Philosophy of Mathematics: Classic and Contemporary Studies (Textbooks in Mathematics)

The philosophy of mathematics is an exciting subject. **__Philosophy of Mathematics: Classic and Contemporary Studies__** explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge. With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic. Most books on philosophy of mathematics have little to no focus on the effects of philosophical views on mathematical practice, and no concern on giving crucial mathematical results and their philosophical relevance, consequences, reasons, etc. This book fills this gap. The book can be used as a textbook for a one-semester or even one-year course on philosophy of mathematics. "Other textbooks on the philosophy of mathematics are aimed at philosophers. This book is aimed at mathematicians. Since the author is a mathematician, it is a valuable addition to the literature." - **Mark Balaguer,** __California State University, Los Angeles__ "There are not many such texts available for mathematics students. I applaud efforts to foster the dialogue between mathematics and philosophy." - Michele Friend, __George Washington University and CNRS, Lille, France__ Cover Half Title Series Page Title Page Copyright Page Contents Preface 1. Introduction 2. Mathematical Preliminaries 2.1. Summary of Propositional and Predicate Logic 2.1.1. Propositional logic 2.1.2. Predicate logic 2.2. Methods of Proof 2.2.1. Direct proof 2.2.2. Proof by contrapositive 2.2.3. Proof by contradiction 2.2.4. Proof by induction 2.2.5. Proof fallacies 2.3. Basic Mathematical Notions 2.3.1. Axioms of ZFC set theory 2.3.2. Ordinal and cardinal numbers 3. Platonism 3.1. Theory of Forms 3.2. Plato’s Epistemological Philosophy 3.3. Aristotelian Realism 3.4. Summary 4. Intuitionism 4.1. Kant 4.2. Brouwer and Constructivism 5. Logicism 5.1. Frege 5.2. Russell 5.3. Carnap and Logical Positivism 6. Formalism 6.1. Term vs. Game Formalism 6.2. Hilbert 6.3. Godel’s Impact 7. Godel’s Incompleteness Theorem and Computability 7.1. Arithmetisation of Syntax 7.2. Primitive Recursive Functions 7.3. Diagonalisation 7.4. Second Incompleteness Theorem 7.5. Speculations on Godel’s Theorems 7.6. “Real” Mathematics vs. “Ideal” Mathematics 7.7. Reasons Behind Incompleteness 7.7.1. Entscheidungsproblem 7.7.2. Irreducible information 8. The Church-Turing Thesis 8.1. Minds and Machines 8.2. Effective Computability 8.3. The New Pythagoreanism 9. Infinity 9.1. Infinity in Ancient Greece 9.2. Middle Ages, the Renaissance, and the Age of Enlightenment 9.3. Cantor’s Set Theory 10. Supertasks 10.1. Transfinite Computability and Continuity 10.2. Infinite Time Turing Machines 10.3. Physical Realisations 11. Models, Completeness, and Skolem’s Paradox 11.1. Godel’s Completeness Theorem 11.2. Lowenheim-Skolem Theorem 12. Axiom of Choice 12.1. Applications of the Axiom of Choice 12.1.1. Russell’s Example 12.1.2. Konig’s lemma 12.2. Which statements are obvious? 12.2.1. Countable unions 12.2.2. Countable pairs of “identical” objects 12.3. Determining the Naturality or Otherwise 12.3.1. Axiom of determinacy 12.4. Concluding Remarks 13. Naturalism 13.1. Godel’s Realism 13.2. Quine 13.3. Maddy 13.3.1. The naturalist second philosopher 14. Structuralism 14.1. Characteristic Properties 14.2. Identification Problem 14.3. Eliminative Structuralism 15. Yablo’s Paradox 15.1. Self-reference and Impredicative Definitions 15.2. What is Yablo’s Paradox? 15.3. Priest’s Inclosure Schema and w-inconsistency 16. Mathematical Pluralism 16.1. Plurality of Models 16.2. Multiverse Conception of Sets 16.3. Liberating the Mathematical Ontology or Blurring the Mathematical Truth? 17. Does Mathematics Need More Axioms? 17.1. Status of the Continuum Hypothesis 17.1.1. Axiom of Constructibility 17.2. Inner Model Programme 17.3. Hyperuniverse Programme 18. Mathematical Nominalism 18.1. Problems of Realism 18.2. Field’s Fictionalism 18.3. Is Mathematics a “Subject with No Object”? 18.4. Deflating Nominalism Bibliography Index This book explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians into thinking about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge.