Logic and mathematics are nothing but specialised linguistic structures.
To express the same idea in still another way, I think that human knowledge is essentially active.
Accommodation of mental structures to reality implies the existence of assimilatory schemata apart from which any structure would be impossible.
During the earliest stages of thought, accommodation remains on the surface of physical as well as social experience.
As far as the game of marbles is concerned, there is therefore no contradiction between the egocentric practice of games and the mystical respect entertained for rules. This respect is the mark of a mentality fashioned, not by free cooperation between equals, but by adult constraint.
Every acquisition of accommodation becomes material for assimilation, but assimilation always resists new accommodations.
The more the schemata are differentiated, the smaller the gap between the new and the familiar becomes, so that novelty, instead of constituting an annoyance avoided by the subject, becomes a problem and invites searching.
Logical positivists have never taken psychology into account in their epistemology, but they affirm that logical beings and mathematical beings are nothing but linguistic structures.
Every response, whether it be an act directed towards the outside world or an act internalized as thought, takes the form of an adaptation or, better, of a re-adaptation.
Children should be able to do their own experimenting and their own research. Teachers, of course, can guide them by providing appropriate materials, but the essential thing is that in order for a child to understand something, he must construct it himself, he must re-invent it. Every time we teach a child something, we keep him from inventing it himself. On the other hand that which we allow him to discover by himself will remain with him visibly for the rest of his life.
Every structure is to be thought of as a particular form of equilibrium, more or less stable within its restricted field and losing its stability on reaching the limits of the field.
Nel, after throwing a stone onto a sloping bank watching the stone rolling said, ‘Look at the stone. It’s afraid of the grass.
We shall simply say then that every action involves an energetic or affective aspect and a structural or cognitive aspect, which, in fact, unites the different points of view already mentioned.
A fact is first an answer to a question. If Sartre had consulted psychologists before judging them in the light of his own genius, he would have learned that they do not wait on the accident but begin by setting themselves problems.
Piaget’s work shows that our concepts of logic, space, time, number, quantity, etc., are not given readymade as Kant thought, but undergo a process of development.
We shall adopt an analogous formula, with the reservation that feelings and cognitive configurations do not depend solely on the existing “field,” but also on the whole previous history of the acting subject.
But if all behaviour, without exception, thus implies an energetics or an “economy”, forming its affective aspect, the interaction with the environment which it instigates likewise requires a form or structure to determine the various possible circuits between subject and object.
Knowledge is not predetermined by heredity; it is not predetermined in the things around us – in knowing things around him the subject always adds to them.
But these structures, forming different levels, are to be regarded as succeeding one another according to a law of development, such that each one brings about a more inclusive and stable equilibrium for the processes that emerge from the preceding level.
So we must start from this dual nature of intelligence as something both biological and logical.
Formal logic, or logistics, is simply the axiomatics of states of equilibrium of thought, and the positive science corresponding to this axiomatics is none other than the psychology of thought.