Objects and Structures in the Formal Sciences

Mathematics, and mechanics conceived as a formal science, have their own proper subject matters, their own proper unities, which ground the characteristic way of constituting problems and solutions in each domain, the discoveries that expand and integrate domains with each other, and so in particular allow them, in the end, to be connected in a partial way with empirical fact. Criticizing both empiricist and structuralist accounts of mathematics, I argue that only an account of the formal sciences which attributes to them objects as well as structure, proper semantics as well as syntax, can do justice to their intelligibility, heuristic force and explanatory power.