this is a purely geometrical exposition of form and structure and plethoration in space.
There are two impolses to measuring:comparison, and congruence. Within congruence is the congruence recogniseable or perceived as derived from parts. The notion of integrity from parts and the idea of pervfect fit. Finally, from all of this , the idea of the appropriate scale.

The monas is, as the point, a spiritual object. As Shunya is to ek, so is the point to the monas.

The monas is perceivable and has part, whereas the semeion has no part.
Compare
Μονας εστιν, καθ' ην  εχαστον των  οντων εν  λεγεται

monas estin, xath' een exaston toon hontoon en legetai
1. A unit is (that) according to which each existing (thing) is said (to be) one.
Monas is, down the each of the relatives of things, "one" defined

Semeion estiin, ou meros outhen

1. A point is that of which there is no part.

In both cases we begin with the subjective experience of the reader. The reader can define anything as monas. The reader can define nothing of a point, it has no part, it does not exist in the sense of the notion "ee exaston". It has no "each", no particular, no substantiation and thus no instance.

A semeion is just that, a supernatural or psychic phenomenon, one to be wary of.

A monas is real, in that it exists and we can define any object or thing that exists, as it. Ton hontoon- the referent-exists and we can call it en-one- according to the monas. The Monas is clearly not an object but a concept, a freedom or liberty to name. Like the Logos it is a power to create order.

The precise greek is "according to the each". The natural relative question is the each of what? We seek a particular referent, expecting to find multiple instances from which we may select one or a single instance. From this single instance we expect to kknow or learn something, by which we can compare all other things: toon hontoon is thaat relative referent.

Toon hontoon is the generality of relativity, it refers to a notion of recursive definition: "the relatives of the(pl)". As a language expression it reflects the economy in language and the noion of reflexivity, relativity and referencability. A pronoun references a noun directly or reflexively. The notion of the common pronoun is extended by the addition of the relevant definite article ending as a suffix. The article ending on its own refers adjectively to "whatever" so the whole form is a self reflexive notion of "anything".

Thus monas refers to the notion of one found in each notion we have within our minds. That notion is clearly used for an infinitude of different circumstances, yet we still recognise the idea of one. This faculty of being able to recognise or define "one" in any siruation is what Monas is.

Both the Semeion and the Monas are the subjective or psychic marks of our interaction with reality, or put another way, signs of the divine spark within us if you hold that conception. The semeion is linked to the gramme, while the Monas is linked to the Meros and through that to the Arithmoi. Thus that which has no meros can only be an anlogy of that which has.and the whole of the 6 books being founded on the semeion are an anlogy of the real forms that have meros, the Arithmoi. The theoretical meets the pragmatic in the Monas.

The arithmoi then enable geometry to impact on place, but through a psychic force or perception called monas. Through that perception geometry or spaciometry impacts through proportion, measurement not counting, through aggregation and perception of the perfect fit, the aritos. And thus-ways to artistic appreciation and aesthetic apprehensions.

Cipherism is the least function of the geometry, with Architecture being the highest. Cipherism has its very important place, but it is not the republic of knowledge, merely its noble slave class.

2. And a number (is) a multitude composed of units.†. An Arithmoi is a gathering together of monads to form a plethos, a crowd. or more precisely a side of a gnomon.

In this conception we see the infinitude of monads that exist in any thing to give it form and to define its sides and boundaries. The inherent calculus has always been there in Euclid, and it is fractal in nature. Pythagoras's harmonium and Kairos are coming to bear, and it is through the mystery of the monas.

When i read the greek myself i understand that the monas is not an arithmoi, because it is a subjective faculty, by which faculty i may call any existing thing "en". Thus , the faculty distinguishes things that exist asw concrete objects and things that i experience as conceptual or perceptual objects. If a thing exists it has parts and it is up to me to name which part or whole i will call en . En by the way is the greek name for one. Alpha is a letter used to count but when counting you can choose to use the alphabet or the names, en, duo , tres, etc.

Therefore it is clear that Monas is not an arithmoi but "en" is the first arithmoi! Through monas i establish an en, and then i can count,BUT Euclids elements do not place counting as the next logical step. but aggregation of monads. Now monads are not ones but units, and units may be of any scale. Thus the aggregation of units to form a side of a gnomon is not describing only univocal units, but any conglomeration of any sorts of units. providing that they fit perfectly into a form. Then and only then are they regarded as arithmoi.

