Observation and Indispensability

This chapter argues that mathematical beliefs have no better claim to being empirically – i.e., a posteriori -- justified than our moral beliefs. It shows that Harman’s influential argument to the contrary is fallacious. It then formulates a better argument for a lack of parity between the cases, in terms of indispensability. It argues that, while the necessity of mathematics is no bar to developing a mathematics-free alternative to empirical science, the contents of our arithmetic beliefs, realistically and even objectively construed, do seem to be indispensable to metalogic. But this at most shows that a subset of our mathematical beliefs have better claim to being empirically justified. Surprisingly, however, the range of moral beliefs that we actually have may be so justified, in a more direct way. The chapter concludes with the prospect that there is no principled distinction between intuition and perception, and, hence, between a priori and a posteriori justification.