On the Philosophical Significance of Frege's Constraint

Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege's Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts that such constraints are ‘toothless’, showing they both assuage Frege's original concerns and accommodate neo-logicist intents by dismissing ‘arrogant’ definitions.