Hipparchus showed that there are forms whose sides do not conform to this perfect fit and tus cannot be classed as arithmoi. Eudoxus showed that any object, if it exists the monas can call a unit and then everything can be measured against that unit, and arithmoi can be found for that unit. The notion of protoarithmoi, equivalence classes of arithmoi was well established, and this extended it into the irrational sides of gnomon. Proto arithmoi and irrationals are linked.

ην εχαστον των οντων   ων
EEn exaston toon hontoon is a continuing study. Etymologically i get back to 2 verbs εχω, ἔχω exoo and ων oon

Both verbs have nominative derivatives,adverbial forms in which they take on the relative role in a sentence structure . In its simplest action these verbs ar possesing and being, but in this case the useage is at the extremum. thus the relative or the referet notion takes priority,because in that sense that is all that is left at the extreme, the abstract notion of possessing being, the final analysis of any object is its mental impression, and the general mental impression is a referent for anything.

The kath preposition though makes the referrent a comparative measure, by which conformity is assessed. And "en" is defined by this mental impress for whatever object is impressed. This impress is itself named or defined "Monas".

## Pythagoras:The Universal Kairos

I have not shied away from the mystical and the mythical roots of mathematics and geometry. My appreciation of the significance of mysticalbthought and philosophy as a driver to scientific research has made that an inane thing to do.having, therefore, full freedom to explore all sources provides many moments of serendipity.

By now I amused yo the myths promoted in childhood education being disappointingly shattered by a more adult investigation, and so it is with Pythagoras and karos.

The notion of kairos indeed justifies the teaching of children fairy tales and myths, but equally coerced the shattering of those myths in the adult! What is " appropriate" as judged by some judge, being an individual or a group or a god or a system is what " kairos" is about. Thus the notion has no single referent , because it is a derived subjective one, and an abstract principle.

How do we come up with such abstractions? The process interested the Greek philosophers who thoroughly researched and expounded upon it, and over the course of time refined it. But such a treatment is not without it's ethnological differentiation, and this in fact is ra key element in abstraction: that is how a word or notion may be separated from it's referent and be given different or additional referents and thus significances.

Kairos would seem to start with a spatial referent which had great and vital significance: the very vital organs of the body. From this would be derived the full balance of life compared with a small region of vulnerability. This comparison is natural and of desperate importance, giving hyperbolic significance to the notion of kairos. By analogy of comparison the proportionate significance is established, and once established is free to be applied in comparative situations. Thus kairos is significantly linked with analogy and comparison with regard to the germane proportions or ratio.

The connection to logos derives solely from it's rhetorical contexts, that is : it's frequent use in rhetoric as a notion to be relied upon to convey proportionate, apt, and appropriate comparisons of significance. It's relation to time relates to its referencing those sequence which are of crucial significance in the sequential outplay of a sequence of events. The significance of such sequence is in the defining moment, that is the encapsulate a moment from which a measure of time may be made or by which an epoch of time may be divided or proportioned. Kairos therefore was a significant rationalising and proportioning notion, and it fell into the hand and mind of Pythagoras with unusual and powerful alacrity, becoming a fundamental principle of order, organisation, summetria, arrangement,architecture and form. As a behavioural principle it conjoined proportionality in action and reaction and promoted ratioed thought, proportionate language, levitate justice, proportionate dictatorship,etc. This received the cognate term " moderation", and thus moderation connotes proportionality not abstinence. There is no equality in kairos, but there is a notion of fairness based on proportion.

Pythagoras, as a mystic, clearly sought to promote this notion of kairos into a principle of living, and he had great influence in his time through his consistent application of this proportional principle. The welfare state is based on a kairotic principle: from each according to his means, to each according to his needs.

Pythagoras's scientific interests were not excepted. He sought and expected kairos in the environment and found it. He demonstrated some relationship between the counting numerals, the units or monads of measure on a string/ tape measure and the juxtaposition of square areas, or square columns of equal height: provided they could be arranged around a right gnomon the areas or volumes summed. In fact he was able to show that this applied to all proportionate figures or columns. These proportionate figures were called similar.

The pythagoreans were interested in these forms called arithmoi and studied there proportions extensively, developing many distinctions such as proto arithmoi, static and dynamic form etc.

I believe that the defining moment for Pythagoras was when he discovered the musical tones within a measuring cord called a mono chord.

The monochord is a measuring cord or a tape measure stretched across a bridge to make a musical instrument. In line with kairos Pythagoras sought to measure proportion in music.

