Introduction to objectual philosophy

X.3.6.2 Conclusions

If a continuous domain of values of the variable x is considered to be, informationally speaking, as non-decomposable, this means that inside it, there are no known singular values except the internal reference value. The non-existence of these internal values is equivalent with the lack of information or with the existence of a non determination. Therefore, the DP 1D elementary object is equivalent with a single AAV (internal reference), where a non determination interval  is being added. The fact that DP contains only a single known value (just like the virtual point, only that the virtual point is not related to the non determination interval any longer) was the reason for choosing the name of dimensional point. The association process to an AAV of a non determination interval is similar with the approximation (truncation) of that AAV (which otherwise would require an infinity of digits for its display), to a realizable value, with a finite number of digits, the difference between them being even that non determination interval. It is clear that this operation was (tacitly) made by people even since the numbers were discovered, but without respecting the definition of the real numbers from the mathematics, and by nonchalantly ignoring the clear difference between these virtual and common numbers.

Any interval of a domain which is major than DP may be synthesized through the adjacent concatenation (joining) of a finite number of DP, according to the following relation:

(X.3.6.4.1)

where the object one-dimensional interval is represented according to the mathematical syntax as an union of elementary object intervals DP_i (demi-closed) in which:

(X.3.6.4.2)

It may come out that is also an object, with the boundaries x₀ and x_n, the internal reference x₀ and the size of internal domain . If you have already read chapter 3, you will recognize in the object from the relation X.3.6.4.1 the external composition of the 1D DP-type objects.

An informed reader could say that, so far, there is nothing new in this approach; the relation X.3.6.4.1 is nothing else but a simple integration ranging from x₀ to x_n, in case that the point interval is replaced by the well-known infinitesimal interval dx from the integral calculus. Indeed so, but the objectual approach on the mathematics which was initiated in this work eliminates the logical contradictions which existed so far and which were not included within the textbooks. Such a contradiction is given by the assertion “A curve is made-up from an ordered set (a series) of concatenated points”. However, according to the classic mathematics, the point’s size is zero and no matter how many zero-dimensional elements would be concatenated, a zero-dimensional result would be obtained anyway. On the other hand, the integral calculus solves the problem correctly, in order to conceive a curve, a series of curve elements being concatenated, but this time, the curve’s elements have a certain dimension. However, one little formality is still required: the recognition of the existence of dimensioned points as realizable objects, and of the virtual points (dimensionless points) as non-realizable objects (virtual, asymptotic models of the realizable points).
In case of 2D and 3D domains, with two or three dimensions, DP will also be either two- or three-dimensional. The model of this kind of objects is a synthetic object (complex one, but this time, by means of internal composition within the model) which consists of two or three one-dimensional objects which were above mentioned, associated with a common reference system - a virtual point which represents the two or three RAV with a simultaneous existence, each of them with its own non determination interval, the internal references of each associated one-dimensional interval.
DP model has been also adopted due to other reasons, namely because the continuity and consistency of the systemic organization of the geometrical objects is therefore achieved, which means that any finite realizable curve (or straight line) segment may be decomposed into a finite number of adjacent DP, any finite surface is decomposable into a finite number of adjacent curves (which are also decomposable), any finite volume is decomposable in adjacent areas etc. This systemic continuity is also required by the existing mathematic system (integral calculus, calculus with finite differences) which operates only with dimensioned elements (infinitesimal elements have the number of dimensions specific to the integration domain).

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