Objectivity, Realism, and Proof: FilMat Studies in the Philosophy of Mathematics (Boston Studies in the Philosophy and History of Science, 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 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 Pincock's 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 Front Matter....Pages i-xl Mathematics in Philosophy, Philosophy in Mathematics: Three Case Studies....Pages 1-9 Front Matter....Pages 11-11 Semantic Nominalism: How I Learned to Stop Worrying and Love Universals....Pages 13-32 Discussion Note On: “Semantic Nominalism: How I Learned to Stop Worrying and Love Universals” by G. Aldo Antonelli....Pages 33-41 Semantic Assumptions in the Philosophy of Mathematics....Pages 43-65 The Modal Status of Contextually A Priori Arithmetical Truths....Pages 67-79 Epistemology, Ontology and Application in Pincock’s Account....Pages 81-99 Front Matter....Pages 101-101 Absolute Infinity in Class Theory and in Theology....Pages 103-122 Sets and Descent....Pages 123-142 True V or Not True V, That Is the Question....Pages 143-164 The Search for New Axioms in the Hyperuniverse Programme....Pages 165-188 Multiversism and Concepts of Set: How Much Relativism Is Acceptable?....Pages 189-209 Forcing, Multiverse and Realism....Pages 211-241 Front Matter....Pages 243-243 What’s so Special About the Gödel Sentence \(\mathcal {G}\) ?....Pages 245-263 More on Systems of Truth and Predicative Comprehension....Pages 265-290 A Critical Overview of the Most Recent Logics of Grounding....Pages 291-309 Computability, Finiteness and the Standard Model of Arithmetic....Pages 311-318 The Significance of a Categoricity Theorem for Formal Theories and Informal Beliefs....Pages 319-344 Objectivity, Realism And Proof In The Philosophy Of Mathematics: An Introduction / Francesca Boccuni And Andrea Sereni -- In Memoriam Of Aldo Antonelli / Andrew Arana, Curtis Franks -- Mathematics In Philosophy, Philosophy In Mathematics: Three Case Studies / Stewart Shapiro -- Part I. Ways Of Mathematical Objectivity: Semantics And Knowledge. Sematic Nominalism: How I Learned To Stop Worrying And Love Universals / G. Aldo Antonelli Edited By Francesca Boccuni And Andrea Sereni. Includes Bibliographical References.