The significance of Kairos is its "vital" nature. Thus it is a vital proportionality that is being conveyed, with such synonyms as just, right, apt. apro pos,correct, judged measured and moderate etc being employed as adjectives. thus the power of the notion of Kairos was in its adjustability to fit: it was not just any proportionality, it was the right, or necessary or apt proportionality, and in this case, the natural or divinely enthused proportinality.

Music was not unexplored before Pythagoras. It, of all the art forms relied on the subjective judgement of a skilled artisan, and the lore that such artisans developed through experience. This subjective skill was called a muse, connecting it to the divine inspiration that originated it: thus music.

By this subjective judgement and skill and religious and mystical sympathies and analogies the 7 stringed instrument was adopted as the aesthetic standard against which to measure all stringed instruments, and the music derived therby was the muses prompting toward a pleasing sound, and the conveyance of mood and emotion. The tuning also was at the whim of the muse.Like wise rhythm and tempo, dynamical range were similarly governed.

Pythagoras therefore, prompted by his belief in the universal principle of Kairos was the first to carry out a scientific investigation of musical sound. That he started with a measuring cord is not an accident. The idea was to find out what proportions of a musical string are involved in producing the pleasing musical sounds, the sounds of the muse. What rules of proportion did the muse use?

We have a direct connection between the physical instrument and the divine agency that inspires its aesthetic use. There is no irrationality therefore in applying these rules to the treatment of mood "disorders", to realign the proportions of the mood faculty . The word mood here is the greek psyche also translated soul. Thus this is a psychotherapy devised by Pythagoras on a basis of kairos and the ratios derived by empirical scientific means.

Pythagoras had a mythical apprehension of the naming of things. Thus to him the number names had significance, the main part of their significance was in the geometrical arithmoi. However this made number subject to geometry. When Pythagoras found the relationships of the muse acting on a measuring chord, he found that they were whole monads in the proportions. This immediately raised the status of the number names to a divine significance, and made them prior to geometry. Thus Pythagoras had an epiphany, in which the muse communicated to him the absolute primacy of numbers, and thus arithmatic, and within numbers the whole numbers were fundamental. He belieed he was shown that all things would have a ratio expressible in whole numbers. The number names became principles, abstracted from any earthly or (geometrical) significance and given a divine provenance and authority. That it seems has never been challenged fundamentally to this day.

However, i frequently challenge it. giving the preeminence to spaciometry and the human interaction with space through the Logos Summetria Response. I would modify the Logos to the Logos-Kairos summetria response, or even the logos Kairos Response, since Kairos implies a summetria.

Although it may seem that number took primacy, this in no way endorses the primacy of the numberline concept a later idea pioneered by Wallis an Dedekind. For pythagoras and all geometers ratio and proportion are the fundamental role of number, tha is numbers apply to comparisons and measurements. The measurinf cord was an instrument t provide numbrs, based on monads. Numbers did not exist without some monad, some divine inspiration of a unit, and that could be anything. Thus number was a principle and it was possible to divine the principles involved in everyday forms and relationships. In this sense information was encoded in forms and relationships, and this formed the the proper study of the more esoteric pythagoreans.

One can easily see , now, how many scientific, philosophic, metaphysic, and geometric, psychic, psychotherapeutic principles may be laid at the feet of pythagoras. Pythagoras did one thing with his measuring chord of music, he measured how the muse adjusted the natural sounds in a string to make pleasant music. He made one significant change in the natural tonal pattern within an octave to produce our familiar tonic scale, but more fundamentally he added an 8th string. This alone made him more than conqueror of nations! Establishing the octave as the fundamental tonal structure, he was able to show audibly the fractal structure of tones used by the muse. He was able to show the infinite iteration of the octave from before hearing to above hearing. He showed how tones outside the octave had counterparts within the octave. He showed how the octave contained everything in proportion that was needed for pleasant and affective music. It also contained horrendous discord, but the ratios of whole numbers were the secret code that kept you in tune with the muse.

The notion of Kairos informed Nwton, and everyone who seeks universal laws: they seek the correct proportionality to derive cosmos out of chaos. The success of the Newtonian philosophy and proportioning made many scientists forget that order is being picked out of chaos, that a mechanical universe is not all there is, but rather a space in which a kairos derives order, and not just any order, but a fractal order based on the music of the monochord.

When i began my research into the fractal foundations of mathematics, it was to reconceive modern mathematics in the light of fractal geometry. In many ways that has been a refiguring of Pythagorean philosophy, reinterpreting and apprehending misleading comprehensions of secret knowledge from the past. All in all Pythagoras left us a great legacy, but his conclusions, like mine are open to question and reevaluation.
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