Because we do not know what Arithmoi are; because we do not recognise the Gnomonic "Algebra" in Euclid's Stoikeioon(Stoikeia) book 2, that is the gnomonic rhetoric for thought; that is, because the connection between books 2 and 7 are obscured to us: WE have misunderstood the Pythagorean school of thought.
The fundamental subjective process we engage in is distinguishing space by conjugation. This is done unconsciously through our sensors, which necessarily must partition our experience of space and sequence our apprehension of it,too.
Consider the lowly mosaic: an epipedos on the floor of some house or courtyard, a humble epiphaneia upon which sunlight shines projecting its shadows of objects like dynamic vectors, onto every part of that mosaic, that sperepipedos. In this illumination, as the shadows play may your mind also be illuminated by the realisation that now you can do gematria, or geometry; that now the movement s of the stars and sun draw out for you a shaddow world of their dynamics, and now you may hope to do Astronomy!
Thusly did pythagoras convey the mystery of the Spheres, the balls of fiery light which cast shadows on the mosaic floor, and inspired the mind as does the Musai.
But what of music? Into every space thus katametresei was placed, and that measuring toll, the mosaic gave to the mind divine proportions, from which Pythagoras using a single chord, it is said. unraveled the harmonies of the Musai. the divine proportions of each "blessed" note, embedded in his single cord stretched taught. And also he heard the sound as the proportions of the humble mosaic , which the sunlight projected his string upon, learning how the same mosaic pebble gave a different sound as the sun moved round, because he had to move upon the string in accordance with the shadow cast!
Without this humble mosaic , embedded in space, neither music nor Gematria could be apprehended, nor the motions of the sun, moon and stars recorded, and thus tyme created in our tables of Babylon!
But for each Mosaic there was a different count; the form of it shaped the apprehended truth. Thus as time progressed it was decided upon, that certain mosaic forms best held certain truths, and into them all he triangle went the most, but none could equal the perfection of that disc which the sphere cast as its shadow on the epiphaneia. Thus the Pythagorean lore was what the Grassmann and Apollonius perceived from Euclid as a Formenlehre. a theory of forms or Ideas that the inner eye might light upon.
This Formenlehre held still, an instantaneous shadow of a slowly changing one, for all as Herakleitos did observe flowed like a river through our apprehending eyes, slipped through our comprehending hands like the sands in the hour glass. Thus ideas were ideal, complete and to themselves , instantaneously removed from that dynamic motion whic engendered them, that we might apprehend an instant of an everchanging truth; and so learnig to move on to that which is constant in our experience, that being change!
But all of this change in shadows cast, stand out starkly in the mosaic frame! which being rigid, fixed and constant as we can best determine, quickly refers to us any changing shadow, any flow. Thus the embedding of our mosaic, our arithmos in the rock, not on sand, provides for us eternal truths as may be.
The question is: if some forms tessellate our space, while others leave gaps that only smaller and yet smaller scaled forms will fill, what is the truth that this phenomenon reveals about the formalisms of our minds?
The many forms of which our mosaics may be made give us the notion of our Manifold, die Mannig faltigkeit, die Vielenfach from which we may compose our best knowledge of the world in an instant. But the Arithmos ar not so formed. Every part of them is but the same, and is of itself a Monas called. The monad being a grammatical version of the predicate form, and rightly as they are always employed by us in measuring or counting.
The favourite monad was the square, derived from the inner properties of the disc , which itself is but a shadow of the sphere. But Escher shows how many and varied may our monads be even if uniform, and how they transform. from parallelogram to geese and back again.
The fundamental transformation between parallel lines or concentric parallels, by symmetry give us as if an instantaneous form, belieing their dynamic transformative role. And so obscuring for a ime that all is moving in a dynamic proportion relative to our fixed embedded Arithmoi.
The DesCartes happens along on Herons work, on Bombelli's innovative rules, and steals the show from Apollonius! While John Wallis places the final nail into the origin of the coordinates! And by Regiomantus all is lost in formulae, although the triangle comes out emperor of all.
No one sees, no one know the humble mosaic that undepins it all, The Arithmos that gives eternal rules, That gives us such comforting stability in our thought that we declare einstein to be mad to muck about with space and time, unaware that he has just renamed the humble Arithmos and called it Aether or Etheric in its stead, raising it above the ground into the very fabric of our space.
Still it is as the lowly Arithmos it is best known as Grassmann, Hermann will testify anon. For then we can embedd all forms together in an algebra so simple it takes away our breath and soothes are aching brows.
If we but take any rectilineal form, of however many sides and slip them parallel to themselves to some common point, this pont being termed the Origin gives us the general Lagrangian reference frame by the way of Wallis. And taking any conic form expanding and diminishing it from the central axis of the cones, making concentric conics enables us to form the polar coordinate reference frame by rays projecting from this central core. And combining the two gives us the most general vorticular frames of reference that we may need.
The Shunyasutras fall like wriggling snakes upon the Mosaic floor, These "natural and violent curves, sometimes smooth like conics have raken time to be defined. And still are: for they be Splines, or roulettes called, but betimes, which i prefer, Trochoids! These Trochoids that i perceive foreshadowed on an Arithmos net, travesing through the elements of that array, to me the essence of the arbitrary motion do betray.
None yet have understood this fine Arithmos floor, topologically as diverse as any torus may or may not be, covered by any known or studied form. Nor yet have they its solidity apprehended, whether in crystal simplices portrayed, or cubed or blocked as our ancestors engineered in space.
But Benoit Mandelbrot in his Fractal geometry has come upon is nonuniformity, and Hermann Grassmann in his philosophy has captured it notationally, And Hamilton sir william Rowan has it rotations captured in Quaternions, Eight lines in this Arithmos array, if the truth be known.
So what are all these lines and curves locii? They are the bounded spaces , the topoi of each elemental part perceived together as in a motion path, as if drawn by hand or in a moment impressed. And whatever shape each element may be it matters not, to that which is perceived, for it is extracted in the mind into its most perfec form, else put their by the Gods! You decide this old Socratic saw, that Plato by and by has made a Law!
And so our analysis is complete, for into points each element is sundered, giving us the most ephemeral perfection, that ghosting image as all disappears in a flash! These are Newtons evanescent forms , his ultimate ratios in a fluxion born, or mayhap dying away. And Einstein, as if a fool had out of nothing such things appear, but really he meant to say the aether but could not, the Aether being the finest Arithmos of All.
We see before our very eyes
Hold in our clutching hands
Lift to our eager ears
The Arithmoi fulfilled
The mosaics that once instances portrayed
Now before us, who we are is evidently displayed,
In an Iphone
Or some other Irons Apple Gear.