From Constitutive Functions to Equivalence Classes

Abstract

When children set up classifications prior to the constitution of classes based solely on similarities and differences, i.e., on a system of objective equivalences, we observe an initial stage in which there exist only ‘figural collections’² where the following principle applies: not only are the elements of a class spatially arranged in such a way as to give the latter an overall shape (row, rectangle, etc.), but also (at least in the case of simpler forms) in such a way that one element is linked to another by reason of various ‘suitabilities’ (‘convenances’) which are unrelated to similarity. For example, a triangle is placed on a square in order to make a house and its roof, a nail with a hammer, a fir tree with a hut (instead of with another fir tree), etc. We also know that the early definitions made by children are not made on the basis of ‘kind and specific difference’ (last stage), but rather by ‘usage’ as evidenced by the use (in all languages) of the words ‘it’s for’: a mountain ‘is for climbing’, a snail ‘is for crushing’, etc. It would therefore seem clear that prior to operatory classifications based on additive class inclusions in extensions and on objective equivalences of different orders in intension, there exists a mode of classification based on the relationship between actions which are functional in two ways, i.e. as the applications of schemes of actions and as the expressions of dependences.

With the collaboration of E. Schmid-Kitzikis.