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270 Jean Piaget
more systematically. The first is the multiplication of separate systems of logic,
all coherent but without any direct aiations between them. A fact which emer-
ges out of this multiplicity of logics is that no single one of them is comprehen-
sive enough to provide the substructure for the whole, while their diversity as
such is so great that it prevents them from achieving this aim. The logician is
then reduced to wondering how he builds up his logic, which means studying
the psychology of the logicians themselves and the history of their constructions.
But in this case, as always, history presupposes a more general psychology,
examining, for instance, by what methods of abstraction and construction the
logician’s logic is worked out, starting with the most intuitive method and going
on to the most formalized one, etc. Consequently, sooner or later, it is impossible
not to come up against the essential problem of the:psychology of the intelligence,
that is to say, the problem of the operative structures which are constituted by
a subject in the course of his activities, but are not at the conscious level and are
not to be confused with the deceptive intuitions based on appearances etc.,
which characterize the conscious level.
The second development is fraught with even more important consequen-
ces: it is the discovery of the boundaries of formalization. Goedel’s theorems
have shown that a theory cannot demonstrate its own non-contradiction by its
own means alone or by the weaker means of the other theories which it presup-
poses: in order to establish such a demonstration, we must resort to ‘stronger’
means, in other words, build up a richer theory which would encompass and
go beyond the preceding one, and so on. It follows from this that, for a human
intellect, by which is meant one that creates science and not one that projects it -
once created- into the realm of Platonic Ideas, the system of deductive theories
no longer assumes the shape of a pyramid resting on an unshakable, or at least
completed base, with a number of storeys, each one in final form, superimposed,
but rather a gradual construction, where the gaps on each landing are filled in on
the one above it, always providing that the process is continued ad infinitum.
Hence, two consequences of considerable importance.
The kt is that logic can no longer form a closed circuit. It is the science of
formalization, but formalization now has its limits : reality which looms beyond
this boundary thus appears to be simply intuitive, which leads us to think that
there is also a domain ‘within’ these boundaries which logic would have to
formalize and axiomatize and which would consist not of the subject’s con-
scious though, but of his operative structures. This does not of course mean
that logic stops at that point, but only that it starts from it (as Aristotle did with
syllogistics) and then extends its axiomatics as it sees fit. Hence, the second con-
sequence which emerges from what has gone before is that logical structuralism
is not static but constructivist; this progressive construction, which consists in
endlessly filling gaps which are constantly appearing at higher levels, is strangely
similar to the psychological development of intelligence itself, with its building
up of structures balancing themselves one after the other, but always relying
on those that follow to fill their gaps and rebalance them on broader ground.
For this reason, several young logicians have recently begun to concern
themselves with the psychology of development, in order not to discover formal