Let the points A, B, C be distinguished in space. Let them not be in any particular or special order.
I want to set up an arithmetic for points. The only realistic arithmetic for points is that which Adds them. Therefore A + B =2 Points.
Let us suppose that two points is represented by 2P. Then we may adopt the convention that P stands for a complex point.
If P is a complex point then it must be equal to at least A + B divided by two.
P = A + B / 2
Suppose now that C is added to P.
Then C plus P is equal to 2 points. But one of the points P is a complex point. Thus we need to write 2P' equals C + P
We can now see that P' equals C + P /2
We can continue in this way adding as many different distinguished points as we wish. Each new complex point will be a construction of all the other complex points. This set of complex points will have a relationship that will show us how the points are sequenced in space.
This sequence of complex points can of course be added into itself. This P' can be added to P" and the sum can be said to be 2P"'
When we begin to add the points sequentially we noticed that we do so in a binary finish. This is why to always appears in the definition of a complex point. However when we begin to add the complex points then we find that the two complex points develop into binary power sequences. This is a reflection of our inability to add except in a binary way.
Nature's not so limited. It can effect in multiple. When we see this we are unable to apprehended. Therefore Our limitation is what confines us i binary sequences in our additives and summation algorithms.
This limitation has its benefits. We notice the rhythm and the flow of these binary sequences. This is why will we use 4 time 2 time 8 time these reflect our ability to add and to subtract in our natural sequence algorithm.
Our natural number system that is 123,… Does not reflect this binary. The names are based on the syllabary from which the alphabet is derived. The early alphabet is a complex etymological development which is based on both the syllabary and also the script forms which were developed to denote Specific sound patterns or phonemes. The sequential pattern of the Syllabary was also often used for counting. So the names of our numbers are derived from this practice.
While the names do not reflect this binary development the so-called successor function, or the +1 cultural count does reflect this binary behaviour.
It is in fact the behaviour that is captured by the success of function and also by the plus one cultural count. This behaviour is the sequential addition of individual points.
It is the behaviour which is of fundamental importance to the development of mathematical structures. The names of the numbers or the names of the numerals is not significant. The sequential behaviour and the sequential behaviour pattern is.
The sequential behaviour is best described by the ordinal numbers. The cardinal numbers which are a later development are of less significance. When we create a structured in mathematics we use the Ordinal behaviour of numbers.
The other aspect of the complex point is that it it amalgamates the sequential addition of single points. However it is possible to sequentially add complex points. In so doing we aggregate regional counts of additional points. This serves to identify spatial regions, and boundaries to different areas or volumes of space.
Well the identification of these regions or volume Spaces is not definite, it does how ever serve to develop the idea of spatial or regional density. This is simply a notion of how intensive the count was in a particular region or volume of space with the associated orientation or direction.
We've therefore see that the notion of points and the counting of points is at the basis of not only our sequential or order sense is also a basis of our spatial sense and the sense of form.
The complex point therefore that is the summation of points is somehow the notion or the model of space region and form.
With that in mind we can return to our initial idea of three points A, B, C. These three points form a complex point which we identify as a triangle and we identify its region and its direction at the same time.
The response to a scatter of points is to sequentially complexity the point notion into a region or form, and to develop the ordinal behaviour we call sequencing and counting. These are fundamental subjective behaviours we bring o mathematics or philosophy through the agency or portal of the seemeioon..
In contrast, the multiple form of 2 points is defined as the straight line. The multiple form of 3 points I define as the curved open line or closed line.
Since we have yo specify which , open or closed curve, not much work has been done on specifying this notational tool.
Norman uses this device in Hyprbolic geometry to define a triangle, and 3 lines to define a trilateral. However, imeouldbdefinena circle or closed form by a single line that goes through all the 3 points.
To be brief, the product of general sequences of points will be defined as an open or closed MULTI lineal form. The outer boundary of a closed multilineal form is the fundamental product of the points, but many adjugates products will be contained within that boundary.
For an open product, the main product is a sequential line . Examples range from a straight line to a spiral even in 3d space.
Traditionally the regionality has been ascribed to these product boundaries, but as we have seen, and defined, this is to be ascribed to thes ummation or aggregation of points into complex points.
We. Can now look at the Arithmetic of lines.