 Ante rem structuralism, as well as the modal nominalist structuralist interpretation of mathematics, could benefit from a second-order formulation. If the ante rem structuralist wants to insists that the natural number structure is fixed up to isomorphism by the Peano axioms, then she will want to formulate the Peano axioms in second-order logic. And the modal nominalist structuralist will want to insist that the relevant concrete systems for arithmetic are those that make the second-order Peano axioms true (Hellman 1989). Similarly for real analysis and set theory. Thus the appeal to second-order logic appears as the final step in the structuralist project of isolating the intended models of mathematics.