Sebastian G.W. Speitel

February 8th until April 7th, 2025

Affiliation: University of Bonn

Research for a study about:

Carnap’s Problem and Mathematical Determinacy

In a somewhat less well-known work, Carnap (1943) demonstrated that the model-theoretic values of the logical constants of classical propositional and first-order logic are underdetermined by their usual inferential characterizations. This constitutes Carnap’s (Categoricity) Problem. The ‘underdetermination of semantics by syntax’ uncovered by Carnap has consequences for a range of philosophical projects. Any position that claims that access to model-theoretic semantics is achieved on the basis of the inferential behaviour of expressions, for example, will have to contend with this particular type of underdetermination.
    This project is concerned with the repercussions of Carnap’s Problem for questions concerning mathematical determinacy. It has been a long-standing issue for so-called moderate mathematical realists to justify how determinate reference to mathematical structures can be achieved on naturalistically acceptable grounds given the possibility of non-standard models for relevant theories. While such indeterminacy can be reduced by adopting formalisms that enable categoricity proofs for important mathematical theories the challenge consists in explaining on what basis access to the resources needed to articulate such logics can be justified. After all, the determinacy of crucial notions such as ‘all subsets’, ‘infinitely many’, etc. might be just as out of reach as the structures the moderate realist attempts  to pin down by means of them.
    The underdetermination Carnap discovered for classical logic proves helpful for the moderate mathematical realist, for a solution to Carnap’s Problem includes an account of what it takes for a notion to be ‘fully determined’. This standard of determinacy is not restricted to the limited setting of purely logical notions, but holds for any formal concept access to which is mediated by inference. In particular, then, the moderate mathematical realist’s determinacy problem can be seen as an extension of Carnap’s original problem for logical notions, and resolving underdetermination in the logical case might help with reducing underdetermination in the mathematical case as well.

Lecture

Mathematical Determinacy

Philosophy of Science Colloquium Talk

Date: March 06, 2025

Time: 16h45 - 18h15

Meetings are usually held on Mondays from 16:45 to 18:15 in HS 3A

Abstract:

The existence of non-standard models of important mathematical theories, such as first-order Peano-Arithmetic, threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to mathematical structures is possible on the basis of our best mathematical theories. The move to frameworks stronger than FOL to articulate ‘better’ versions of these theories is denied to the moderate realist on the grounds that it merely shifts the indeterminacy ‘one level up’ into the meta-theory by — illegitimately, — assuming determinacy of the notions needed to formulate such logics.
    In this talk I want to outline the beginnings of a response to the determinacy challenge facing the moderate mathematical realist. I argue that the unique determinability of notions that enable categorical characterizations of important mathematical structures provides grounds for claiming naturalistically acceptable access to these structures, sufficient to resolve the determinacy challenge. I will illustrate the idea by showing how the mathematical realist may achieve arithmetical determinacy, and discuss ways to extend this approach to richer mathematical theories