Mathematics as Calculus and as Grammar

Severin Schroeder’s book Wittgenstein on Mathematics is reviewed and at the same time critically discussed by concentrating on its main aim: to show the coherence of Wittgenstein’s mature philosophy of mathematics. Although Schroeder is dealing with Wittgenstein’s philosophy of mathematics in its entirety, he is mainly interested in the mature philosophy which he sees as dominated by two central ideas: that mathematics is essentially algorithmic, called the calculus view, and that the results of mathematical proofs are grammatical propositions, called the grammar view. According to Schroeder there is an initial tension between these two ideas which he then resolves in two steps: he first clarifies the specific sense of »grammatical« with respect to mathematical propositions and, secondly, he emphasizes the application of mathematical propositions. The reviewer’s criticism of Schroeder’s approach is mainly concerned with this second point. Discussing several specific cases, the reviewer shows that Wittgenstein is much less oriented towards applicability than is assumed by Schroeder.