Foundations of Mathematics, Truth, and Implicit Commitments, University of Warsaw, Warsaw, 11 – 13 April 2024

Conference Description

In recent years, the notion of implicit commitments has received new attention in the Philosophy of Mathematics. Focusing on theories of foundational interest, in which substantial parts of mathematics can be reconstructed, philosophers, mathematicians, and logicians have been trying to determine the extent of the commitments (if there are any) implicit in foundational theories. This investigation started in the 60s with the work of Solomon Feferman and others on the so-called’ reflection principles’, statements expressing, for a given theory S, that S is sound. Famously, Feferman investigated whether, for a foundational theory S, such reflection principles are implicit commitments of S. Since the 60s, Feferman’s investigation generated an enormous amount of literature and research programmes. Although much progress has been made in our understanding of implicit commitments, much work is still needed.

Our conference aims to provide a platform to gather philosophers, mathematicians, and logicians working on implicit commitments and related notions in the context of philosophy and the foundation of mathematics.

Speakers 

Andrea Cantini (Università degli Studi di Firenze)
Ahmet Çevik (Gendarmerie and Coast Guard Academy)
Martin Fischer (LMU Munich & MCMP)
Kentaro Fujimoto (University of Bristol)
Volker Halbach (University of Oxford)
Leon Horsten (University of Konstanz)
Graham Leigh (University of Gothenburg)
J. Miguel Lopez Munive (University of Oxford)
Carlo Nicolai (King’s College London) 
Simon Schmitt (University of Turin)
Michael Sheard (St. Lawrence University)
Albert Visser (Utrecht University) 

Conference Program

11 April

09:50–10:00 Cezary Cieśliński, Welcome
10:00 –11:00 Andrea Cantini, A Fixed Point Theory over Stratified Truth
11:00–11:30 Coffee break
11:30–12:30 Graham Leigh, Truth, Proof & Realizability 
12:30–14:30 Lunch
14:30–15:30 Carlo Nicolai, TBA
15:30–16:00 Coffee break
16:00–16:45 Simon Schmitt, Set-theoretic Bicontextualism
16:45–17:00 Coffee break
17:00–17:45 J. Miguel Lopez Munive, Non-classical semantics for necessity conceived as a predicate

12 April

10:00–11:00 Volker Halbach, Soundness, Completeness, and Reflection 
11:00–11:30 Coffee break
11:30–12:15 Martin Fischer, Implicit Commitment, Reflective Closure and Conceptual Consequences
12:15–12:30 Coffee break
12:30–13:15 Michael Sheard, Proof Theoretic Circularity and the Liar
13:15–15:00 Lunch
15:00–16:00 Kentaro Fujimoto, Some variants of the theory of classical determinate truth
16:00–16:15 Coffee break
16:15–17:15 Open Problem Section
19:00 Conference dinner

13 April

10:00–11:00 Albert Visser, TBA
11:00–11:30 Coffee break
11:30–12:15 Ahmet Çevik, Structuralism and Choice-free Intuitionistic Theories
12:15–12:30  Coffee break
12:30–13:30  Leon Horsten, Implicit commitments of theory acceptance versus implicit commitments of acceptance of concepts

Abstracts

Andrea Cantini, A Fixed Point Theory over Stratified Truth. We look back to μ-calculus and take (loose) inspiration from work on intensional fixed points over 1st-order arithmetic, where one is allowed to build up fixed points in a very nested and entangled way. We experiment with strong systems and address the question: to what extent is a stratified (implicitly type theoretic) discipline compatible with self-reference or unfoundedness.

Ahmet Çevik, Structuralism and Choice-free Intuitionistic Theories. A raw object is a relational entity which is not equipped with a well-ordering. I introduce a novel constructivist account of structuralism by developing a theory of raw objects that are produced in case one does not assume any version of the Axiom of Choice (AC). I then argue, so as to demonstrate how constructive/intuitionistic positions of structuralism are affected on Choice-free intuitionistic philosophies of mathematical practice, that raw objects constitute in structuralism an intermediate ontology between ante-structures and systems. I also discuss the relationship between raw objects, unlabeled graphs, and Fine’s arbitrary objects, along with Horsten’s theory of generic structures. Consequently, raw objects seem to naturally appear in constructive/intuitionistic (hence, Choiceless) versions of structuralism.

Volker Halbach, Soundness, Completeness, and Reflection. I define logical consequence for standard first-order logic using an axiomatized primitive truth predicate and argue that this is the proper setting for discussing sound and completeness theorems, whose proofs are then compared to their usual model-theoretic versions. The soundness 
theorem is the global reflection principle for logic. I discuss then reflection principles for mathematical and semantic theories and their connection to truth-theoretic principles.

Graham Leigh, Truth, Proof & Realizability. Compositional theories of truth tend to follow the Tarskian tradition, either the model-theoretic semantics for classical predicate logic or its generalisation to many-value and possible-worlds. Other realisations of truth, which includes computational interpretations and game semantics, have been largely ignored by truth theorists. In this talk I will present axiomatic rendering of Krivine’s “classical realisability”, a ‘truth as programs’ semantics for classical logic that validates extensions of Peano arithmetic (and generalises the Tarskian model). What sets this conception of truth apart from the Tarskian view is its treatment of falsity as primitive and truth as a derived notion. The traditional compositional theories of truth arise as special cases of classical realisability. This is joint work with Daichi Hayashi (Hokkaido University, Japan). 

Michael Sheard, Proof Theoretic Circularity and the Liar. The notion of circularity is intimately connected with our understanding of the Liar paradox.  Even just the formulation of the Liar sentence requires circularity of reference; attempts to assign a truth value to the Liar sentence results in semantic circularity; proposals to resolve or explain away the paradox have resulted in what might be called circularity of resolution, in the form of the Strengthened Liar paradox and other “revenge” phenomena.  In this talk I will add a form of proof-theoretic circularity to this list.  Specifically, I will show how an attempt to apply the process of cut elimination to a particular derivation of the Liar paradox leads to yet another kind of circularity.  I will conclude the presentation with some thoughts about the philosophical implications of proof-theoretic circularity, and what it might tell us about the essence of the Liar paradox and our attempts to create a robust theory of type-free truth.