Revolutions and Paradigms in Logic. The Case of Proof-Theoretic Semantics

• Applicant: Dr. Antonio Piccolomini d Aragona
• Subject Area: Theoretical Philosophy ; History of Science
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 525967005

Kuhn’s theories on scientific revolutions and paradigms, and Lakatos’ theories on scientific research programmes constitute two major contributions to the philosophical reconstruction of the historical-conceptual development of science. Both these approaches have been fruitfully applied to empirical sciences such as physics and chemistry; it is on the contrary much debated whether they can be employed also for the assessment of formal sciences. While the applicability of Kuhn’s and Lakatos’ theories to mathematics is somewhat debated, the issue is instead mostly unexplored when referred to the more specific field of (formal) logic. In my project, I aim at addressing the topic of a potential Kuhnian and Lakatosian rendering of post-Fregean and post-Hilbertian logic. Although this general question is of course expected to be dealt with as a starting point of my investigation, I shall otherwise endorse a sort of case-wise strategy, trying to draw broad conclusions from smaller but significant samples. In particular, I shall focus on the crucial case-study of the logical opposition between realism and (anti-realist) constructivism. My pivotal claim is the following: logic is currently dominated by a realist Kuhnian paradigm, constituted by the combination of model theory and set theory. Against this, one can detect a (currently) minor but well-established constructivist Lakatosian research programme, which I shall mainly understand as given by the combination of Prawitz’s semantics and Martin-Löf’s intuitionistic type theory. The realist paradigm arose in response to a critical moment in the history of logic, i.e. the discovery of semantic and set-theoretic paradoxes and the proof of limiting results such as Gödel’s incompleteness theorems. This determined the abandonment of previous foundational attitudes, and the adoption of new core-concepts for the analysis of validity and the foundations of mathematics. However, some ingredients from the previous frameworks survived, and merged into a sequence of theories which led to today’s wide field of proof-theoretic semantics. The proposed research project, besides being innovative in itself, is also seemingly fruitful for a number of reasons. First, it permits to ground in a philosophical-historical perspective one of the classical themes of philosophy, otherwise addressed in usually metaphysical terms, i.e. the fight between realists and anti-realists. Second, via Kuhn’s influence, the perspective may be also sociological; more specifically, one may understand the competition between theories also as a fight between different “geo-theoretical” traditions, each with its places and “champions”. Third (but the list may not be exhaustive), the research may constitute the first step towards a more ambitious project, i.e. a history of general proof-theory from Gentzen to the present time, a crucial, but so far missing piece in the otherwise variegated tradition of proof-theoretic studies.

• DFG Programme: Research Grants