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 less^{8}9
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.

Copyright © 2006-2011 Aurel Rusu. All rights reserved.