Solving the Proper Problem: Wittgenstein, Fictionalism and the Applicability of Mathematics

This thesis proposes a solution to the problem motivating Albert Einstein's question, 'How can it be that mathematics … is so admirably appropriate to the objects of reality?'; the problem of mathematical applicability. It considers Mark Steiner‘s anthropocentric non-naturalist attitude to the problem in the light of Hartry Field's fictionalism, contending that Field's proposed solution, though prima facie naturalist in character, in fact inherits non-naturalist commitments which were problematic in anthropocentric solutions. Specifically, Field's solution retains a view of the grammar of our mathematical expressions as based on the model of object-and-designation. However, Field finds importance in the notion that what mathematicians know is non-propositional in the sense of being concerned with inference rules rather than assertions of existence, and a way of exorcising the rogue grammatical commitment in question from the fictionalist account can develop from a focus on this aspect of our talk about mathematics tout court rather than just epistemologically. The view that emerges when we do this, in the first instance enables a naturalistic and wholly adequate solution to the problem of mathematical applicability, and in the second instance is precisely that articulated by Ludwig Wittgenstein. Einstein's question seems pressing only if we understand the applicability of mathematics on the model of a kind of 'fit' between two realities, the mathematical and the physical/empirical. Once we attain an overview of the grammar of our mathematical talk and thereby recognise it for what it is, the question of fit between mathematics and the world can be seen to be empty. The importance of the notion of an overview of the problem and the general philosophical terrain in which it sits chimes with Wittgenstein's own attitude to the requirement for philosophy of achieving what he calls an 'übersichtliche Darstellung'.