Euclid in his Common inferences (ennoia) rehearses the concepts of duality as they apply to objects. But the Isis notion is of greater applicability than just objects. It pervades all his thinking, and the mind of his fellow greeks in Alexandria. More particularly it is applied to processes, and in fact this is an understated inference found in Euclid, but alluded to perhaps only in the constructions.
The inference is : if two processes are dual then the results are dual. Without this inference it is impossible to say that the line of dual points, defined as straight, bisects the circle in 2, and to demonstrate its veracity in every circle, and indeed to show that a diameter cuts the circle in 2. Finally to show that the ratio of the circumference to the radius is the same in every circle, and thus to found the notion of similarity of Schema using straight lines and arcs.
This inference runs deeper. For it means that underlying Euclid's teaching material and its Rhythmical and poetical structure is a dual which consists in the underpinning processes that construct the objects and schemas of every proposition. That is to say , for every proposition in Euclid, and every definition, there is a dual form of it in which the minutiae of the processes involved in constructing and establishing the notion are rigorously and clearly laid out. In this way for example the line ab is the result of the process of drawing a pencil from a to b against a straight edge. This detail of the process is the dual of the resultant image on the page for example.
I will not labour the point, but the very computer you are using is a grand example of dual proceeses. Thus whwn i click my mouse and draw the line ab on screen, it is the dual proceeses programmed within the application that realise it on the screen.
This means that the programming platform that i use every, day and the programming community have utilised, is a process algebra to enable a mathematical algebra to be placed on screen.
Now clearly a process algebra, like any algebra can be as complex as one likes, bu i wish to remain close to the process algebra we call vector algebra, and the combinatorial process and theory i hope to demonstrate as a rich process algebra which underpins all vector and mathematical algebras.
Suppose there is a motion A which acts on a space b so that A(b) is the set of motion sequents Ai(b), where i is acted on by a successor function S(i) =i+1, which drives the motion A. There is a Frame F(x,y,z) with a method MF which applies it to each motion sequent .
Suppose a motion B which acts on c so that B(c) is the set of motion sequents Bi(c), where i is acted on by a successor function S(i) =i+1, which drives the motion, then i can combine the motions A and B in a number of ways. I can join them by i, so the motion run synchronously. They can run in this way , disjointly, or combined, that is either as A(b) Û B(c) or A(b) n B(c) determined by S(i) =i+1 whether synchronous or sequential. Or it could be arranged by the MFwhich directs the use of the F(x,y,z) which gives a result from each motion sequent, and these can be used to organise the combination, either as relative to each other or as relative to some common object. that is MF(A(b)) Û MF(B(c)), determined by the method MF
A scene is built up by such a combination of motion sequents. The collective sequents with their respective succesor functions, all relative to each other, represent the combination of motion sequents Scene(A,B,C…Z) the set of combined motion seQuents Scenei(A,B,C..Z), Where A(b)….Z(&) are motions in the scene combined in some way.
So i begin to notate aspects of the subjective processing we all do, and still we see that i have choice with regard to how i combine the motion sequents. It is also clear that the motion sequents are controlled by the successor function, but no explanation has been given here about how the motion sequents are derived into the sets for the successor function to run through, or how this method arises.
Thus we may posit a meta method that applies some processes to achieve this. And then of course the very same situation relates to the meta method, and we acknowledge the fractal nature of the tautologies involved in setting up a level of description.
Returning to this state level , i will therefore need to clarify the elements that the meta method controls , and seek to establish their relational structure and their combinatorial theory. For me, this has been substantially done by Euclid, Grassmann in particular, and Hamilton. While Newton has a great contribution o make, i have yet to find a text by him on the Foundations of Algebraic combinatorics.
Shifting levels now, i draw attention to programming languages, particularly HTML% and XML. These notations are called languages because i can use them to communicate very precisely with an intelligent processor or another programmer who studies the language.
However, the language is a series of state signals, that precisely initialise and direct the state of a machine that runs automatically and has cybernetic feedback and feed forward loops. Thus we have a motion that acts on a region in the machine which is driven automatically by a successor function. We have a frame that physically stores the output for each successor driven motion sequent, and it too is driven by some subroutine that performs a method and stores the output in a sequence driven by its own version of the successor function. The store too has a method for processing the stored information, again driven by its own successor function.
The complexity of the action of doing all this processing is hidden by a simple device that accesses only the output stores for all the processes. and processes their sequential data into a visual form, by means of a complex signal processing decoder.
Underlying this motion and apparent motion is the ceaseless successor function and its various cousins, and by precisely keeping these all in step an amazing machine automatically processes all manner of signal intensities.
Thus every aspect of this process algebra relies on the essential motion of a successor function or a collection of related successor functions. Each successor function is locked in to a method, and that method controls a flowing substance, or just space in motion by attracting or repelling different types of space, condensing or evaporating. space in some combinatorial sequence dependent on the surrounding motion structure and relative motion.
In this view, again, the attributes we sek and observe emerge from the larger scale structures which depend on the organisation of simple dynamic relationships. But some successor function is essential to perceive this organisation at my level of consciousness.
This raises the intriguing connection between successor function and perception. There is not just one sucessor function in this analysis but a whole family that are fractally linked as well as freely related. What puts the methods together in such a way as to create complex conscious response as opposed to conformaion of molecular structure, is an unanswerable question. We respond to it with faith
The faith in the evolutionary possibikity of complex motions arriving at a complex arrangement of regions which becomes sturdy enough to maintain a consistent behaviour, with this process repeated at every level resulting in bacterial life at the phyto plankton level, which reproduces, and there is overwhelming evidence for this fractal organisation in evolution; or the faith in some higher level of fractal organised immateriality that interfered with the material world to shape its processes.
Both faiths are valid and not mutually exclusive. Fractal principles allow them to sit nicely together a different scale levels.
So the collection of Successor functions are key to the process algebra, especially as it is a model of subjective processing as it acts on space. As it is a model, it is worth pointing ou that the successor function is an inadequate model of motion , it models sequential relation only. It is not time, because the time measured by a clock or pendulum is a motion. The successor function moves through the motion sequents that describe that motion. To call it time introduces confusion in the model / However, time emerges from the model through the agency of the successor functions.
Some may perceive n avoided tautology in this way of describing things, and they would be correct. The tautology however is not avoided, it is acknowledged and used as the basis o develop a better notation. Even in our thinking we need these cycles of reference to establish a stable base for further development. WE have to accept that we do not really know anything beyond a certain system of referents, and that is probably good enough for most of us.