The Pirah\~a and the cognitive gap in Frege's theorem: Hume's principle without the #
Frege's theorem proves that Hume's principle, in second-order logic, yields all of arithmetic. Yet the Pirah\~a people show one-to-one correspondence (equinumerosity) only where pairing can be enacted, with its range extended under local training, and still have no counting or arithmetic. We argue this is not a paradox but a matter of precise localization. Hume's principle includes a cardinality operator # that names cardinals as objects (often modeled as equivalence classes of equinumerous concepts), and what the Pirah\~a lack is not the relation but this operator. We identify the number-word practice as the cognitive realization of #, which recasts the "number-as-cognitive-technology" thesis in formal terms and locates the cognitive boundary at symbolization, not recursion. The identification is generative, not decorative: the reach of # tracks the reach of the token practice that carries it, so across languages and cultures we see a gradient, not a sharp cliff. And number words are not special as words; what # needs is any stable, reusable marker that can preserve exact cardinal identity across absence, rearrangement, delay, or modality shift: a spoken numeral, a scratch on a stick, or a knot in a cord. So the thesis is about having some symbolic token-practice, not about language specifically. It is supported by converging evidence from Nicaraguan homesigners, numerate adults under verbal interference, and cross-linguistic numeral gradients. We make no causal, acquisition, or neural claim; the identification is constitutive.