Abstract: While it is relatively clear what the subject matter of empirical sciences is, puzzles persist about the proper subject matter of mathematics. The logicists took mathematics to be concerned solely with deductive arguments. Their programme attempted to combine three incompatible claims: that mathematics is a body of truths, that it is non-empirical, and that it employs proofs obeying the rules of classical logic. By giving up the third contention, it becomes possible to salvage the logicist programme and to explain better mathematical practice, including the fact of its application in empirical disciplines.

Keywords: conservative extension, constructivism, Hartry Field, Frege, logicism, mathematics, real numbers,