General approach
Physical phenomena sometimes happen to be put into mathematical formula the elements of which are worked out by abstract logical – mathematical thinking often years and ages before discovering their physical interpretation. But it is well known that empirical experimental truth (for example physical constants) can not be achieved by way of pure deduction without being observed. This way we can not attribute their origin to the pure human intelligence. On the contrary some logical – mathematical results for example the theory of "n”-dimensions geometry can not be achieved by way of experimenting in physical sphere and this allows us to attribute its origin only to human way of thinking. These two independent types of knowledge can not be considered as reducible to one another. However in spite of this difference there is a wonderful conformity between them and sometimes isomorphism.
The problem is to find out how mathematical logical units and physical reality perceived by us correlate with each other, what does their correlation depend on? Either on the activity of the subject, the unity of Nature or synthetic structures a priori, or their commonality to God and so on. Modern science discovered that the same logical- mathematical structures (or general systems) can be found in all spheres of life and regulate the interaction between the object and physical or social reality. These general normative systems everywhere have one function - it is adaptation of an object to the environment. There are common mechanisms of formation and developing such systems that influence the growth and amount of knowledge. The basic mechanism that regulates the relations between biological structures in adapted organism and general structures in cognitive system is the balance or equilibration i.e. regulation of the reverse connection. Thus there is a general way of evolution from the adaptation of the animal getting used to it’s special "ecological niche", up to a science comprising the knowledge not only material (macro- and micro-) but also ideal (multidimensional geometry, etc.) worlds. Thus the new knowledge appears possible so far as the new logic patterns are formed which presumes to "seize" new experience. In cognitive plan these structures are formed by systems of intellectual processes which develop as the schemes of expedient (adapted) activity of an organism in an external world.
These cognitive structures (groups or groupements ) differ in complexity, articulation, and hierarchization. Groupement is a system of concepts (classes or relations) implying a coordination of points of view and a pooling of thought. An operatory groupement is a system of operations with compositions exempt from contradiction, reversible, and leading to the conservation of the totalities envisioned.
These structures are well-ordered and possess characteristics associated with successive groups of order one, two, three, and so on. Each successive group structure contains the "precipitate" structures of antecedent groups, so the higher contains the lower. The complementary theory of cognitive operations describes a set of basic operations which may be iteratively applied to successive levels of cognitive organization. The significance and meaning of operations depends upon the level of cognitive organization, and so logic itself undergoes successive transformation and embellishment.
A genetic basis of these structures and also the guarantee of conformity of scientific knowledge to the reality is, finally, the logical – mathematical experience of a child improved by means of the mechanism of regulation. Initial adequacy is constantly supported owing to that the subsequent designs, generalizing the former models gradually select "general" coordination of any possible action which are common for the activity of the biological organisms and for the interaction of physical objects.
Logical-mathematical designs transcend the frames of reality, so to achieve the maximum of possibility that physical systems realize only a part, and thus become adapted to these physical systems long before. The sequence of the structures is ruled by a definite law: every new achievement is the result of a rational structure, according to materials, gained as a result of a former structure, which reorganizing them makes them still valid, thus, their synthesis not being preformed in elements, appears nevertheless necessary. The arbitrariness of conventions of the subject here is replaced with the law of a rational design by which the perception of casual influences of environment is ordered in the sense that data which can be integrated into logical –mathematic frameworks of the subject depend on the cognition of the last. New and unexpected experience will be immediately assimilated and understood or, on the contrary, distorted, depending on the fact whether the object has the ready- made mathematical patterns to which this experience can fit or it has to reorganize everything it gains or finally the structure of the experience differs very much from the ones the object has at the moment.
Recognition of the implicit laws of individual thinking and public agreement to make them the rules of the game for cooperative searching the truth turn simple functional equalities immanent every cognitive and life activity into "normal" in full meaning of the words. The mechanism of this turning is bringing to balance social exchanges with some peculiar logic. This (isomorphism) correlation between logical-mathematical structures and physical reality certifies functional uninterrupted link between life and mentality, in other words in all the cases biological evolution, intelligence, social life and so on, we talk about the necessity of explaining regular formation of material and not material structures, that are linked inside and adapted to the environment.
This way mathematical–logical structures are the reason for the organization of cognitive activity not only in the prospect of universal rules or law for biological adaptation but even more in the view of interaction and unity of the material and spiritual organization of the Universe.
Mathematics and Social Reality
There are the similar correspondence between logic and mathematics and the social reality.
The logical progress goes hand in hand with progress in socialization, is it because the child becomes capable of rational operations due to the fact that social development makes him capable of cooperation; and at the same time, is it because his individual logical acquisitions permit him to understand other people and thus lead to cooperation. The two sorts of progress go completely hand in hand, they constitute two indivisible aspects of a single reality that is at once social and individual.
Considered from the point of view of their psychological development, logical operations constitute the final equilibrial form of actions reached when they are 'grouped' into mobile systems that are both indefinitely composable and rigorously reversible. Now, social cooperation is also a system of actions, interpersonal rather than simply individual, but actions all the same and consequently subject to the laws of action. One can say, therefore, that social actions that end in cooperation are themselves ruled by laws of equilibrium and that they will, like individual actions, only attain equilibrium on condition of becoming organized into composable and reversible systems. Then, the laws of the groupement become simultaneously the laws of cooperation and of individual actions on the physical world. And cooperation is therefore, to be conceived according to the very etymological meaning of the term that designates it, that is, as a set of co-operations.
