Gottlob Frege's foundational philosophy of mathematics derived the concept of natural numbers from pure logic via Hume's Principle; cognitive linguistic evidence from the Amazonian Pirahã language reveals a profound conceptual gap in logicist foundations.
In 1884, Gottlob Frege sought to establish that arithmetic is an extension of pure logic, proving in Frege's Theorem that natural numbers can be derived from one-to-one conceptual correspondence (Hume's Principle).
Frege assumed that the human capacity to identify equinumerosity—matching two sets one-to-one—naturally and universally produces the concept of discrete cardinal numbers.
This interdisciplinary philosophical and cognitive study examines the Pirahã tribe of the Amazon, whose language possesses no discrete number words or grammatical counting markers. By showing that one-to-one matching can function without ever giving rise to discrete cardinal numbers, the author exposes an unacknowledged cognitive gap in Frege's logicist program.
This work reshapes the philosophy of mathematics, proving that arithmetic cannot be reduced to abstract logic alone without incorporating culturally mediated cognitive technologies like linguistic counting routines.
The Pirahã 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 Pirahã 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 Pirahã 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.
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