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RETHINKING SPATIOTEMPORAL EXTENSION: HUSSERL’S CONTRIBUTION TO THE DEBATE ON THE CONTINUUM HYPOTHESIS

Abstract
In this text, I intend to demonstrate the relevance of Husserl’s phenomenology for the debate on Cantor’s continuum hypothesis. Once described the classical formulation of this problem by Cantor, Dedekind, Zermelo-Fraenkel, and Hilbert, I observe that the current discussion about this issue is characterized by the opposition between a Platonist (Gödel) and a formalist (Cohen) solution. Although this latter is widespread among mathematicians, a few of them still think that the continuum conjecture is relevant for a philosophical foundation of set theory and, in general, for a scientific description of reality. Most of them have been somehow inspired by Husserl's phenomenology. This is the case, for instance, for Weyl and Gödel himself, even if both of them gradually abandoned phenomenology for, respectively, constructivism/predicativism and Platonism. My aim in this text is to reconstruct this “minor” history, in order to show how Husserl’s account of the continuum, developed in different ways by Weyl and Gödel, remains the unique radical attempt to found mathematical formalization on intuition. Although the continuum, namely the phenomenological condition of both the flux of the lived-experiences and the flowing of the intuitive data, is a real leitmotiv of the phenomenological method as a whole, it plays a peculiar role in the early Husserl, notably in his lectures of 1891 on Philosophy of Arithmetic, those of 1905-1908 On the Phenomenology of the Consciousness of Internal Time, and those of 1907 on Things and Space. In these texts, there emerges a theory of how the concept of the continuum originates in the intuition of concrete data: more precisely, the intuition of continuity is conceived as the phenomenological condition for any mathematical formalization of the continuum. This does not entail that Husserl is not committed to the problem of a rigorous formalization of the continuum. Rather, as demonstrated by his in-depth inspection of spatial perception and time-consciousness, he is fully aware of the limits of any attempt of formalizing continuity (the same limits Weyl will emphasize concerning Cantor-Dedekind’s axiom). Accordingly, it is precisely for its attempt to keep together intuition and formalization that transcendental phenomenology still plays a relevant role in the current debate about the foundation of mathematics.

Key words
Continuum, set theory, platonism, formalism, intuition, time-consciousness.

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