Educational implications of some of wittgenstein’s remarks on mathematics: proposition, inference and proof

Inspired by Rousseau’s ideas, the different tendencies of the new school have emphasised educational practices centred on the pupil’s activity, presupposing in the learning process a natural rationality being developed as much as possible through observation and experimentation, and not by means of verbal instructions. The goal in this paper is to point out some misleading practices which arise from these empiric directives, by using some of Wittgenstein’s remarks on the nature of mathematics, particularly those published in Philosophical Remarks (PR). As one of the major representatives of the linguistic turn, the Austrian philosopher presents a new conception of language where the senses are not developed from cognitive structures or abstracted from some extra linguistic reality; but rather they are formed within language, which has a multiplicity of functions. In particular, the mathematical propositions perform a normative function, very similar to rules, as they allow inferences and determine what makes sense and what does not make sense. From this perspective, we defend that learning mathematics involves essentially a training, process in which the student forms an accepted rationality, among many other effective and possible ones