The thesis according to which the meaning of a mathematical sentence is given by its proof was held by both Wittgenstein and the intuitionists, following Heyting and Dummett. In this paper, we clarify the meaning of this thesis for Wittgenstein, showing how his position differs from that of the intuitionists. We show how the thesis originates in his thoughts, from the middle period, about proofs by induction, and we sketch his answers to a number of objections, including the idea that, (...) given the particular meaning he gives to this thesis, he cannot account for mathematical conjectures. We conclude by showing how his views find a favourable echo today in the paradigm of “proposition-as-type” and extensions of the Curry-Howard isomorphism from which this paradigm originates. (shrink)

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to (...) appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument. (shrink)

A survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937–1939, and later testimony of Goodstein and Kreisel; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g., Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of Tarski's semantical (...) work, both for our understanding of Wittgenstein's remarks on Gödel, and our understanding of Gödel's theorem itself. (shrink)

Both John Langshaw Austin and Willard Van Orman Quine were critical of the traditional division of propositions into the two categories: analytic and synthetic. their criticism has, however, a di erent character. Quine questions the usefulness of the notion of analyticity, whereas Austin does not accept the view that every proposition should be considered either analytic or synthetic. According to Quine, we have to abandon the notion of analyticity because we cannot de ne it in a satisfactory way. Quine’s criticism (...) is based on his conviction that the very notion of meaning is suspicious om the scienti c point of view. this general outlook is supported by arguments the point of which is to show that we cannot avoid an indeterminacy of translation. Austin criticises the distinction for di erent reasons. According to him, it is not the notion of meaning which is suspicious, but a certain model of this notion — a model which is based on false analogies. In my text, I compare these two approaches and point out that they have di erent metaphilosophical sources. the main di erence lies in the fact that, according to Austin, statements about linguistic meaning usually have a descriptive character, whereas Quine claims that linguistic meanings are theoretical entities. In the last part of my article, I discuss the thesis of indeterminacy of translation and assess its credibility, as it plays a key role in Quine’s criticism of the notion of meaning. (shrink)

My concern in this thesis is with the transcendental question concerning the condition of possibility for social science. I argue that for social scientific knowledge to obtain one must: have a conception of knowledge formation as theoretically mediated and fallible; and, social scientific knowledge claims must be about an object of study which conceptualises social structure as an enablement as well as an external constraint upon agency. This means: arguing for an anti-foundational epistemology, which avoids becoming truth-relativism, by being complemented (...) with a metaphysical realist ontology ; and, using a social realist meta-theory of emergent properties to explain how methodology has a conceptually mediated and fallible access to social reality. Developing a critical examination of the presuppositions of social scientific knowledge also means, afortiori, using realism as an underlabourer. The negative underlabouring role is to proscribe theories based on some form of epistemic immediacy, or being-knowing identity. It therefore means rejecting positivist, empiricist and essentialist versions of social science. The form of essentialism dealt with is called the sociological logic of immediacy, and this pertains to definitive ontologies of social structures or human being. Whereas the use of positivist and empiricist epistemology as a positive underlabourer produces a methodology that conflates the real into the 'actual', the use of an essentialist ontology makes methodology either redundant, or an exercise in arbitrary verificationism. Against this, realist anti-foundationalism can act as apositive underlabourer for the social sciences if it is complemented by a social realist ontology of emergent properties, to act as a metatheory which guides methodology. In developing this argument my chief concern is to show that realism is a more adequate position than post-Wittgensteinian positions which focus on 'practices' and how people 'go on' in 'forms of life'. 'Adequacy' in this sense pertains to epistemological discussions about the status of knowledge, together with ramifications of post-Wittgensteinianism for knowledge of the socio-political realm. This means providing a critique of Rorty and Giddens, after dealing with the issue of empiricism. Although Rorty's critique of 'postmodernism' as essentialist is accepted. Whereas realism can explain how we have a conceptually mediated and fallible knowledge of reality, including social reality, post-Wittgensteinian positions fall into truth-relativism and essentialist conceptions of human being and social structures. (shrink)