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.