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

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 !

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.

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.

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.

Grassmann gives me hope, but i am still troubled.

The notion of a reference frame is only a basis for constructing a reference frame system, It is the reference frame system that holds the information about reality in invariant forms. In other words we have a system of reference frames and the invariant interference of their influence on our subjective experience is what we recognise or regard as eternal verity, truth, reality. Grassmann's linear algebra method and praxis makes it so straightforward that it is hard to believe that we have had to struggle to comprehend it. It is an apprehension of the interaction i have with space, the relationship i have in deriving a model (FS) of notFS from interacting with notFS.

It is at once far more general than any other conception while being far more specific. It convolutes with the wau i have to perceive things remarkably well, so that for the first time i feel i can notate my experience concisely but transparently. The reason why i believe is that Grassmann looked at the convolution and realised that we had a way of assessing it through combinatorics. Doing the work made clear to him the limits of the applicability of his conceptio. He then designed rules to stay safely within these limits and in that way provided us with a map of a land he had already explored in general . We are now able as he attempted , to fill in the detail.

That it is a map and not reality, and not the last word on reality, nor even the most sensitive map is shown by Cliffords immediate generalisation of Grassmann into Bi quaternions. linking immediatelyy Graswsmann's "map" with Hamilton Sketch map. I put it that way because Hamilton derived an abstract notion and applied it as much as he could to geometry, whereas Grassman Started with the full Euclidean geometry and "streched it". And before doing it he worked out all the ways he could sensibly stretch it. Thus his notion of a "vector" is indeed a elastic geometrical fo, a dynamic euclidean form. What his genius is was to take the "found" or inherent vectors in Euclidean geometry, the natural ones geometers us without realising and to make them explicit and accessible. by doing that he revealed the language we use when we manipulate space as being notable: that is we can denote a spatial manipulation action or intention explicitly and so write down precisely what we are doing or intend to do concisely and what that results in. "Numbers" or rather trig ratios then become coefficient or adjectives and adverbs of these new verbs or s we call them vectors.

This is Grassmann's conception, and this is why he was not understood in his time. His 'Zweig" was in fact not a twig or a branch but a foundational trunk of geometrical thinking specifically but logical and linguistic thinking generally.

For example it took millenia to end up with 26 letters in the english alphabet, but combinatorially and including punctuation and rules of grammar we can describe every aspect of our experiential continuum. However , we still know that this facility as flexible and fluid and essential as it is is still only providing approximate communication information, and different rules provide different maps of the same thing, ie the different languages!

The link to language is fundamental and deep, but as i have explained the Logos Summetria response undelies it all, so that it is not what is a subset of what but rather what is the relativistic structural foundation our neurology supplies in which we may begin to compare and contrast and distinguish? For it is that which leads to both language and geometrical mathematics, or as i refer to it spaciometry.

I have discussed before that in a reference frame parallelism is the notion of independent reference , not orthogonality, which carries the same idea but relates it to our perpendicularity sensory organ: it senses perpendicularity. Cartesian coordinates rely on the independence of the measurements and it is was natural to think that was related to the unique orthogonality of the 3 axes, but in fact Grassmann demonstrates powerfully that it is in fact parallelism that confers independence, and thus any reference frame can be established using that as an ordering rule for measurement. Thus we establish any number of non parallel directions, take the point of intersection of these orientations as a centre of rotation and an origin, and then order the measurements by rotation writing them down in an ordered set. This set of n measurements is now a generalised coordinate system relative to that centre of rotation and origin, We can then make independent measurement anywhere in space by measuring parallel to any or all of these reference orientations

This is why Grassmann is so straight forward, and why his method is so general, and of course there are geometrical relationships based on trigonometric ratios between all of the orientations.

Now the ordering of the set of coefficients/coordinates of the orientations actually follows a locus. In the plane it is a closed boundary,usually but not always a circle. It could also be an open boundary like a spiral and the interplay between the 2 is precisely what defines modularity. Now in 3d space we need 2 things a surface closed or open that surrounds the centre of rotation/origin and a path closed or open on that surface. Thus we could have a spheroidal surface on which a circle is drawn through the poles. but in that case we need other such loci to ensure we can reference all of space. The choice then is usually either great circles hat intersect at the poles ,ie radials from the poles or parallel circles parallel to a reference great circle. These choices section the coordinates into ordered sets which themselves are relatively ordered reflecting the geometry and the reference system chosen.

There is another alternative set of loci and that is the set of trochoids. One in particular the cornu spiral or the spherical helix going from pole o pole provides the possibility of an infinite set as a coordinate system: that is one long string of coefficients as it orders all the radials. Of course it seems clear that we could only ever approximate such system, and it remains to be shown that a single helix could order all the radials. Peanno's curve could do it and so that is another possible locus for a coordinate system.

Its funny, but Peanno was one of the first mathematicians to really grasp what Grassmann was expositing.

So parallelism and postulate 5 in Euclid really have a bearing on Reference systems that hold information about reality.

The minimum such system consists of my subjective gnomon and a local gnomon/ Although a reference frame requires a second arbitrarily small region to establish an axis in point of fact reference frames are mutual things and that is what is meant by relativity: my subjective gnomon and the local gnomon could equally well be referenced by each other,BUT the subjective gnomon is unique in that every reference gnomon relates to it and has a relative position that moves as it moves. This is the problem of parallax and incidentally why i think Einstein has got part of his theory wrong because he excludes the subjective gnomon. I tend to use gnomon because it relates directly to euclid and indeed carries the implication of trigonometric ratios in the reference frame.

So the problem with postulate 5 is parallax and perspective. I am sure that Euclid had not intended to say anything about infinity, but rather about an iterative process, Thus if the local measurement is that the lines are parallel that was good enough, He was well aware that lines which are locally parallel appear to meet in a point in the distance, his point was that they may meet, but they do not cross. Thus postulate 5 really states that any lines that meet bur do not cross are parallel. This applies equally well on a spherical surface and with regard to a straigtht line and a curve. The notion of parallel like the notion of an"angle" is much more general in Euclid than we are commonly taught. The tangent was devised just for this purpose of dealing with parallelism in curves, and the behaviour of tangents define the limit of parallelism. Thus if a curve has a tangent that never crosses a another line or tangent then that curve is parallel at the pont of that tangent.

The sophistication of the thought of the greek fathers is often underestimated. Very very often Newton borrowed ideas directly from the greeks in his formulation of the calculus, and indeed the calculus is applied trigonometry, and extension to trigonometry, just as Grassmann's Ausdehnungen are his elastic forms,stretchable geometry,bendable Euclid!

Thus the issue is that to apprehend reality we need not just one vector reference frame but a system of vector reference frames which define the notion of relativity and, most important this ,include the effect of the subjective gnomon.

Grassmann seems to me to be the best and safest way forward. As Clifford has combined it with Hamilton it seems Clifford algebras will light the path.However the light shines more brightly from Grassmann.