Jakub Macha from
Masaryk University, Brno, Czech Republic, visits the Wittgenstein Archives in the period 3-11 December 2013. In this context, a seminar is organized:
- Tuesday, December 10, 2013, 12.15-14.00, Sydnesplassen 9 (Sydneshaugen skole), rom N 230:
Pictorial Aspects of Mathematical Notation in Wittgenstein: Numbers and Proofs
The core in Wittgenstein’s conception of mathematics can be summed up in the motto that “arithmetical rules are statements of internal relations.” (PPO, p. 390) I am going to focus on Wittgenstein’s insistence on a certain pictorial aspect of mathematical notation, which is, of course, his Tractarian heritage. Mathematical notation must always be capable to depicture a state of affairs. This is true of numbers, but also of mathematical proofs. Numbers and proofs are for Wittgenstein a sort of prototypes of certain activities. (1) The pictorial aspect of numerals is expressed in the key definition of a cardinal number: “A cardinal number is an internal property of a list.” (PR, p. 140) Wittgenstein’s concrete and finitistic approach takes numeral for concrete objects as opposed to Frege-Russell’s approach based on abstract sets. The decisive advantage of Wittgenstein’s conception of numbers over Frege and Russell’s is that numbers are rooted in our primitive activities like children’s finger counting or counting with the abacus. (2) Mathematical propositions are statements of internal relations as well. A proof of a mathematical proposition aims to picture or rather lay down its internal relatedness to a system of other mathematical rules. We may say that “the completely analysed mathematical proposition is its own proof.” (PR, p. 192) Proof is so a picture of an experiment, even more “it can be thought of as a cinematographic picture” (RFM, p. 159).
