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520 Jean Piaget
matical structures. If we accept the nominalist hypothesis, according to which
these structures are a mere language used to express experiential data, the
relations between their syntax and their semantic should obey the general laws
that govern their synchronic and diachronic relations. And at first sight, this
indeed appears to be the case: syntactical rules are continuous in time, while
meanings vary. The theorems of Euclidian geometry are true today, even though
they have changed in meaning, mainly for two reasons: first, they do not seem
to us today to be the expression of a unique, necessary form of space, as Kant
still thought; we see them as one of several systems of measurement, and this
undoubtedly alters their meaning, enriching it, moreover, with all the possible
transitions between Euclidian and non-Euclidian structures; the other reason,
which is still more general, is that spatial forms do not appear to us today as
static figures, but as the results of transformations, so that each form of geome-
try is subordinated to a basic ‘group’ of transformations, and that these groups
give birth to each other in the same way as sub-groups can be differentiated
within a main group. But although these meanings depend, at every point in
history, on the synchronic system of knowledge under consideration at that
moment, they do not succeed each other at random, as if they were the result of
accidents or exogenous factors; proceeding by reflective abstraction from pre-
vious states of construction, new inventions which alter meanings are in line
with a progressive equilibration in which the synchronic balance is at once
the result and the starting-point of new constructive processes. In this respect,
therefore, the situation is considerably different from that of the ‘natural’
languages, in which synchronic balance is a question of re-equilibrations that
are governed by a great many external and internal factors.
This problem of the relationship between synchronic balance and diachronic
evolution gives rise to another closely allied problem - that of the nature of
the innovations which change human behaviour in the course of history and
necessitate re-equilibrations. Here we may identify three possible types of
innovation, which play a very different r61e in the relations of approximate
continuity or discontinuity between present equilibrium and previous equilibra-
tion processes. The first of these types of innovation is that of ‘discoveries’,
which bring to light what was already in existence, independently of the subject,
but which was not known or perceived before (for instance, the discovery of
America). Obviously, in such a case, the necessary re-equilibration, are not
determined solely by the previous states of the system. Secondly, we speak of
‘inventions’, when new combinations emerge as a result of the actions of the
human subject (without going back to what some biologists have called organic
‘inventions’ in relation to highly differentiated organs that are specially adapted
to a new situation). It is the property of an invention that, however well the
components that are combined may have been known (so that the innovation
is only a matter of making the actual combination for the first time), yet the
invention could have been a different one; to invent a new symbolism, for in-
stance, does not imply that others could not have been invented instead. It is
obvious that in such cases also present re-equilibrations and past history are
relatively independent. There is, however, a third type of innovation in human