Does the concept of natural numbers truly require reference to abstract objects? This philosophical paper critically examines Hofweber's argument for 'internalism,' a thesis asserting that expressions referring to natural numbers do not, in fact, refer to any real objects. The analysis challenges the notion of 'restricted nominalism,' which posits that natural numbers do not exist as abstract entities. The authors contest Hofweber's claim that internalism necessitates restricted nominalism. The analysis asserts that Hofweber's central argument for this position is flawed, and the proposed analysis of quantification risks collapsing internalism into either a traditional error theory or a form of realism, thereby undermining its initial premise. This study offers a robust critique of internalism as a basis for restricted nominalism about natural numbers. By questioning the validity of Hofweber's arguments and highlighting potential inconsistencies, the authors contribute to the ongoing debate regarding the ontological status of numbers and the nature of mathematical discourse.
As a theoretical exploration in the Synthese journal, this work aligns with the publication's broad scope in philosophy and social sciences. It engages in a deep investigation of ontological and semantic claims, contributing to the publication's ongoing dialogue about fundamental questions in philosophy.