Objectivity, Realism, and Proof

1. Front Matter

   Pages i-xl

   PDF

2. Mathematics in Philosophy, Philosophy in Mathematics: Three Case Studies

   • Stewart Shapiro

   Pages 1-9

3. The Ways of Mathematical Objectivity: Semantics and Knowledge

   1. Front Matter

      Pages 11-11

      PDF

   2. Semantic Nominalism: How I Learned to Stop Worrying and Love Universals

      • G. Aldo Antonelli

      Pages 13-32

   3. Discussion Note On: “Semantic Nominalism: How I Learned to Stop Worrying and Love Universals” by G. Aldo Antonelli

      • Robert C. May, Marco Panza

      Pages 33-41

   4. Semantic Assumptions in the Philosophy of Mathematics

      • Robert Knowles

      Pages 43-65

   5. The Modal Status of Contextually A Priori Arithmetical Truths

      • Markus Pantsar

      Pages 67-79

   6. Epistemology, Ontology and Application in Pincock’s Account

      • Marina Imocrante

      Pages 81-99

4. Realism in a World of Sets: from Classes to the Hyperuniverse

   1. Front Matter

      Pages 101-101

      PDF

   2. Absolute Infinity in Class Theory and in Theology

      • Leon Horsten

      Pages 103-122

   3. Sets and Descent

      • Brice Halimi

      Pages 123-142

   4. True V or Not True V, That Is the Question

      • Gianluigi Oliveri

      Pages 143-164

   5. The Search for New Axioms in the Hyperuniverse Programme

      • Sy-David Friedman, Claudio Ternullo

      Pages 165-188

   6. Multiversism and Concepts of Set: How Much Relativism Is Acceptable?

      • Neil Barton

      Pages 189-209

   7. Forcing, Multiverse and Realism

      • Giorgio Venturi

      Pages 211-241

5. The Logic Behind Mathematics: Proof, Truth, and Formal Analysis

   1. Front Matter

      Pages 243-243

      PDF

   2. What’s so Special About the Gödel Sentence \(\mathcal {G}\)?

      • Mario Piazza, Gabriele Pulcini

      Pages 245-263

   3. More on Systems of Truth and Predicative Comprehension

      • Carlo Nicolai

      Pages 265-290

   4. A Critical Overview of the Most Recent Logics of Grounding

      • Francesca Poggiolesi

      Pages 291-309

   5. Computability, Finiteness and the Standard Model of Arithmetic

      • Massimiliano Carrara, Matteo Plebani, Enrico Martino

      Pages 311-318

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

• Formal Analysis
• Formal Theories
• Foundations of Mathematics
• Informal Notions
• Ontology in Mathematics
• Philosophy of Mathematics and Logic

• San Raffaele University, Milan, Italy

  Francesca Boccuni

• Institute of Advanced Studies, Pavia, Italy

  Andrea Sereni

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 Plato’s 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).

Policies and ethics