Lecture

Extremal Assumptions in the Foundations of Mathematics

Logic Café Hybrid Lecture
The Philosophy Department and the Institute Vienna Circle are jointly organizing a series of talks this term

Date: 19/06/2023

Time: 16h45- 18h15

Meetings are usually held on Mondays from 16:45 to 18:15 in Raum 3D (D0316), 3. floor Department of Philosophy

This talk can be followed via online Plattform.

Online Plattform access:

univienna.zoom.us/j/99422395420

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Abstract:

Extremal axioms impose a condition of either minimality or maximality on the admissible models of an axiomatic theory. For example, while the axiom of Induction enforces the requirement of minimality on the models of Peano arithmetic, the axiom of Line Completeness imposes the condition of maximality over the models of Hilbert geometry. Similar extremal assumptions have been adopted in real analysis, non-standard analysis and set theory. The idea of extremal assumptions is potentially much more important than has been pointed out in the literature -- with the exceptions of Carnap & Bachmann (1936), Hintikka (1998) and Schiemer (2012). In this talk, I will address two main questions: (i) How to formulate extremal assumptions in a suitable logical language? (ii) Under what conditions do extremal assumptions entail the categoricity of the theory?

Report

During the six-month fellowship at the Institute Vienna Circle, I completed the first chapter of my PhD thesis on extremal assumptions in the foundations of mathematics. Specifically, the first chapter investigates how the restrictions of minimality and maximality can be implemented in precise and general terms. After presenting the initial draft of the chapter at the Logik Cafe seminar (University of Vienna) in May 2023, I have extensively revised the work.

At present, the paper provides a historically informed presentation of the so-called extremalist program in the foundations of mathematics, guided by two main questions: (i) how to formulate the extremal properties of minimality and maximality in a suitable metatheory? (ii) under what conditions do extremal properties characterize up to isomorphism the intended structure of, for instance, the natural and real numbers? After assessing the proposal of Carnap & Bachmann (1936), I claim that the extremalist program can be carried out within two alternative frameworks -- i.e. model-theoretic and category-theoretic structuralism. Ultimately, I point out that the extremalist program sheds new light on the commonly accepted distinction between intended and algebraic structures.

I presented the revised work at the 11th European Congress of Analytic Philosophy in August 2023 and at the 15th Conference of the Italian Society for Analytic Philosophy in September 2023. I'm currently planning to send the work for publication in a top-tier logic journal.