Search

This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his method to a subset of first-order formulas, namely, formulas that do not contain dyadic sentential connectives within the scope of any quantifier. [ABSTRACT FROM AUTHOR]

Newton claims to have proven the heterogeneity of light through his experimentum crucis. However, Olaf Müller has worked out in detail Goethe’s idea that one could likewise prove the heterogeneity of darkness by inverting Newton’s famous experiment. Müller concludes that this invalidates Newton’s claim of proof. Yet this conclusion only holds if the heterogeneity of light and the heterogeneity of darkness is logically incompatible. This paper shows that this is not the case. Instead, in Quine’s terms, we have two logically compatible theories based on mutually irreducible theoretical terms. From a Quinean point of view, this does no harm to the provability of the corresponding statements. [ABSTRACT FROM PUBLISHER]

It is controversial whether Wittgenstein's philosophy of mathematics is of critical importance for mathematical proofs, or is only concerned with the adequate philosophical interpretation of mathematics. Wittgenstein's remarks on the infinity of prime numbers provide a helpful example which will be used to clarify this question. His antiplatonistic view of mathematics contradicts the widespread understanding of proofs as logical derivations from a set of axioms or assumptions. Wittgenstein's critique of traditional proofs of the infinity of prime numbers, specifically those of Euler and Euclid, not only offers philosophical insight but also suggests substantive improvements. A careful examination of his comments leads to a deeper understanding of what proves the infinity of primes. [ABSTRACT FROM AUTHOR]