## X.3.6 Dimensional points

The point notion is a basic concept in mathematics, mostly in geometry, representing a graphical substitute (a representation) for the singular numerical value from an 1D domain, further extended to the domains 2D, 3D so on, by means of a proper association of singular values (obtaining new objects as a result of internal model composition, as we have noticed in chapter 3). These singular numerical values from mathematics mostly belong to the well-known set of real numbers {R}. The major problem related to the singular values which belong to this set is that these singular values are not realizable. As it was mentioned in chapter 2 and in the annex X.3.1, the singular values from {R} are absolute accurate values (AAV), which means that they contain an infinite information amount, therefore, they are not realizable under an abstract form and neither the less89 under a material form, the points which correspond with these values being considered as virtual points (theoretic, imaginary, mathematic, dimensionless). For overcoming this problem, people always did what it had to be done, namely, the AAV truncation up to a value with an information finite contents which could be represented through a reasonable number of digits. But, this operation is informationally similar with the association of a known non determination interval with an AAV, that is an interval which includes the rest of the digits up to infinity. This interval which is currently known as an error, tolerance, uncertainty interval etc., with a known amount, makes that the information contained into AAV to which it is related to be finite. But an interval known as an amount, even it has non-determinated values represents a dimension, therefore, the point which corresponds to a truncated (approximated) value is not dimensionless any longer. Therefore, we may find that even since the ancient times people have operated with dimensional points thinking that they are dimensionless.

Another aspect is also worth noticing: in case we are dealing with values of some concrete objects (which can be numerically expressed), either they are material or abstract, truncated (therefore, the ones with dimensions) values are used for the calculation, and when these values are mentally projected by decreasing to zero the error interval, virtual objects (as asymptotic limits) are obtained - dimensionless points. Thus, it seems to be very clear the difference between the objects called dimensional and dimensionless points - the contained information amount. This clear distinction between the two point types which is revealed to the reader by means of the objectual philosophy is only one of the numerous examples which will underline the dichotomy in the world of abstract objects, dichotomy which clearly divides this abstract world in two complementary parts: world of the abstract realizable objects (objects with finite information content) out of which, some of them may belong to the known reality, and the world of virtual (objects with an infinite information content), abstract objects which derive from the first category as a result of an extreme generalization, and which may belong (but not always) to the absolute reality (the two reality types are described in chapter 9).

89 The material realizability requires much more strict conditions than the abstract one; we may design on a computer an invar bar with a length of 0.543218964387 m, but we would never be able to realize it because the accuracy required by that number (abstract realizable) is under the threshold of the atomic dimensions.