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Psychology 2 7 I
or logical truth in it, but to understand more fully the epistemology of their
own science.
11. The relations between mathematics and psychology are of a similar nature,
though with an additional and somewhat closer connexion resulting from
mathematicians’ concern with the teaching of modern mathematics in the elementary
grades, and therefore with the necessity of taking the laws of development
into account.
In fact the teaching of mathematics depends to a large extent on the idea that
one has of the subject and consequently of its epistemology. Although a mathematician
will naturally never consult a psychologist in order to know how to
prove a theorem, in other words to intervene in the technical aspect of his particular
branch of study, the question of the ‘foundations’ of mathematics is quite
different; in this epistemological field - but one which now forms part of
mathematical problems themselves - mathematicians have always been divided
into three schools of thought, though in actual fact they can be reduced to two:
those who regard logic as the basis of mathematics (to whom we can add the
Platonists, as their eternal universals include logic as an elementary component),
and those who resort to operative or subject focused activities like PoincarB,
Enriques, Brouwer and many others (including E. Borers physicalism, for to
consider functions as if they were inferred from physical entities is meaningful
only if the real reference is to the actions whereby the physicist makes the phenomenon
vary until it can be invested with a mathematicizable structure).
Now, from this epistemological point of view, the present trends in mathematics
are towards a definitely constructivist structuralism. The principal ideas
of the Bourbaki school are well known; at the base of the mathematical edifice,
there are three major ‘parent structures’ (algebraic, order and topological
structures), and by means of differentiations and combinations, countless particular
structures (or sets) are derived from them. Now, psychological research
done in Geneva has revealed that these three parent structures correspond, in
concrete and limited forms, to the three elementary operative structures found
in the child with the formation, around the age of 7 or 8, of the first logicomathematical
mental processes. The notion ‘structures’ as it is understood by the
Bourbaki school is gradually being complemented or even superseded by that
of ‘category’ (a set of objects and all their functions), but S. Papert has pointed
out that there is a tendency to replace the operations ‘of mathematics’ by those
‘of mathematicians’, and, here again, we find that the notion of category has
fairly deep psychological or ‘natural’ roots. Lichnerowitz has shown that mathematical
‘entities’, in the ordinary or even metaphysical sense of the word ‘entity’,
do not exist, but that these so-called entities are isomorphisms and later become
isomorphisms among isomorphisms, etc., according to a scheme of construction
in which some psychologists rediscover that ‘reflecting abstraction’ which
rebuilds unremittingly at higher levels what it has taken from actions or
operations at lower levels, this being the common natural method of forming
logico-mathematical structures. For their part, psychologists, studying this
formation, which may be spontaneous or guided by teaching methods, are,