Abstract

The debate between formalism and intuitionism has often been interpreted as a quarrel internal to the philosophy of mathematics. Such a reading nevertheless underestimates its epistemological scope. The respective positions of Hilbert, Brouwer, and Gödel in fact involve a more general question: what is an intelligible content, and under what conditions does a symbolic procedure become knowledge rather than a mere manipulation of signs? This article argues that Gödel’s incompleteness theorems should not be understood merely as a technical limitation of formal systems. They bring to light a structure of thought that exceeds calculation, not because thought would stand outside it, but because it presupposes the articulation of three irreducible moments: a moment of intuitive and schematic apprehension, a moment of critical reworking, and a moment of communicable formalization. Formalization is indispensable; yet it is not sufficient to generate meaning, to account for discovery, or to secure reference. If formalization is not sufficient to produce meaning, illuminate discovery, and ground reference, then scientific discovery must be understood as a synthetic reconfiguration rather than as a simple sequential chain, the effectivity of a theory must be distinguished from the truth it aims at, and the question of artificial intelligence must be reconsidered on new grounds. What artificial systems still lack is not combinatorial complexity, but the concrete inscription of their statements within an experience of the world, and the reflexivity that allows a subject to relate to its own operations as to something that concerns it. Thought thus appears as a circulation between several levels of meaning, and not as the mere execution of rules within a closed universe. This article refers to a broader project entitled Philosophy of Meaning, by the same author, available on PhilPapers. Keywords: Gödel, Hilbert, Brouwer, Kant, Popper, Turing, schematism, scientific discovery, meaning, artificial intelligence, reflexivity, effectivity.