Wittgensteins philosophy and the language of mathematics

Ladislav Kvasz

Academy of Sciences of Czech Republic

Philosophy of mathematics played in Wittgenstein’s thought an important role. From the Tractatus, as well as from the Remarks on the Foundations of Mathematics, it is possible to reconstruct two different, but nevertheless in many respects analogous interpretations of mathematics. We can call these Wittgenstein’s explicit philosophies of mathematics. Both of them were developed in a broader framework of philosophy of language. This framework concentrated in the times of the Tractatus around the distinction of that what can be said and what can be only shown, while in the later period it was the issue of rule following. For Wittgenstein (as for Kant) mathematics was not the subject of primary interest; it served rather as an illustration of some more fundamental problems. Wittgenstein’s explicit philosophies of mathematics received in the literature broad attention. We can mention Crispin Wright (Wright 1980), Pasquale Frascollu (Frascolla 1994) or Esther Ramharter and Anji Weiberg (Ramharter and Weiberg 2006). The two philosophies of mathematics that can be reconstructed from the explicit statements contained in Wittgenstein’s work do not, however, exhaust the wealth of incentives which Wittgenstein’s ideas offers for the philosophy of mathematics. In addition to the efforts to understand precisely what Wittgenstein in the particular period of his development thought about mathematics, we can try to apply his theory of language to the language of mathematics. Thus first, we can apply to mathematics the Tractarian distinction of what can be said and what is only shown in the language. It turns out that such a distinction is in the language of geometry or in the language of algebra a meaningful one and it allows a deeper understanding of the semantics of these areas of mathematics. By applying the Tractarian theory of language to the language of mathematics we get theory, which I shall refer to as Wittgenstein’s implicit philosophy of mathematics of the period of the Tractatus. Considerations of this kind led to papers (Kvasz 1998, and 2006) and were summarized in the second chapter of (Kvasz 2008). Similarly we can apply to mathematics the concept of a language game and see mathematics itself as a set of interrelated language games. So we get Wittgenstein’s implicit philosophy of mathematics of the period of the Philosophical Investigations. The theory presented in (Kvasz 2000) and summarized in the first chapter of (Kvasz 2008) can be interpreted in this way. The structures described as “tools of symbolic and iconic representations”, can be interpreted as language games. Experts on Wittgenstein will perhaps stress that the mentioned implicit philosophies of mathematics contradict Wittgenstein’s explicit philosophy of mathematics and thus give a distorted picture of his views. Nevertheless, an understanding of the tension between the explicit and the implicit philosophies of mathematics of Wittgenstein seems to be important from the point of view of the history of philosophy of mathematics.