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# 8. Euclid spaces

The one-dimensional (1D) sentence states, ‘under order αβ, addition a1 is on place p1.’ There is no discussing this elemental fact. The two-dimensional (2D) sentences states, ‘under order αβ, addition a1 is on place p1’ and ‘under order γδ, addition a1 is on place p2.’ The contradiction can be made visible on the orders, the amounts and the places. The number of such contradictions is a meta-argument. To reach a compromise, we merge the statements into, ‘a plane with rectangular axes αβ, γδ exists; on this plane, addition a1 has the coordinates x=p1, y=p2.’ The re-translation of the 2D statement into 1D involves two assertions. The hypothesis about the existence of a plane is supported by two one-dimensional sentences; the strictness of the implication can be understood as a measure of probability that a plane exists, on which a1 has the coordinates (p1,p2). One can calculate the ‘costs’ of the existence of a plane by using the metric (a1|p2-p1) while evaluating alternatives of placing the same amount on different places.

The indecision about whether the plane exists on which (a1|x,y) is the case, and how it is called, is a metric about the remaining contradiction about what is and what is not the case. The triangular translocation of the unit movement allows using it as a unit of logical fulfilment. Knowing that (a1,p1,p2) is the case delineates classes of possibilities of what can, will, may, may not, will not be or are not the case.

Assembling planes

We construct a Descartes type space by using CA, KA, QK sorting orders as its common axes. (8.g.3.)  We can interpret the 1D places of addition a1, in CA p1, in KA p2, in QK p3 as z, x, y coordinates of a point in a three-dimensional (3D) Euclid (Descartes) space.

There is a second 3D space to be assembled of the standard reorders. This is the b-oriented space and its axes are CB_KQ, KQ_QA, QA_CB. The central element has the coordinates 70, 70, 70 in the a-space and 67,67,67 in the b-space.

The re-translation of a static 3D property (x=p1, y=p2, z=p3) into 1D statements needs six statements, of which two versions are identical, these referring to the point’s position in the a- resp. b-space.

Please note that two planes are also of standard characteristics. The central element is on place 63 in these.

Amount-, place- and order-oriented statements

The implications of ‘under order αβ, amount a1 is on place p1’ can be read off as ‘if αβ then (a1,p1),’ ‘if (a1,p1) then αβ,’ ‘if a1 then (αβ, p1),’ ‘if p1 then (αβ, a1).’ The strictness of the implications is evidently different. The human brain is used to thinking in the direction of principle → realization, so we are inclined to use the idea of order as a premise and deduct where something belongs to as a conclusion. The relationship of the numbers shows no such prejudice. We can deduct the existence of the order from the Sachverhalt that (a1,p1), but we cannot be sure of the kind of order of which (a1,p1) is a part of the realization.

It appears that expressions of the type (order, amount, place) can be built and calculated for each combination of the arguments. The axes of the logical-accounting Euclid space are order, amount and place, and the cells of the matrix contain numerical values that can be added up into a grand total over all re-orderings, additions and places.

The imprecision vs. strictness of the implication of {αβ|a1|p1} varies greatly.

Mastering Time

We have seen that it is possible to assign each of the additions a place in a space in a static way. Each of the additions can be given a planar position by any two orders, and a spatial position by means of its standard aspects. The translation from the 3D realization into 2D and 1D sequences involves three markers per logical statement, of which there are two. The two versions of saying the same are distinguishable by place; they are identical in content. We have points with differing properties on distinguishable places in a space – there twice -, in planes and sequences. The Wittgenstein tautology in the Minkowski space-time confluence exists. We have shown the accounting mechanism dealing with the translation of 1D into 3D and 3D into 1D to be functional.

To find the mechanism used by Nature in the copying back and forth between one and three dimensional ways of putting what is where (if a complete order exists), we have to expand the tautology without leaving it. If all the standard movements are carried out, the tautology is complete and the result is trivial. We can make use of roughly two thirds of the statements that are ‘presently’ not the case, and free up some logical accounting wiggle room to manoeuvre between what can be the case and what will be the case. We use the standard chains to connect the points being the case on the planes.

Heretofore, we used a1 as premise to reach a conclusion about which of the pi is that p1 that is the most congruent with order o1 being the case. Now we use ai to conclude, which order is that order o1 that is the most congruent with place p1 being the case. The chains connect orders, and the standard chains create spaces and give locations – coordinates – in the spaces. Inasmuch as only the treadmill of standard retranslations takes place, this space is devoid of anything but the central elements and their respective positions.  The elements are consumed by their relentless reorderings between X,Y and Z axes of space and have no amounts nor distances to make mischief. This is the state of the world, if everything that is the case can be and will be the case. Biology teaches us to look into the complications, where two versions of what can be the case exist and only in cooperation between the two versions happens that what will be the case. In the next step we shall fuse the two Euclid spaces into one.