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Added by: Viviane FairbankAbstract:
In Remarks on the Foundations of Mathematics, and elsewhere, Wittgenstein made some general statements about mathematical proof which can only be described as paradoxical. What he says is directed to proofs for which there is no general method of solution such as there is for statements of the form a x b = c where a simple calculation is sufficient. His concern is with propositions for which one cannot in advance of proof describe any procedure which will demonstrate them... In an earlier study,' 'Proof and the Theorem Proved', I singled out a group of propositions for which his account of proof seemed correct, and not paradoxical.... But I did not succeed in making his account of deductions of connections between established concepts paradox-free. I should like to return to his account as applied to such demonstrations, with the aim of explaining what justification there is for it, and what his account comes to.
Comment: This paper is a follow-up to Ambrose's earlier "Proof and the Theorem Proved" (which is also listed in the DRL).
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Added by: Viviane FairbankAbstract:
There is something curiously opaque about Wittgenstein's assertions concerning the connection between proof of a proposition p, sense of the sentence " p ", and understanding " p ". The difficulties inherent in his account are apparent, and he has
himself mentioned them, while leaving their resolution obscure. I shall now detail them, so that it will be clear what difficulties must be got round and what clarification is required.Comment: This is a useful work of Wittgenstein exegesis by one of his students and collaborators, concerning his views on mathematical proof with discussion of several examples. It could be used in an intermediate/advanced course on the philosophy of mathematics or on Wittgenstein's philosophy more generally.
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Added by: Jie GaoPublisher's Note: Does scepticism threaten our common sense picture of the world? Does it really undermine our deep-rooted certainties? This book offers an answer to these questions through a comparative study of the epistemological work of two key figures in the history of analytic philosophy: G. E. Moore and Ludwig Wittgenstein. While historically accurate and engaging with scholarly work in this area, the book also puts forward novel interpretations of their works and brings out their relevance to present-day debates both in epistemology and philosophy of language.Comment: This book is a useful and sustained examination of a variety of themes in Wittgenstein's On Certainty, the very late compilation of remarks inspired by G.E. Moore's engagement with scepticism and idealism in "A Defence of Common Sense," "Proof of an External World" and a few other papers. Among the topics considered are the strategies of Moore's arguments, ordinary and philosophical uses of language, differing interpretations of Moore, externalism, internalism and contextualism, Wittgenstein's objections to Moore, meaning and use, language games, Cartesian and Humean sceptical arguments, the epistemic and semantic status of so-called "hinge" propositions, epistemic relativism, and a comparison of Wittgenstein's and Moore's views with those of subsequent philosophers. It thus constitutes a very good reading or even central text for a course on Moore's epistemology, Wittgenstein's epistemology and external world skepticism.
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Added by: Fenner Stanley TanswellAbstract:
The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem.Comment (from this Blueprint): Secco and Pereira discuss the famous proof of the Four Colour Theorem, which involved the essential use of a computer to check a huge number of combinations. They look at whether this constitutes a real proof or whether it is more akin to a mathematical experiment, a distinction that they draw from Wittgenstein.