My first response to experience is to order. I guess this is an adjugate part of a much more general organising approhn. But I do not organise globally, At first I organise locally and internally. From there I may gradulally extend my reach and range.
What is my favourite or cultural way of ordering? It is , in common with birds, singing a tune. We call it counting. We have developed two standard tunes: the alphabet, which is a very much honed down syllabary; and the numbers, which is a free wheeling symbolic labelling system derived from the syllabaries also.
Why do we order?
I can trace this innate response back to the evolutionary advantage given by catalytic biochemical reactions and interactions in an environmental substrate. What this means is that the brute forces of the elecctromagnetic fields are mitigated by the rotational fractals they develop. Like transducers they dapper down the excesses of curvilinear force fields, twistorque accelerations, making them into gently expanding waves of fractal vortices. This is fractal electromagnetic vorticular space, as I experience it, the curvature innate within it I experience as conic sectional gravitational motion to first order approximation, and yet to be fully explored, complex logarithmic motion as an exact rule.( an idea intimated by sir Roger Cotes)
This gentler scenario of bounded storm cells we call particulate density of space, nevertheless interacts in the Ed Lorenzian aperiodic water clock system of fractal quantum space, from the Planck length scale upwards.
At the particular biochemical level we have data access to the interaction is sticky and colloidal. This is due to the molecular dispositions forming substrates against which different sized molecular compounds slide or kinetic ally interact. We observe catalytic preferences that fractally distribute certain types of reactions, which in synthesis display to us recognise able patterns. These patterns are in the main trochoidal in nature, and dynamic. They display a spherical wave like disposition.
The catalytic substrate activity leads to a sequential process, which when synchronous and spatially distributed appears dispersive and wavelike. In the great scheme of things these radial wave like phenomena are only experience in one part of the wave front, and consequently I dimension this phenomenon as a linear sequence of change. In response I emit a radial wave but I produce it sequentially I" believe". In point of fact my sensory mesh and mesh of actuators produces a comparative radial response. It is just my belief system that causes me to think and describe these events as lineal, when they are spatially disposed.
So I count, because the catalytic sequence of events is aperiodically and sequential. But it is not lineal, it is iterative and cyclical.
Over time the song has been tied to certain physical and boundaries objects, but the most useful has been the cord or string, which developed into the tape measure. The fractal disposition of marks on this measuring cord were associated with the song, and various sub intervals were coven new verse like descriptions. This measuring line was destined to take on an uncalled for role as the actual definition of number itself!
Gone were the free flowing trills of the bird! Now staid funerary music and rote "rounds" we're put in its place, and the quicksilver flitting mind was forced into a dank cage!
Outside the cage were the Alice in Wonderland imaginaries. Would they save the day?
They turned out to hold the key to make the cage shrink down to everso small, but the cage had one last trick! Like the wicked witch of the west in the wizard of oz a spell was cast on all the imaginaries. Henceforward they would be called numbers!
Such songs and lyrics as are to be had in the so called complex vector notation defy the descriptive title number!
All the golden birds will break free and trill when this spell is broken!
In the formulation of the couples and the quaternions Hamilton is careful to identify each label for what it is, and each conjunctive or adjugative relation for what it is. Consequently the labell 3 usually called a coefficient in the combinatorial form, is in fact not a coefficient, but part of a complex label. The label carries information about the relative disposition of the step to which it is attached. The step itself is marked by a label, but this label is to identify a metron. A metron is not free, because it must " bind " to it's context and define the conjugacy relation it evokes and the adjugates it contains. But by labelling the metron in this way one may roam among all contextualised metrons that apply.
The question arises? Can a function have complex coefficients! The flat answer is no!
However we may define a metaphor to a function in which the complex notation plays a role with a suitable metron. It has to be put in this way because confusion reigns when it is not possible to sing the song of ordered events.
We may establish a framework which allows us to use the song as a way of describing the elements in an array in an ordered fractal pattern. This order reflects the spatial and radial disposition of sequences in space, and this is how we make use of the complex labels to describe our reality.
This is why, in the end, the " imaginary " have triumphed, because they describe our real experience in contra distinction to the "Doctors" of the old board of Mathematicians!
Hamilton, when he chose to embark on his defence of the imaginaries carefully chose time as his medium of is course, for all agreed that time had but one property and that was that it progressed! By the time Hamilton finished with it it was clear that this was no longer a tenable description of time! But in fact Hamilton had fallen into a trap of smilies. Time progresses in analogy to spatial motion or flow, and thus his description in fact is the abstraction from spatial mechanics of what he wished to discourse upon!. Nevertheless, the main attribute he sought to avoid as problematical to a clear and simple development was orientation. This is why he bowed to Grassmann as his superior, because grassmann discussed the whole gamut of our intellectual apprehension of space including time and orientation.
Nevertheless, the brute force uncovering of the quaternions is no less remarkable despite his twisting of the very roots of unity to his desire! Fortunately it is not too bad an outcome amongst the infinite fractal possibilities. The loss of commutativity is a small price to pay.
The algebraic marking and structuring around a few metrons of the infinite orientations in space is indeed a great feat no matter how imperfect! Of course, yo be able to give an algorithm to mark them in as many distinctions as we please is the triumph that is accorded to grassmann and to him alone. This did not sit well with Gauss, who clearly understood the magnitude of what grassmann had attempted to do more than even grassmann himself! Still, events have taken their course, and now I must seek to delve deeper into Grassmann's mind.