Thinking about Mathematics: The Philosophy of Mathematics

TLDR: The author revealed that within the framework of this book, three main themes - structuralism, identity, and ontological realism - had emerged: structure, identity and structure, and orientation and ontology.

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