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This paper interprets John von Neumann's views on the proper role of mathematics in science, with attention to its implications for economics. The evolution of von Neumann's thought over his lifespan, an examination of which is greatly aided by some self-conscious appraisals of von Neumann himself, suggests that von Neumann ended by taking a very pragmatic approach to the use of mathematics. One can almost characterize it as an engineering approach to the philosophy of mathematics since von Neumann takes mathematics to be justified by its applications. This approach, which is surprisingly non-technical and hence accessible to general readers, is then placed in the context of his wider world-view.