New Directions in the Foundations of Mathematics: Truth, Commitments and its Philosophy

This book brings together new research-level papers in the philosophy of mathematics on truth and commitments in the foundations of mathematics. It provides a comprehensive and current context for researchers, both specialists and non-specialists, in these fields. From both logical and philosophical perspectives, this collection addresses the relationships between the concept of truth, the notion of (implicit and explicit) commitments to foundational theories, and their connection and impact on philosophy and the foundations of mathematics. With this, the book offers new insights into these research areas and fosters discussions that bridge gaps between recent advances in logic and their philosophical implications. Inhaltsverzeichnis Preface (Cezary Cieslinski and Matteo Zicchetti).- Part I: Implicit Commitment and Philosophical Foundations.- 1. More Implicit Commitment (Leon Horsten).- 2. Implicit commitments and the infinitary rules (Bartosz Wcislo).- 3. Truth, Reflection, and Implicit Commitment (Richard Kimberly Heck).- Part II: Truth, Incompletess, and Classical Metamathematics.- 4. The Mostowski Bridge (Ali Enayat).- 5. All you need is the Liar (Albert Visser).- 6. Reflection Inversion (Mateusz Lelyk).- Part III: Reflection, Implicit Commitment and Truth.- 7. Implicit commitment and the concept of truth (Martin Fischer).- 8. Reflecting, unfolding and implicit commitment (Andrea Cantini and Riccardo Bruni).- 9. On Feferman's Schematic Reflective Closure (Kentaro Fujimoto).- Part IV: Truth, Realisability, and Computational Commitments.- 10. Truth classes via approximations (Luca Castaldo and Cezary Cieslinski).- 11. A Friedman-Sheard-style Theory for Classical Realisability (Daichi Hayashi and Graham E. Leigh). Autorenportrait Cezary Cieslinski is an Associate Professor and Head of the Logic Department at the Faculty of Philosophy at the University of Warsaw. He is currently PI of the projects Epistemic and semantic commitments of foundational theories (MAESTRO project number 2019/34/A/HS1/00399) and Reflection principles and their epistemic warrant (OPUS, 2024/55/I/HS1/02974). His research interests are mathematical and philosophical logic, model theory, truth theory (deflationism, theories of self-referential truth) and philosophy of language. Amongst his research outputs, he is the author of the book The Epistemic Lightness of Truth. Deflationism and its Logic, Cambridge University Press, 2017. Matteo Zicchetti is an Assistant Professor at the Faculty of Philosophy at the University of Warsaw. He is interested in Epistemology, Philosophy of Mathematics, and theories of truth. He is the PI of the project Mathematical justification and entitlement in the foundations of mathematics funded by NCN (National Science Center OPUS grant, grant number 2024/53/B/HS1/03927) and a co investigator in the project Reflection principles and their epistemic warrant (OPUS, 2024/55/I/HS1/02974).