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date: 20 October 2021

Abstract and Keywords

David Hilbert claimed that all mathematical problems can be solved. While Ludwig Wittgenstein agrees, he also wants to distinguish between different uses of the word ‘problem’. ‘Open problems’ in mathematics are often understood as being expressed by mathematical propositions that are undecided in the sense that they are not known to be provable or disprovable within currently accepted systems of mathematics. Wittgenstein thinks, controversially, that it is nonsensical to treat such ‘propositions’ as genuine mathematical propositions. On the other hand, he does not want to claim that it is illegitimate for mathematicians to concern themselves with problems such as Goldbach's conjecture, or Fermat's last theorem. This article explores the way Wittgenstein dealt with these issues in the early 1930s, focusing on his treatment of Fermat's theorem. It also looks at how Wittgenstein's treatment of these issues reflects a development towards a less dogmatic view of what is involved in mathematicians' claims to understand such ‘propositions’, which is an important part of the development of his philosophical views from the ‘middle’ to the ‘late’ philosophy.

Keywords: Ludwig Wittgenstein, open problems, mathematics, philosophy, propositions, Fermat's theorem

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