This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge.

The essays collected here  explore  the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe.  They consider fundamental philosophical notions such as set, cardinal number, truth, ground,  finiteness and infinity, examining  how their informal conceptions can best be captured in formal theories.

The philosophy of mathematics is an extremely lively  field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns.

Promoted by the Italian Network for the Philosophy of Mathematics FilMat

Andrea Sereni is Associate Professor in Epistemology and Philosophy of Mathematics at the Institute of Advanced Studies (IUSS) in Pavia. He was a Lecturer at Vita-Salute San Raffaele University, Milan, Italy, where he still teaches. He obtained his PhD at Bologna University and was a visiting scholar at the Arche Centre (St.Andrews, Scotland), MCMP Munich, and University of Oslo. His teaching, research and publications cover issues in the philosophy of mathematics, epistemology, and the philosophy of language. He is co-author, with M. Panza, of Platos Problem, An Introduction to Mathematical Platonism (Palgrave, 2013) and co-editor (with F. Pataut and D. Molinini) of Synthese special issue Indispensability and Explanation. He is member of COGITO, CRESA and NeTS research centres, and coordinator of the promoting committee of the Italian Network for the Philosophy of Mathematics (FilMat). Francesca Boccuni is a Lecturer in Logic and Philosophy of Logic and Science at Vita-Salute San Raffaele University, Milan, Italy. She was a visiting scholar at the University of Oxford, Institute Jean Nicod CNRS (Paris), Ohio State University, University of Bristol, University of London, and University of Oslo. Her research interests and main publications (Studia Logica, Erkenntnis) concern the abstractionist programme in the philosophy of mathematics and Frege's Logicism, second-order logic and plural logic, and the issue of reference in mathematics. She is member of COGITO and CRESA research centres, and a member of the promoting committee of the Italian Network for the Philosophy of Mathematics (FilMat).

SECTION I: MATHEMATICAL OBJECTS AND AXIOMATIZATION.- PART I: THE VARIETIES OF MATHEMATICAL OBJECTS.- Chapter 1: Semantic Nominalism: How I learned to Stop Worrying and Love Universals; Aldo Antonelli.- Chapter 2: Modality, Abstract Structures and Second-Order Logic; Robert Black.- Chapter 3: Category Theory and Set Theory: Algebraic Set Theory as an Example of their Interaction; Brice Halimi.- PART II: AXIOMS AND SET THEORY.- Chapter 4: Absolute Infinity; Leon Horsten.- Chapter 5: Forcing, multiverse and realism; Giorgio Venturi.- Chapter 6: True V or not True V, that is the Question; Gianluigi Oliveri.- SECTION II: REFERENCE AND EPISTEMOLOGY.- PART III: THE PROBLEM OF REFERENCE.- Chapter 7: Numbering Planets and Equating Facts; Robert Knowles.- Chapter 8: Multiversism and the Problem of Reference: How much Relativism is Acceptable? Neil Barton.- PART IV: MATHEMATICAL EPISTEMOLOGY AND COGNITION.- Chapter 9: The modal status of arithmetical truths in a contextual a priori framework; Markus Pantsar.- Chapter 10: Epistemology, Ontology and Application in Pincocks Account: A Weak Link? Marina Imocrante.- Chapter 11: Bootstrapping Rebooted; Mario Santos-Sousa.- SECTION III: FORMAL THEORIES AND THEIR PHILOSOPHY.- PART V: TRUTH AND FORMAL THEORIES.- Chapter 12: Incompleteness and the Flow of Truth; Mario Piazza.- Chapter 13: Notes on Axiomatic Truth and Predicative Comprehension; Carlo Nicolai.- PART VI: INFORMAL NOTIONS AND FORMAL ANALYSIS.- Chapter 14: Logic of Grounding: An Alternative Approach; Francesca Poggiolesi.- Chapter 15: Computability, Finiteness and the Standard Model of Arithmetic; Massimiliano Carrara and Enrico Martino and Matteo Plebani.- Chapter 16: The Significance of Categoricity for Formal Theories and Informal Beliefs; Samantha Pollock.