Bingo. Mathematics is a consistent model which can be applied to do useful things and make useful predictions, but like any logical system it springs from axioms that must simply be taken as given. Most of these axioms (Modus Ponens[1], for example) are so simple and obvious that it seems bizarre to construct a way of thinking without them, but that does not intrinsically make them "true".
[1] http://en.wikipedia.org/wiki/Modus_ponens