Thinking about Mathematics

Professor of Philosophy, Department of Philosophy, Ohio State University at Newark and Professorial Fellow, Department of Logic and Metaphysics at the University of St Andrews, Scotland.

• 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

"Stewart Shapiro admirably provides an accessible introduction to contemporary thinking in mathematics, while avoiding caricature of the technicalities. His ease with the subject and lucid style makes this book a succinct introduction to a fascinating intellectual discipline." Times Literary Supplement