We continue with Piaget and Inhelder:

“There is a second difference between physical experience and logico-mathematical experience or deduction. Whilst the latter, proceeding by means of reflective abstractions, leads to progressive purification (whose final stages are today those of the formalization peculiar to "pure" mathematics), physical experience is always a sort of "mixture". There is in fact no "pure" experience in the sense of a simple recording of external factors, without endogenous activity on the part of the subject. All physical experience results from actions on objects, for without actions modifying objects the latter would remain inaccessible even to our perception (since perception itself supposes a series of activities such as establishing relationships, etc.). If this is so, the actions which enable us to experiment on objects will always be dependent on the general coordinations outside of which they would lose all coherence. This means that physical experience is always indissociable from the logico-mathematical "framework"³ which is necessary for its "structuralization" This logico-mathematical device is in no way restricted to translating the experience into formal language—as if it were possible to have on the one side, the experience itself and, on the other, its verbal translation.”

[footnote 3. Establishing relationships or logical classes, functions, counting and measuring, etc.]

“This brings us back to the central argument of empiricism: that all knowledge should be related as closely as possible to observable facts.”

“In reality, in every field—from physics to psychology, sociology or linguistics—the essence of scientific knowledge consists in going beyond what is observable in order to relate it to subjacent structures. Firstly, logico-mathematical structures must go outside the scope of what is observable, i.e. what is furnished by physical experience in the broad sense (including biological, psychological experience, etc.). Infinity, continuity, logical necessity, the hierarchy of constructions and of reflective abstractions are all unobservable realities according to the empiricist, and if they had to be attributed to the simple powers of a "language", this language would have the surprising property of being infinitely richer than that which it describes. Secondly, in physics we might just be justified in regarding as observable features the repeatable relations which functional analysis strives to translate into "laws", but on examination of the actual work of scientists—and not the philosophical statements to which they so often limit themselves—we have to recognize that their systematic and unceasing need to discover why things happen forces them to break through the barriers of the observable. In these last decades, measurement has become a problem and researchers have often sought to identify the structures before attempting measurement. To take just one classical example, no one would dispute that the very widespread success of the application of the group structures in physics means that physicists also subordinate what is observable to systems or models which are not. Present-day achievements of structuralism in biology also provide an example of this and almost all the social sciences are proceeding along the same lines.”


“To sum up, the innumerable problems continually being raised by the nature of mathematics and its application to experimental science have moved us further away from, rather than towards, the empiricist ideal of scientific knowledge.”

[This ends Piaget for the moment.]
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