### X.3.2.1 The objectual definition of distributions

While the continuum model was the obsession for the mathematicians and for the idealist philosophers for centuries, the objectual philosophy sets a more pragmatic approach, the discrete one, based on the concept of object as a finite information entity, with which a realizable IPS is able to operate with. This approach, which due to this reason is also denominated as objectual, is based on the fundamental properties of the objects class: invariance (as model), (de)composability and mostly, their distinguishability. These properties are also self-understood (but without being defined) in the mathematics field, in case of some objects, such as the sets, whose elements are referred to as “objects” of the set. On the other hand, the classic (continuous) mathematic approach may be understood due to two reasons:

• Continuity of some basic amounts (such as the spatial position of a material object), closely related to the infinite divisibility of space and of the abiotic material systems, it is upheld by the present paper as well, but with the note that this aspect (of continuity) belongs to the absolute reality (referred to as objective reality according to other papers, that is a virtual abstract object) which is inaccessible to realizable IPS, because it contains an infinite information amount. Due to the finite information processing ability, the realizable IPS must focus oneself to only on a part of the absolute reality, therefore on a discontinuous approach of the continuous amounts.

• The objectual approach on the knowledge may be made only as a result of the understanding which has been occurring over the last century, regarding the information processing processes, which can be applied only on the discrete objects (deliberate pleonasm).

According to the objectual philosophy, the distributions are considered as ordered sets of distinct assignment relations, between the elements of other two sets: the ordered set of the independent variable’s values (set which represents the distribution support) and the set of the dependent variable’s values (which in this paper is also named the distributed attribute). The three sets which are made-up from distinct abstract objects (belonging to the relations, support values and distributed values) are obviously equipotent. This way of defining the distributions determines in its turn some changes in the definition of the terms derived from the differential calculus:

• Another definition of the term continuity (required for defining the continuous functions), namely that the continuity is regarded as an invariance (a continuous unchanged maintenance) of the symbolical assignment relation on the support domain (continuity domain). In the most general distribution case (such as the lists, tables, matrix etc.), the concrete assignment relations do not have a general symbolic representation any longer (an invariant function), but they are specific to each support element, and in this case, the distributions, unlike the algebraic functions, can be used.

• Organization of the distributions as complex objects, decomposable up to the level of the elementary object, determine the occurrence of many distribution types depending on the structure of this elementary object:

1. Primary distributions, where the fundamental element is made-up from a singular value of the dependent variable, assigned by means of a concrete (local) relation to a support singular value. If the set of the concrete assignment relations of a primary distribution has a symbolic representation which does not depend on the concrete support value, then, this distribution replaces the classic continuous functions even if there is a discrete support.

2. Derived distributions of a primary distribution, where the fundamental element is a finite variation (difference) of a specific rank (of the same dependent variable from the primary distribution), assigned through a relation to a variation belonging to the support variable. Just as it was above mentioned, if there is an unique symbolic relation for the set of actual assignment relations of the elements belonging to the derived distribution, that distribution replaces the derivative functions (of any rank) of the primary distribution (they are also valid for discrete supports).

• The introduction of the density term for the ratio between the finite variation of distributed amount and the support variation, density which is equivalent to the local derivative of a function, but not in a single point (a singular value) as it is defined in the classic differential calculus, but on a support interval with the internal reference at a specific singular value.