The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the notion of “same meaning” (...) is itself context-sensitive, depending on the purposes of a given conversation. (shrink)

Wittgenstein characterises ‘necessary truths’ as rules of representation that do not answer to reality. The invocation of rules of representation has led many to compare his work with Kant's. This comparison is illuminating, but it can also be misleading. Some go as far as casting Wittgenstein's later philosophy as a specie of transcendental idealism, an interpretation that continues to gather support despite scholars pointing to its limitations. To understand the temptation of this interpretation, attention must be paid to a distinction (...) Bernard Williams (1981) makes, echoing Kant, between empirical and transcendental idealism. Williams claims that the move to transcendental idealism is an effort to block empirical idealism. The latter, in this context, amounts to a scepticism regarding necessity and a denial of its objectivity. To show what is wrong with the transcendental interpretation, it is important to challenge the mistaken presumption that Wittgenstein's philosophy would otherwise lead to empirical idealism. The fundamental mistake common to both attributions is that they misunderstand the relation that obtains between our rules of representation and the contingent facts that condition those rules of representation. Once this relation has been clarified, Wittgenstein's philosophy can be understood as providing a genuine alternative to realism and idealism. (shrink)

The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.