A sound philosophy of mathematical physics balances a philosophy of mathematics with a philosophy of physics, sharpening the general applicability problem of mathematics by also taking care of: (i) the early modern `mathematization of the world picture'; (ii) the theory-laden character of the targets of mathematical models of modern physics; and (iii): Wigner's `unreasonable effectiveness of mathematics in the natural sciences'. Guided by a historical survey, I propose that theories of mathematical physics are meaning-constitutive a priori constructions, conventional but far from arbitrary and best described as hypothetical. Their models subsequently mediate between theory and nature, that is, between the a priori and the a posteriori. Models mediate by playing the role of Wittgensteinian yardsticks or objects of comparison to be held against nature as represented by data models, where the comparison is made via surrogative inference. This balancing act compromises realism.
