The Lineal Arithmetic; Logos Analogos with lineal symbols

The fundamental aspect of a point is location and orientation. So when Euclid writes that the point is that which has no part he is not referring to these aspics but to the spatial content of a point.

If we think a little deeper, and pay attention to the underlying Greek, we will understand that Euclid is indicating something else. He is indicated that the point, or the seemeioon is the end of spatial analysis. It is the beginning of spatial synthesis. It is the boundary between our subjective contribution to space and our objective behaviours in space. What we contribute subjectively to space comes through the portal of this seemeioon. What we contribute physically comes through the complexification of the seemeia

The complexification of the seemeia is what we describe using the aggregation of points and the product of points . The aggregation of points gives us the sequence of ordinal numbers ordinal counting ordinal arranging and it also produces the complex regional orientation , and regionalisation that we associate with aspects of space.

There is one other important aspect that aggregation of points or complexifircation from point aggregation gives us, that is the density of space. This is an undefined concept but we will get to it later.

There is one aspect of the complexification of points that we pick out and call the product of two or three points. This product is the aggregation of points which produces a line. The aggregation of three points in general I define as producing a curve. But in this case that is of three points there is two options. The first option is at the curve is open. The second option is that the curve is closed.

In general curve could be a straight line. In the case of two points AB the product curve is generally accepted as a straight line. But definitely in the case of three points ABC the Line is definitely a curve which is either made up of straight lines or of curved lines.

If the line is closed that is ABC = AA , then the lines are the boundary of some oriented region of points, including variably oriented surfaces.

If the line is open then the curve is some variably oriented combination of points in space. Examples vary from straight lines to spirals.

Now we have the concept of a line as a product of points we can define a related concept to orientation.

Location and orientation are what we subjectively contribute to space . A curve is our first objective contribution. We usually trace, draw, drag something in the ground to mark out a curve. We can also re draw it mentally, making a curve a crossover idea,both objective and subjective.

In drawing a curve we physically indicate direction( orientation in motion) and displacement( location in motion) . Often we confusingly drop the dynamism of these terms and so confuse them with fixed experiences!.

Thus AB is dynamic! It is a direction and dilacement, via the product of points starting at A and reaching as far as B.

All products are dynamic

Now the product of two point A,B is the line AB which we can write as a .

We need to realise thatA+A cannot give a complex point 2P. Thus A+A can be written as 0P

We also need to recognise that if A is a point and there is a line which is CD, that we can consider adding the point A to the line CD. The result of adding such a distinguished point to a distinguished line, CD is a combination. The combination can only be resolved when we know whether that point A is on the line or adjacent to the last point of the line. In this case we can consider the result to be a line whether it is CD or CD'.

In general however the combination will not be resolvable into a line but into a complex point. The complex point were clearly have a structure different to that produced by an aggregation of scattered points. This means in practice that the complex point will be a combination of A + CD/2. This is a new structure which has not been observed before and will need to be worked out fully.

We know that the degenerate case that is A plus the line AB or B added to the line AB will result in just the line AB. When it comes to considering adding two lines AB and CD we need to take into account that though the lines are dynamic the reference points from which they are drawn aremnot dynamic.

These points are our subjective contribution to space, and though they are referenced relative to our own proprioceptive sense, and so rotatebascwevrotate, they arevfixedbrelative to our senses. We can also assign this fixity to objects in space and imagine our reference frame results if fixed there!

However, clearly to add lines we need to bring them together point wise. We can best do that if the point that begins the second line ends the first line. The only pragmatic way to do this is by parallel lines in space. We need a notion of parallel lines, and symbolic switching.

The idea of projecting lines onto points, so that lines can be combined through a common point is not an easy spatial concept. It has several pre requisites. The first is a notion of true or undeviating orientation in the line. The second is a flat plane or urface that does not deviate its orientation. The third is that thesev2 parallel lines must be in the same plane.

Given these things the requirement or tools to construct these lines objectively also exists.

It has undoubtedly taken skilled artisans thousands of years onhone these requirements to the truest approximation of their ideal.. Thus it is with confidence that philosophers felt that they could anticipate parallel lines that never meet, and as a coolly, the projection of lines to any point in the plane.

These projected lines came to be called Vectors by Hamilton, and they preserved the magnitude and direction of a line segment , and projected it to ny point in space.

Grassmann came at the issue from a different tack.. Hevrecognisedbthe parallelogram as a form of multiplication, andvthevtrisngle as a form of addition. From both forms he picked out the rules of arithmetic with lines. Later he came to the projective aspect of his lineal arithmetic. It is because he wanted arithmetic to apply so closely to discrete lineal magnitudes that he persevered by ironing out the bugs. In so doing he came to the notion of a plane of parallels in grid form . The actual mechanics of addition were finessed., but we can say that the lines can be projected o to the point at either end of another segment. These joins we're used to signify 2 joins: one an aggregation, the other a multiplication.

I woke this morning with the realisation that my understanding of commutativity and anti-commutativity was in error.

I suddenly realised that what was causing my tooth was that as a child I was taught that the page itself was a reference frame with horizontal and vertical axes. I was then taught that the neatest way to layout our work was to use these axes. Therefore if we were given a construction problem it was expected that we would place the first line horizontally and then use what was called an angle in fact to place the second-line relative to the first.

This seems very logical but the consequence of the ones that without realising it this process was embedded unconsciously in my mind and in my understanding of the relationship between lines. Thus if I was asked to draw two lines a and b relative to each other then unconsciously I've processed this as one horizontal and the other at an angle. It therefore seems strange that if you reverse the instruction by using the same letters in a different order that this was not the same construction .

Many constructions soon convinced you that these were not the same construction but in fact a rotation symmetry of each other.

Thus when I was introduced to vectors in that strange and very confusing way:that they were magnitudes with direction. And then later they were lines. And then even later it was said that they were parts of a parallelogram: that is the sides of the diagonals of a parallelogram it is no wonder that I was fundamentally confused. The lines had an unexplained notion called direction in addition to the expected quantity called length. So what the heck was magnitude?

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