Applied mathematics often operates by way of shakily rationalizedexpedients 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 ourmathematical descriptions that is alien to the physical world. In this vein the mathematical opportunistopenly 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 physicalsolutions. Following (...) John von Neumann, the present paper argues that the axiomatization of a scientifictheory can be performed in a rather opportunistic fashion, such that optimism and opportunism appear as twomodes of a single strategy whose relative weight is determined by the status of the field to beinvestigated. Wilson's promising approach may thus be reformulated so as to avoid precarious talk about a physicalworld that is void of mathematical structure. This also makes the appraisal of the axiomatic method inapplied matthematics less dependent upon foundationalist issues. (shrink)