 A second article from the 1960s that was influential in the rise of structuralism is Hilary Putnam’s “Mathematics without Foundations” (1967). Like in Benacerraf’s case, for Putnam the foil was a set-theoretic foundationalist position. This position is sometimes, although not always, understood in a realist sense (e.g., by Gödel), i.e., as the description of an independent realm of abstract objects, namely the universe of sets characterized by the Zermelo-Fraenkel axioms. In opposition to that position, Putnam suggests a form of “if-then-ism” (which can be traced back to Russell).  That alternative can again be illustrated in terms of the natural numbers. How should an arithmetic theorem, say “\(2+3=5\)”,  be understood now? It should be analyzed as having this form: 