The Limits of Formalism: A Phenomenological Critique of Mathematical Meaning and Intelligibility
Abstract
This article presents a comprehensive phenomenological critique of formalism in the philosophy of mathematics, arguing that purely syntactic interpretations fail to account for the genesis and structure of mathematical meaning. Formalism, primarily associated with David Hilbert, conceives of mathematics as the manipulation of symbols according to explicit rules; however, it omits examining the conditions under which these symbols acquire meaning. Drawing on Edmund Husserl's transcendental phenomenology and engaging with both Gottlob Frege's logical analyses and Ferdinand de Saussure's linguistic contributions, this study argues that mathematical intelligibility is grounded in a pre-symbolic realm of ideal structures, constituted within acts of consciousness.
The article also proposes a tripartite model in which forms act as the ideal ground of meaning, symbols serve as mediators of formal operations, and words provide discursive expression. Through this conceptual framework, it is demonstrated that formalism presupposes precisely what it seeks to eliminate: a prior realm of intelligibility that cannot be reduced to mere symbolic manipulation. By reconstructing the fundamental processes of emergence, stabilization, and communicability of mathematical meaning, this work establishes both the limitations of formalism and the need for a phenomenological alternative.
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