Variations of Intuitionistic Revisionism

(Logic and Philosophy of Mathematics, Englisch)

t is well known that intuitionists reject classical reasoning over infinite domains. But their arguments against it are not homogenous throughout different proponents of the issue and they yield further philosophical and technical differences down the line. In this paper, I will differentiate three types of intuitionistic revisionism, which I attribute to Weyl (with an argument shared with Wittgenstein), Dummett, and Bishop, and compare their relative merits. I will particularly focus on their justification of induction.

The respective arguments against classical reasoning over infinite domains center around the following objections:

1. Classical reasoning is unable to make certain distinctions with respect to infinite domains and thus lacks means of differentiation (Bishop).

2. Classical laws for generalisations over infinite domains lack epistemic warranty (Dummett).

3. There is a complete lack of sense attributed to generalisations over infinite domains in general (Weyl, Wittgenstein).

I will argue that Bishop's proposal turns out to be the one that is the least confrontational towards classical mathematics, but, when put to the test, it does not seem to avoid a possible deadlock scenario with the proponent of classical mathematics. Weyl and Wittgenstein pose the most radical critique towards classical reasoning over the infinite, but they overshoot and thereby cause significant problems for a subsequent justification of induction. Finally, Dummett seems to occupy a middle ground. His argument, if successful, forces the classical mathematician to make substantial epistemic or ontological commitments while also managing to retain and justify induction as a legitimate principle.

Chair:

Zeit: 14:00-14:30, 07. September 2022 (Mittwoch)

Ort: SR 1.004

Jann Paul Engler

(University of St Andrews, United Kingdom)

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