Introduction

In On Certainty (OC, 1969) there are several remarks in which Wittgenstein discusses simple mathematical propositions like “2 x2 = 4” or “12 x 12 = 144.” And one may wonder whether he thought that these mathematical trivialities, much like G. E. Moore's truisms in “A defence of common sense” (1925), could be examples of hinge propositions. If they were, they would have to be importantly different from other mathematical propositions, because of the peculiar role they would play. To get a sense of the difference, they would have to play a role like the one of “The Earth has existed for a long time” with respect to ordinary propositions about the past such as “There were dinosaurs on the Earth over 230 million years ago.” While the latter proposition is open to verification and control and may turn out to be true or false, the former is not open to such verification and control and, for Wittgenstein, it is neither true nor false, or, if it is true, it is so in a totally “empty” sense (McGinn 1989: 128). Indeed, according to Wittgenstein, hinges play a rule-like role (OC 95), while ordinary empirical propositions do not. If there were mathematical hinges, they would have to play such a normative function, while other mathematical propositions should not be akin to rules.

The trouble, however, is that for Wittgenstein (at least on the vulgata) all mathematical propositions are rules and allegedly false ones, like “2 + 2 = 5,” are in fact meaningless, not false. Thus, in the realm of mathematics, as Wittgenstein thought of it, it seems we don't have a contrasting class with respect to which we could say that only some mathematical propositions would have a hinge status. Of course, one might reject Wittgenstein's own conception of mathematical propositions as rules, to maintain the more traditional view that they are genuine propositions, subject to truth and falsity. Or else, one may dispute the vulgata and offer a different interpretation of Wittgenstein's views on mathematics. In that case, however, one would still be left with the task of explaining why only some mathematical propositions (or statements), like “2 x 2 = 4” and “12 x 12 = 144,” would be rules, which are neither true nor false, or, if they are true, they are so in an entirely “empty” sense.