[[!meta title="The Psychology of Intelligence"]]

* Author: Jean Piaget

## Logic and psychology

    An axiomatics is an exclusively hypothetico-deductive sci-
    ence, i.e., it reduces to a minimum appeals to experience (it even
    aims to eliminate them entirely) in order freely to reconstruct its
    object by means of undemonstrable propositions (axioms),
    which are to be combined as rigorously as possible and in every
    possible way. In this way geometry has made great progress,
    seeking to liberate itself from all intuition and constructing the
    most diverse spaces simply by defining the primary elements to
    be admitted by hypothesis and the operations to which they are
    subject. The axiomatic method is thus the mathematical method
    par excellence and it has had numerous applications, not only in
    pure mathematics, but in various fields of applied mathematics
    (from theoretical physics to mathematical economics). The use-
    fulness of an axiomatics, in fact, goes beyond that of demonstra-
    tion (although in this field it constitutes the only rigorous
    method); in the face of complex realities, resisting exhaustive
    analysis, it permits us to construct simplified models of reality
    and thus provides the study of the latter with irreplaceable dis-
    secting instruments. To sum up, an axiomatics constitutes a “pat-
    tern” for reality, as F. Gonseth has clearly shown, and, since all
    abstraction leads to a schematization, the axiomatic method in
    the long run extends the scope of intelligence itself.

    But precisely because of its “schematic” character, an axiomat-
    ics cannot claim to be the basis of, and still less to replace, its
    corresponding experimental science, i.e. the science relating to
    that sector of reality for which the axiomatics forms the pattern.
    Thus, axiomatic geometry is incapable of teaching us what the
    space of the real world is like (and “pure economics” in no way
    exhausts the complexity of concrete economic facts). No axi-
    omatics could replace the inductive science which corresponds
    to it, for the essential reason that its own purity is merely a limit
    which is never completely attained. As Gonseth also says, there
    always remains an intuitive residue in the most purified pattern
    (just as there is already an element of schematization in all intu-
    ition). This reason alone is enough to show why an axiomatics
    will never be the basis of an experimental science and why there
    is an experimental science corresponding to every axiomatics
    (and, no doubt, vice versa).

    -- page 30