Practice-dependent realism and mathematics
Julian C. Cole · OhioLink ETD Center (Ohio Library and Information Network) · 2005
I present a solution to a puzzle concerning the interpretation of mathematical practices.Mathematical claims seem to be objective and their surface grammar suggests that they are about objects or structures.Furthermore, these objects or structures do not appear to be located in the spatio-temporal world.For these reasons, Platonists have suggested that mathematical claims concern domains or structures that are abstract and are in no sense constituted by our intellectual activities.Yet if Platonists are correct, then something very peculiar is going on.Items that are abstract and not constituted by our intellectual activities cannot influence spatio-temporally instantiated activities like mathematical practices.Contemporary authors have responded to this observation in two distinct (and incompatible) ways.Some have argued that mathematics should be understood as a fiction that helps us represent and reason about the spatio-temporal world.Others hold that the surface grammar of mathematical claims is misleading; they are really claims about what is logically possible and/or necessary.A philosopher who accepts either suggestion must pay a high price.Specifically, (s)he expresses a problematic lack of respect for actual mathematical practice: either (s)he doesn't take mathematicians' claims literally or (s)he interprets mathematicians as offering extremely misleading characterizations of their subject matter.Neither option is acceptable.