Combination, Comparison and Construction.
Wednesday, January 18, 2012 11:07:22 AM
The experience delivers me a instantaneous given apportionment, a scattering of semeia which i immediately process at different levels to determine intensity, form and rotational motion, This set of outputs then becomes the basis for further processing in which combintorial relationships are determined as rotational or translational dynamic or static forms, with the status of a closed boundary experience or an upen boundary experience.
The combinatorial relationships with the boundary information is further pocessed to deliver a comparison of these relations in boundary rotational terms both static and dynamic. This comparison leads to distinctions called by the greeks, and the western world thereafter "Logos". The apportionment within which this comparison takes place dynamically is called "Kairos".
These are not magnitudinal comparisons, just comparisons of difference and equality.
Thus the combinatorial structures, relationships and arrangements underlie and support the comparative process that is the ground for language development, and the simultaneous development of the notion of magnitude.
The logos and the Kairos enable one to symbolise internally , subjectively using any of the sensory networks any particular combinatorial relationship we focus attention on, thus distinguishing the relationship from the background apportionment. In a wonderful facility of consciousness one is able to subjectively process this particular combination in many alternative relativities, based on the combinatorial information, some of which are spatial, the others are intensity.
Once one has the combinatorial relativities and relationships sufficiently processed one may subjectively elect to emphasise one magnitudinal form above others, to be used as a rotational constant for measurement by a monas/metron. The process of measurement is a complex relational and combinatorial exercise involving rotational motion.
The pre-comparative combinatorial processing and comparison has already distinguished closed forms and motions from open, and thus open motions are part forms of closed ones, let us say. Thus closed motions are analysed into parts and all motions may be characterised as some form of closed motion or rotational motion. For example even the backward and forward motion of train on a track can be considered as a closed motion, and a defunct type of rotational motion, constrained by a track.
Thus comparison shifts scale to utilise one particular form as a metron/monas. This comparison gives rise to special logoi whch we call counting names.
The relationships between the combinatorial formations under this process of measurement gives rise to new and more measurement driven terminology. The terminology changes, not the combinatorial structure in its static or dynamic form.
Thus one may describe a given combinatorial form in a number of different but equally valid comparative terms, with more or less conformity to the actual structure and its combinatorial relationships.
The introduction of monas or metron allows the notion of a multiple form, and from this Pythagoras, and particularly Euclid was able to lay down rules and laws of combination, comparison and calculation of proportions. The laws of combination involve construction from smaller elements or types of elements. It was not until Grassmann that these elements were considered as a separate field of study from the forms they traced out.
All experienced the tracing out and the infilling of form, few recognised the invariant nature of these dynamic tracing motions or there relationship to 3. By taking this tack and exploring it fully Grassmann was able to recast Euclid in the modern mind and to preserve the real power of Euclid's teaching material, which over the millenia had slowly ossified into a static form in mens minds.
Grassmann did not get it all right. For example he mixed up the plus and the minus sign with addition and subtraction, which of course we are all taught to do, and he still felt multiplication was a different operation. In fact multiplication is a convention, but by separating out the combinatorial options Grassmann did a great service in revealing what is actually going on. We use the terms number and +/- and multiplication at our peril. It helps little to recognise these as symbols to be defined later by the expositor. The combinatorial relationships of a form and of motions are what one needs to keep clear in mind.
Of motion there are 2 sorts: closed and open or part of a closed motion, We may safely translate this to rotational and part rotational motion with certain constraints. From these motions one may trace all the elements required to construct and combinatorially arrive at a form or dynamic motion.
The Ausdehnungs groesse lay out in terminology this idea, that any form is constructed from combinations of subforms, and each subform is distinguishable by its rotational relativity in a network of combined subforms. We need only relative information and multiple information to characterise any multiple form. The only combinatorial actions allowed are rotation about a centre, translation relative to that centre and reflection in that centre. Once the structure is established or constructed calculations may then be done.
There is no action which is multiplication. One may arrive at multiplication by dividing forms into multiple parts, or by gathering together like forms from wherever this may be, or by composing a copy of a form from its fundamental elemens, over and over, requiring a supply of copies of these elements from wherever they may be found. The compounding of ingredients into a substance that is placed into a mold is a refined version of this method.
Thus when we multiply, it is by initiating some production process, some search and gather process, or some division process. 2*1 means nothing if it does not mean find, gather, make, form, divide some thing to give 2 like things.
When we say area is length multiplied by breadth, we state nonsense unless we mean we gather like units into a multipleform that is governed by the units arrangement in a gnomonic form. The two sides of the gnomon are divided by the sides of the unit to set the arrangement in order, in rows or column of contiguous units. The number of units is a multiple of the number of units in a row or column by construction. The length and breadth are factors of the multiple form. What one can verily say is the factors of the multiple form is the lengthwise units and the breadthwise units.
So when Grassmann suggests that Srecken can multiply each other, he is talking nonsese. However what he fumbles onto is the combinatorial structure of joining two Strecken, and considering the join to be the edges of a parallelogram, Thus he has not multiplied, he has formed a shape by combinatorics.
Now Grassman follows the idea of multiplication by forming a new shape which has 3 strecken and then combining them in pairs, making 2 overlapping paallelograms, and one given parallelogram. How are these 3 related?
Combinatorially they are related through the given parallelogram, by forming a strecken triangle combination, then the parallelograms are linked by a relationship of the three Strecken.
c + a = b or ca = b
lb = lc + la or lb = lcla ->-clla -> -cl-al =-cal = -bl.
AS you may see the + has no real significance unless specified. But one has to apprehend that one is dealing with strecken, and to signify that Grassman put  around the strecken pair.