 An ongoing debate in the foundations of quantum physics concerns the role of mathematical rigor. The contrasting views of von Neumann and Dirac provide interesting and informative insights concerning two sides of this debate. Von Neumann’s contributions often emphasize mathematical rigor and Dirac’s contributions emphasize pragmatic concerns. The discussion below begins with an assessment of their contributions to the foundations of quantum mechanics. Their contributions to mathematical physics beyond quantum mechanics are then considered, and the focus will be on the influence that these contributions had on subsequent developments in quantum theorizing, particularly with regards to quantum field theory and its foundations. The entry  quantum field theory  provides an overview of a variety of approaches to developing a quantum theory of fields. The purpose of this article is to provide a more detailed discussion of mathematically rigorous approaches to quantum field theory, as opposed to conventional approaches, such as Lagrangian quantum field theory, which are generally portrayed as being more heuristic in character. The current debate concerning whether Lagrangian quantum field theory or axiomatic quantum field theory should serve as the basis for interpretive analysis is then discussed.