It is only by cooperating with others and not beforehand that the individual elaborates his logic. The social relationships themselves present such a logic. The decrees of a dictator do not engender it necessarily, whereas free cooperation leads to that reciprocity of perceptual judgments and representations which, alone, make the objective operation possible. It is, then, a matter of understanding how social relationships end in logic. Individuals' actions on one another, which lie at the basis of every society, only create a logic on the express condition that they themselves acquire a form of equilibrium analogous to the structure whose laws may be defined at the end-point of the development of individual actions. And even that is a matter of course, since individual actions are more and more socialized and since cooperation is a system of actions like any other.
In sum, the social relationships equilibrated into cooperation constitute groupements of operations exactly like the logical actions exercised on the external world by the individual, and the laws of groupements define the form of ideal equilibrium common to both social and individual actions.
***
Individuals' actions on the external world, as we have seen, obey a law of development such that the equilibrium toward which they tend takes on the mobile and reversible form of the groupement. Social relations consist in the actions of one individual on another. In exchanges of thought, such relations also tend toward a form of reciprocity, which implies the reversible mobility belonging to the groupement. Complete reversibility presupposes symbolism, because it is only by reference to the possible evocation of absent objects that the assimilation of things to action schemes and the accommodation of action schemes to things reach permanent equilibrium and thus constitute a reversible mechanism. The symbolism of individual images fluctuates far too much to lead to this result. Language is therefore necessary, and thus we come back to social factors. In other words, cooperation is only a system of operations carried out in common; it is only a matter of co-operation. In order to make the individual capable of constructing groupements, it is first necessary to attribute to him all of the qualities of the socialized person.
The groupement as a logical structure is a form of equilibrium and that form of equilibrium applies necessarily to the process as a whole.
As for logic itself, it goes beyond both of these functions since it arises from the necessarily ideal equilibrium toward which they both tend. This is not to say that there exists a logic in itself that simultaneously rules individual and social actions, since logic is only the form of equilibrium immanent in the processes of development of these actions themselves. But actions, becoming composable among themselves and reversible, acquire, by being elevated to the rank of operations, the power of being substituted for one another. Thus, the groupement is only a system of possible substitutions either within a single individual's thought (operations of intelligence) or within thought exchanges from one individual to another (cooperation). These two sorts of substitutions constitute, therefore, a general logic, at once collective and individual, that characterizes the form of equilibrium common to cooperative as well as to individual actions.
The development of psychic activity from the perception and skills to images and up to sophisticated mental conclusions and formal thinking is the function of the growing communication and balance between the recognition of reality by the organism and accommodation to it. And thus one can say that intelligence with it’s logical operations ensuring balance between the universe and thinking continues and finishes the unity of the process of adaptation. Homo sapiens is the organism that began to comprehend ( by means of communicating with the environment) the basic features of the structure of life that remained unchanged from unicellular to the human being making use of the way of self organization as the way of thinking. Along with this means of self organization of organic material reproduce the characteristics of the self regulation of the Universe as it is.
Application: The nature of the groupement
What is the specific nature of the groupement (group structure) alleged to be central to the formal operations of cognition? Although this group (Klein-four) may be manifested in a variety of different situations, it is defined in terms of transformations on logical propositions.
There are many general transformations which transform particular operators into others. Thus an operator such as (p v q) can be transformed by inversion or negation into (p*q), a transformation that we may designate by N, so that N(p v q) = (p* q). But (p v q) can also be transformed by reciprocity R, so that R(p v q) = ˉp v ˉq = p/q. Again (p v q) can be transformed, by correlativity С (i.e., by permuting the v and the *.), so that С (p v q) = p* q. Finally, the operator (p v q) may be transformed into itself by Identical transformation I, so that I (p v q) = (pvq). Thus, one can see that I, N, R, and С form a commutative group of four transformations among themselves, for the correlative С is the inverse N of the reciprocal R, so that С = NR (and С = RN as well). Likewise, we have R = CN (or NC) and N = CR (or RC). Finally, we have I = RCN (or CRN, etc.).
{I(p→q)=p→q; N(p→q)=p*q;
R(p→q)=q→p; C(p→q)= -(q→p)}.
This group is of psychological importance because it actually corresponds to certain fundamental structures of thought at the formal level, for inversion N expresses Negation, Reciprocity R expresses symmetry (equivalent transformations oriented in opposite directions), and Correlativity is symmetric with negation.
What is meant here is that subject can perform mental transformations upon mental transformations or operations upon operations —all in terms of his thinking about the possibilities in a situation, for example, the balance. For instance, with reference to the logical operation (p * q) the subject can perform any one of four transformations or operations upon it. If this is selected as an example the subject can negate it logically, that is, he can think of the negation of (p* q), i.e. N(p* q), where (logically) N(p* q) = ˉpv ˉq. Similarly, the reciprocal R(p•q) is a possibility that the subject can think of, where R(p* q) = ˉp*ˉq and the correlative C(p*q) can be thought of, where C(p*q)= p v q. This works similarly for I(p v q), R(p v q) and C(p v q). For example, N(pvq) = ˉp*ˉq means 'the negation of either increase the weight or increase the distance or both' is 'decrease the weight and decrease the distance' (on one side of the balance). The INRC group is a group of transformations upon logical operations (and transformations upon a physical system), by which the subject can express the possibilities of solving particular problems (in this case the balance). It is this group that is to be central to formal thought, even more so perhaps than the sixteen combinatorial operations because N and R, the forms of reversibility, are now supposedly united in this single closed structure.
Ernest GRIGORIAN, Institute for Social Sciences