This Element discusses the problem of mathematical knowledge, and
its broader philosophical ramifications. It argues that the
challenge to explain the (defeasible) justification of our
mathematical beliefs ('the justificatory challenge'), arises
insofar as disagreement over axioms bottoms out in disagreement
over intuitions. And it argues that the challenge to explain their
reliability ('the reliability challenge'), arises to the extent
that we could have easily had different beliefs. The Element shows
that mathematical facts are not, in general, empirically
accessible, contra Quine, and that they cannot be dispensed with,
contra Field. However, it argues that they might be so plentiful
that our knowledge of them is unmysterious. The Element concludes
with a complementary 'pluralism' about modality, logic and
normative theory, highlighting its surprising implications.
Metaphysically, pluralism engenders a kind of perspectivalism and
indeterminacy. Methodologically, it vindicates Carnap's pragmatism,
transposed to the key of realism.
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