Grassmann next moves into the notion of processing motion sequents. The idea "Stoss" is the root of it for him, while for me it was Hamilton's "pure time" analogy. In any case the idea of morion sequents processing is functionally different. When you aggregate motion sequents you get a motion, as in the case of a film, or a video frame aggregation. The film is a subjective experience when the frames are aggregated in a special way. The rules of the way frames are aggregated are called editing, and in editing the meet and the join of film frames is crucially important.
Now all of this is am analogy for a much wider application of motion sequents: motion sequents as "frames" of reality, of real experiential continuum as opposed to, but hardly distinguishable, dream experiences.
This processing analogy holds good for all motion sequent processing and is the general basis for explaining Grassmann's more philosophical explanation of Ausdehnungs groesse, their combinations and terminology.
The first thing to realise is that combinatorics, editing and constructing a reality of form in motion are all related, and distinguishable. The terminology to distinguish is important, and subject related. Sometimes different terms are used for the same process, and i am going to examine meet and join as examples of this. suffice it to say that contiguity and continuity play n important role in distinguishing, and out of the frame of mathematical musings, have an immediate and natural meaning that carries through into any subject. The distinction suffices to support thr distinction between multiplication and addition, and at this level entirely explains the subjective difference , it also explains why in general
ab =-ba i a functional terminology of process that has sense and in particular applies to Strecken.
Suppose i take an object a, that object actually is an output from a subjective processing sequence, and is called a compass multivector network, which as a related set of vectors is the equivalent of an Ausdehnungsgroesse. The spatial position relative to every other object b, c, d, e, … is recorded by a common vector or Strecken. The objects may actually be bonded by a path or some other relation, physical or otherwise. Using transformatiional geometry i can describe relationships between symbols as objects.
Transformational geometry deals with affine transformations: rotations,translations and reflections. In addition to this i would need to add notions of meet and join.
Suppose i move a to meet b a->b = a+b
then the relative motion is to b
If i move b to meet a a<-b =a+b
Then the relative motion is to a.
Thus meeting obscures two different cases of relative motion.
Suppose a rotates around b with b as a centre of rotation a<?>b or a<¿>b, with ? and ¿ indicating contra rotations, then i have an action on a relative to b wherever a and b are relative to each other.
Thus a<?>b(a….b)= b….a means the action has rotated a about b by a half rotation
a<¿>b(a….b)= b….a means the action has rotated a about b by a half rotation but in the contra direction.
Thus half rotation obscures 2 different cases of relative action.
To be rigorous i would include other cases, particularly the rotation of a relative to itself, so that i do not have to distinguish just yet "a" from "upside down a" or any other variant.
If a meets b and then i perform a rotation on the meet relative to one a<¿>b(a+b)= b+a
a<?>b(a+b)= b+a then the meet is not the same
Because the relativities are now contra each other by rotation. a moves to b but the resulting meet is different by rotation.
Thus if i wish to write a+b = -(b+a), the = must be defined as "gives the same meet as" , and the minus must be defined as "rotate second relative to the first by a half circle", or "rotate relative to the centre of relativity".
There is another consideration: if the centre of rotation is not specified, and a process of half circle rotations takes place around the 2 possible centres arbitrarily, then the meet(a+b) may have a translation relative to some third object c. Thus the motion sequents have to consider inter relationships with all objects to have universal application.
The join is now a stronger form of the meet, in which the objects that meet are "glued together" to form a new object.
Thus a^b = joined(a+b)
This affects everything. Although i can still rotatate relative to a or b it is now a^b that has to be considered as rotating, and "upside down a" cannot be avoided and a<?>b(a^b) is null.
However there is a weaker case which we generally allow and that is a+b rotating relative to a or b. In this case upside down a can be reversed by a half rotation relative to itself. The outcome of allowing this type of system of rotations is the same as just rotating one object relative to the other, so why do it? The reason is illogical , but there, to make writing notation easier! Consequently we start with a join, loosen it to a meet, rotate it as a join , then rotate the individual objects relative to themselves , and then rejoin.
Alternatively w e may start with a join, rotate the join, loosen the result to a meet, rotate the individual objects relative to themselves and rejoin.
There are other alternative systems, but the end result is the same. Thus our notation loses coherence with real objects and we have to inspect the results of manipulation of notation against actions on objects in space.
If we regard this symbolically we are setting up the superstructure of relationships. A building analogy helps to clarify: we first establish the blue prints before we take action. ab =-ba is such a blue print instruction. When these two instructions are established, there combination is defined to lead to a null result. Thus moving the object a to the place of object b is the same as moving object b to the place of object a, unless b rotates relative to a by a half circle, in which case the spatial motion results in the opposite or contra motion with a different outcome. Using a third object as a common relative we can then define this resulting situation as being contra to each other, and relative to this third object, performing both actions sequentially results in a null action
In every way, by hitting upon the relationship in 3 points and 3 Strecken Grassmann had a natural and sturdy platform to develop his general construction blueprints for the Ausdehnungslehre, in which every important relationship was laid out clearly on the table before him. Hamilton, in his couples laid out the deductive relationships between 2 moments of time, and more, but would have missed this invariance because his scheme was rooted in objective relationships, not subjective invariants, ie analogous forms that could hold more than one set of "meanings" at the same time.
The importance of triangles, the so called triangle numbers and the tetractys were known to the Pythagoreans. To impeach their intelligence as mystical meanderings in this matter shows a misunderstanding of the keen empirical sensibility hey possessed. These relations, carefully worked on by Grassmann, establish the importance of having a regard to the subjective-objective interface between man and space, and the magic within the gematria of many mystical philosophies, from the Yi Ching, Through the Brahmaputsidhanta,and the Pythagorean Theurgy to modern Quantum and String theory.
There are 2 other aspects of an "object"/symbol to consider, now this rough plan has been laid out. The first is the internal relativity of an object, in compass multivector terms the subjective experience of the network for an object, thus its own relative right and left, up and down etc; The second is the subjective memory of an object, in motion sequent terms its frames as sequenced in the subjective memory, Both these aspects have direct bearing on processing relevant information, ie extracting and abstracting what is deemed relevant or important.
The consequence of considering these things is the increase in complexity, but also the realisation that the accepted terminology is barely accurate, but it is sufficient for the "wise" one to achieve a limited goal: the relative position of any assigned point in space after an action assuming continuity of motion: the relative intensity of any volume in space assuming a continuous variation.
No object should be considered stationary either rotationally or translationally. Thus a static form is only relatively static to a certain scale, determined by the output processing accuracy. It often happens that as we increase accuracy, ie magnify the output, that local variations become apprehensible. Thus what one apprehends as one object may resolve itself into a multiple form of objects that meet or join, and are in relative motion. The stability of the system, dynamic or static, is in major part what contributed to it being apprehensible as single form.
The ausdehnungsgroesse and the compass multivector networks are applicable to this fractal state of affairs.