Finitism

Abstract: PART I. PERSPECTIVE CHAPTER 1. WHAT IS SO INTERESTING ABOUT MATHEMATICS (FOR PHILOSOPHER)? ATTRACTION - OF OPPOSITES? PHILOSOPHY AND MATHEMATICS: CHICKEN OR EGG? NATURALISM AND MATHEMATICS CHAPTER 2. A POTPOURRI OF QUESTIONS AND ATTEMPTED ANSWERS NECESSITY AND A PRIORI KNOWLEDGE GLOBAL MATTERS: OBJECTS AND OBJECTIVITY THE MATHEMATICAL AND THE PHYSICAL LOCAL MATERS: THEOREMS, THEORIES, AND CONCEPTS PART II. HISTORY CHAPTER 3. PLATO'S RATIONALISM, AND ARISTOTLE THE WORLD OF BEING PLATO ON MATHEMATICS MATHEMATICS ON PLATO ARISTOTLE, THE WORTHY OPPONENT FURTHER READING CHAPTER 4. NEAR OPPOSITES: KANT AND MILL REORIENTATION KANT MILL FURTHER READING PART III. THE BIG THREE CHAPTER 5. LOGICISM: IS MATHEMATICS (JUST) LOGIC? FREGE RUSSELL CARNAP AND LOGICAL POSITIVISM CONTEMPORARY VIEWS FURTHER READING CHAPTER 6. FORMALISM: DO MATHEMATICAL STATEMENTS MEAN ANYTHING? BASIC VIEWS: FREG'S ONSLAUGHT DEDUCTIVISM: HILBERT'S GRUNDLAGEN DER GEOMETRIE FINITISM: THE HILBERT PROGRAM INCOMPLETENESS CURRY FURTHER READING CHAPTER 7. INTUITIONISM: IS SOMETHING WRONG WITH OUR LOGIC? 1. REVISING CLASSICAL LOGIC 2. THE TEACHER, BROUWER 3. THE STUDENT, HEYTING 4. DUMMETT 5. FURTHER READING PART IV. THE CONTEMPORARY SCENE CHAPTER 8. NUMBERS EXIST GODEL THE WEB OF BELIEF SET-THEORETIC REALISM FURTHER READING CHAPTER 9. NO THEY DON'T FICTIONALISM MODAL CONSTRUCTION WHAT SHOULD WE MAKE OF ALL THIS? ADDENDUM: YOUNG TURKS FURTHER READING CHAPTER 10. STRUCTURALISM THE UNDERLYING IDEA ANTE REM STRUCTURES, AND OBJECTS STRUCTURALISM WITHOUT STRUCTURES KNOWLEDGE OF STRUCTURES FURTHER READING REFERENCES INDEX

Abstract: 1: Introduction 2: The Realism Debate 3: Irrealism 4: Against Meaning Skepticism 5: Avoiding Strict Finitism 6: Meaning as Graspable 7: Truth as Knowable 8: Analyticity and Syntheticity 9: Finding the Right Logic 10: Cognitive Significance Regained 11: Defeasibility and Constructive Falsifiability

Abstract: Preface Abbreviations I. The Philosophy of Arithmetic of the Tractatus 1. Preliminaries 2. Systematic Exposition 3. The "Knowledge" of Forms: Vision and Calculation 4. Foundations of Mathematics (I) II. Verificationism and its Limits. The Intermediate Phase (1929-'33) 1. Introduction 2. Finite Cardinal Numbers: the Arithmetic of Strokes 3. Mathematical Propositions 4. The Mathematical Infinite 4.1 Quantifiers in Mathematics 4.2 Recursive Arithmetic and Algebra 4.3 Real Numbers 4.4 Set Theory 5. Foundations of Mathematics (II) III. From Facts to Concepts. The Later Writings on Mathematics (1934-'44) 1. The Crisis of Verificationism: Rule-Following 2. Mathematical Proofs as Paradigms 3. The Problem of Strict Finitism 4. Wittgenstein's Quasi-Revisionism 4.1 Cantor's Diagonal Proof and Transfinite Cardinals 4.2 The Law of Excluded Middle 4.3 Consistency References

Abstract: Publisher Summary In this chapter, constructive is meant as finitism, constructive recursive analysis, and intuitionism. The chapter discusses constructivism as the study of a special area in the whole of mathematical experience. The principal aspects of constructivism discussed in the chapter are the role of logic and abstract concepts; reductions to quantifier-free statements; and interpretation of the logical operations; intensional aspects; the validity of Church's thesis; continuity axioms, the possibility of a theory of continuous; usefulness of the subjectivistic interpretation; and the quest for explicit definability. The chapter also discusses Markov's schema; connection between validity for intuitionistic predicate logic and mathematical assumptions. It presents the existence of classical counterparts to problems of constructive mathematics and systematic procedures for constructivizing classical theorems.