Applied mathematics often operates by way of shakily rationalized expedients that can neither be understood in a deductive-nomological nor in an anti-realist setting.Rather do these complexities, so a recent paper of Mark Wilson argues, indicate some element in our mathematical descriptions that is alien to the physical world. In this vein the “mathematical opportunist ” openly seeks or engineers appropriate conditions for mathematics to get hold on a given problem.Honest “mathematical optimists”, instead, try to liberalize mathematical ontology so as to include all physical solutions. Following John von Neumann, the present paper argues that the axiomatization of a scientific theory can be performed in a rather opportunistic fashion, such that optimism and opportunism appear as two modes of a single strategy whose relative weight is determined by the status of the field to be investigated. Wilson's promising approach may thus be reformulated so as to avoid precarious talk about a physical world that is void of mathematical structure. This also makes the appraisal of the axiomatic method in applied matthematics less dependent upon foundationalist issues.