Linguistic realism in mathematical epistemology

One project in the epistemology of mathematics is to find a defensible account of what passes for mathematical knowledge. This study contributes to this project by examining philosophical theories of mathematics governed by certain basic assumptions. Foremost amongst these is the “linguistic realism” of the title. Roughly put, this is the view that the semantics of mathematical sentences should be taken at face value. Two approaches to mathematics are considered, realist and fictionalist. Mathematical realism affirms the existence of mathematical objects, taking much of what passes for mathematical knowledge as knowledge of such things. It faces the challenge of explaining how such knowledge is possible. The main strategies here are to appeal to the faculty of reason, to a faculty of intuition or to the faculty of sense perception. Recent examples of each strategy are considered and it is argued that the prospects for a satisfactory mathematical realism are limited. Mathematical fictionalism does not affirm the existence of mathematical objects, claiming that mathematics is, or should be considered to be, a form of pretence. It faces the challenge of explaining how a form of pretence can discharge the roles mathematics has in empirical applications. Strategies here are to argue that mathematics is an eliminable convenience or, acknowledging that this may not be the case, that the roles played by mathematics in empirical applications are played in similar contexts by acknowledged forms of pretence. It is argued that the first strategy is not promising but that there is a version of the second that can be defended against objections. In closing, consequences of the conclusions reached are explored and directions for future research indicated.