Thinking about Mathematics The Philosophy of Mathematics

by Shapiro, Stewart

• ISBN13: 9780192893062
• ISBN10: 0192893068
• Format: Paperback
• Copyright: 2000-10-05
• Publisher: Oxford University Press

Summary

This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy.

Author Biography

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

Table of Contents

PART I. PERSPECTIVE
What Is So Interesting About Mathematics (for a Philosopher)?	3	(18)
Attraction---of Opposites?	3	(4)
Philosophy and Mathematics: Chicken or Egg?	7	(9)
Naturalism and Mathematics	16	(5)
A Potpourri of Questions and Attempted Answers	21	(28)
Necessity and A Priori Knowledge	21	(3)
Global Matters: Objects and Objectivity	24	(9)
Object	25	(4)
Truth	29	(4)
The Mathematical and the Physical	33	(6)
Local Matters: Theorems, Theories, and Concepts	39	(10)
PART II. HISTORY
Plato's Rationalism, and Aristotle	49	(24)
The World of Being	51	(1)
Plato on Mathematics	52	(8)
Mathematics on Plato	60	(3)
Aristotle, the Worthy Opponent	63	(8)
Further Reading	71	(2)
Near Opposites: Kant and Mill	73	(34)
Reorientation	73	(3)
Kant	76	(15)
Mill	91	(11)
Further Reading	102	(5)
PART III. THE BIG THREE
Logicism: Is Mathematics (Just) Logic?	107	(33)
Frege	108	(7)
Russell	115	(9)
Carnap and Logical Positivism	124	(9)
Contemporary Views	133	(5)
Further Reading	138	(2)
Formalism: Do Mathematical Statements Mean Anything?	140	(32)
Basic views; Frege's Onslaught	141	(7)
Terms	142	(2)
Games	144	(4)
Deductivism: Hilbert's Grundlagen Der Geometrie	148	(10)
Finitism: The Hilbert Programme	158	(7)
Incompleteness	165	(3)
Curry	168	(3)
Further Reading	171	(1)
Intuitionism: Is Something Wrong with Our Logic?	172	(29)
Revising Classical Logic	172	(2)
The Teacher, Brouwer	174	(11)
The Student, Heyting	185	(5)
Dummett	190	(7)
Further Reading	197	(4)
PART IV. THE CONTEMPORARY SCENE
Numbers Exist	201	(25)
Godel	202	(10)
The Web of Belief	212	(8)
Set-Theoretic Realism	220	(5)
Further Reading	225	(1)
No They Don't	226	(31)
Fictionalism	227	(10)
Modal Construction	237	(6)
What Should We Make of All This?	243	(6)
Addendum: Young Turks	249	(7)
Further Reading	256	(1)
Structuralism	257	(33)
The Underlying Idea	257	(6)
Ante Rem Structures, and Objects	263	(7)
Structuralism Without Structures	270	(5)
Knowledge of Structures	275	(14)
Pattern Recognition and Other Abstraction	276	(7)
Implicit Definition	283	(6)
Further Reading	289	(1)
References	290	(13)
Index	303