Mirko Engler PhD

November 1st 2019 - June 30th, 2020

Research for a study about:

To investigate the intellectual engagement of the mathematician Paul Bernays with the later philosophy of Rudolf Carnap during a 7-month study visit at the Institute Vienna Circle

Report

The axiomatic method as conceived by David Hilbert and developed in collaboration with Paul Bernays from 1919 to 1939 is generally recognized as having a great impact on the philosophy of the Vienna Circle, especially on the thinking of Rudolf Carnap. Both Carnap's early work in the philosophy of mathematics, which can be seen as the approach to reconcile logicism and formalism, and his general philosophical attempt to extend the axiomatic method from mathematics to the sciences in general can be regarded as evidence for that claim. While this direction of intellectual interaction, so to speak from Göttingen to Vienna, is extensively studied, the opposite direction still deserves closer attention. The purpose of the study visit has been to contribute to that. In the course of the stay at the Institute Vienna Circle, the study of a letter exchange between Carnap and Bernays as well as a few talks given by Bernays in his later years has revealed several topics in Bernays thinking which are directly and indirectly influenced by Carnap. To mention two important ones:

- Following Carnap, Bernays believed - in contrast to the spirit of the 1960's - in the general possibility of separating the sciences into an empirical part and an analytic part. 

- Influenced by Carnap's expansion of the axiomatic method to the sciences in general, the later Bernays developed a severe skepticism concerning the suitability of the formal axiomatic method - even inside mathematics.

A talk given at the online Logic Café discussed some more technical issues of meaning and reduction of theories - topics that are both present in Carnap's and Bernays' work.

Podcast Lecture at Logic Café

Title

"Aspects of Inter-Theoretic Reduction"

Discussion of Podcast:

Date: 25/05/2020

Time: 17h15-18h00

Venue: Logic Café on ZOOM

siehe logik-cafe.univie.ac.at

Abstract:

The talk considers inter-theoretic reduction relations that extend relative interpretability and discusses their preservation characteristics of some formal properties which are relevant to the meaning of theories.

Thereby, we intend to characterize "reduction relation" by certain intuitive criteria of reducing theories with the help of translations functions. A prominent, special case of such a reduction relation is relative interpretability. By definition, relative interpretations preserve provability of sentences under translation. We are pushing forward the issue of how to construct extensions of relative interpretability in order to preserve certain semantical properties of theories under translation. The main idea in accomplishing this task is to consider strong notions of equality of theories and consider their corresponding reduction relations. In the final analysis, we propose a corresponding reduction relation to the notion of bi-interpretability and investigate its preservation characteristics with regard to kappa-categoricity and automorphism groups.

Lecture (cancelled due to Corvid-19)

Lecture at : Logik Café

Aspects of Inter-Theoretic Reduction

 

Date: April 20th, 2020

Time: 17:15 - 18:45

Venue: NIG, HS 3A