Thinking about Mathematics - The Philosophy of Mathematics
Shapiro (Stewart)
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Cover Blurb

  1. This unique text by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. The book tackles questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. In a historical survey the author discusses the role of mathematics in such thinkers as Plato, Aristotle, Kant, and Mill, and covers the three major positions throughout the twentieth century:
    … that mathematics is logic (logicism),
    … 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, Shapiro looks at contemporary positions and work which brings the reader up-to-date on the discipline.
  2. Thinking about mathematics is accessible to those with little background in either mathematics or philosophy. It will be enjoyed by students and professionals in mathematics who have little contact with academic philosophy and equally by philosophy students and other philosophers who forgot much of their mathematics.
Contents
    Part I. Perspective
  1. What Is So Interesting About Mathematics (For A Philosopher)? – 3
  2. A Potpourri Of Questions And Attempted Answers – 21
    Part II. History
  3. Plato's Rationalism, And Aristotle – 49
  4. Near Opposites: Kant And Mill – 73
    Part III. The Big Three
  5. Logicism: Is Mathematics (Just) Logic? – 107
  6. Formalism: Do Mathematical Statements Mean Anything? – 140
  7. Intuitionism: Is Something Wrong With Our Logic? – 172
    Part IV. The Contemporary Scene
  8. Numbers Exist – 201
  9. No They Don't – 226
  10. Structuralism – 257
    References – 290
    Index – 303


"Shapiro (Stewart) - Thinking about Mathematics: Preface"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Preface


Preface (Full Text)
  1. This is a philosophy book about mathematics. There are, first, matters of metaphysics: What is mathematics all about? Does it have a subject-matter? What is this subject-matter? What are numbers, sets, points, lines, functions, and so on? Then there are semantic matters: What do mathematical statements mean? What is the nature of mathematical truth? And epistemology: How is mathematics known? What is its methodology? Is observation involved, or is it a purely mental exercise? How are disputes among mathematicians adjudicated? What is a proof? Are proofs absolutely certain, immune from rational doubt? What is the logic of mathematics? Are there unknowable mathematical truths?
  2. Mathematics has a reputation for being a cut-and-dried discipline, about as far from philosophy (in this respect) as can be imagined. Here things seem to get settled, once and for all, on a routine basis. Is this so? Have there been any revolutions in mathematics, where long-standing beliefs were abandoned? Consider the depth of mathematics used — and required — in the natural and social sciences. How is it that mathematics, which appears to be primarily a mental activity, sheds light on the physical, human, and social world studied in science? Why is it that we cannot get very far in understanding the world (in scientific terms) if we do not understand a lot of mathematics? What does this say about mathematics? What does this say about the physical, human, and social world?
  3. Philosophy of mathematics belongs to a genre that includes philosophy of physics, philosophy of biology, philosophy of psychology, philosophy of language, philosophy of logic, and even philosophy of philosophy. The theme is to deal with philosophical questions that concern an academic discipline, issues about the metaphysics, epistemology, semantics1, logic, and methodology of the discipline. Typically, philosophy of X is pursued by those who care about X, and want to illuminate its place in the overall intellectual enterprise. Ideally, someone who practises X should gain something by adopting a philosophy of X: an appreciation of her discipline, an orientation toward it, and a vision of its role in understanding the world. The philosopher of mathematics needs to say something about mathematics itself, something about the human mathematician, and something about the world where mathematics gets applied. A tall order.
  4. The book is divided into four parts. The first, ‘Perspective', provides an overview of the philosophy of mathematics.
  5. Part II, ‘History', sketches the views of some historical philosophers concerning mathematics, and indicates the importance of mathematics in their general philosophical development.
    • Chapter 3 ("Shapiro (Stewart) - Plato's Rationalism, and Aristotle") deals with Plato and Aristotle in the ancient world, and
    • Chapter 4 ("Shapiro (Stewart) - Near Opposites: Kant and Mill") moves forward to the so-called ‘modern period', and considers primarily Immanuel Kant and John Stuart Mill. The idea behind this part of the book is to illustrate an unrelenting rationalist (Plato) — a philosopher who holds that the unaided human mind is capable of substantial knowledge of the world — and an unrelenting empiricist (Mill) — a philosopher who grounds all, or almost all, knowledge in observation. Kant attempted a heroic synthesis between rationalism and empiricism, adopting the strengths and avoiding the weaknesses of each. These philosophers are precursors to much of the contemporary thinking on mathematics.
  6. The next part, ‘The Big Three', covers the major philosophical positions that dominated debates earlier this century, and still provide many battle-lines in the contemporary literature.
  7. Part IV is entitled ‘The Contemporary Scene'.
    • Chapter 8 ("Shapiro (Stewart) - Numbers Exist") is about views that take mathematical language literally, at face value, and hold that the bulk of the assertions of mathematicians are true. These philosophers hold that numbers, functions, points, and so on exist independent of the mathematician. They then try to show how we can have knowledge about such items, and how mathematics, so interpreted, relates to the physical world.
    • Chapter 9 ("Shapiro (Stewart) - No They Don't") concerns philosophers who deny the existence of specifically mathematical objects. The authors covered here either reinterpret mathematical assertions so that they come out true without presupposing the existence of mathematical objects, or else they delimit a serious role for mathematics other than asserting truths and denying falsehoods.
    • Chapter 10 ("Shapiro (Stewart) - Structuralism") is about structuralism, the view that mathematics is about patterns rather than individual objects. This is my own position (Shapiro 1997), so one might say that I have saved the best for last. With the exception of this temporary chutzpah, I have tried be non-partisan throughout the book.
  8. The plan all along was to try to write a book that would offer something to those interested in mathematics who have little background in philosophy, as well as those interested in philosophy who have little background in mathematics. For the most part, some familiarity with high-school or early college-level mathematics, and perhaps an introduction to philosophy should suffice. I avoided excessive symbolization, and tried to explain the symbols I do use. In some places, I may have assumed too much for those uninitiated in university-level mathematics, and in other places too much for those unfamiliar with philosophical terminology, but I hope those places are few and far between, and do not interrupt the flow of the book. The Oxford Dictionary of Philosophy (Blackburn 1994) might prove to be a handy source for those new to academic philosophy.
    [ … snip … ]


"Shapiro (Stewart) - What Is So Interesting about Mathematics (for a Philosopher?)"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 1


  1. Attraction — of Opposites? – 3
  2. Philosophy and Mathematics: Chicken or Egg? – 7
  3. Naturalism and Mathematics – 16


"Shapiro (Stewart) - A Potpourri of Questions and Attempted Answers"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 2


  1. Necessity and A Priori Knowledge – 21
  2. Global Matters: Objects and Objectivity – 24
    … 2.1. Object – 25
    … 2.2. Truth – 29
  3. The Mathematical and the Physical – 33
  4. Local Matters: Theorems, Theories, and Concepts – 39


"Shapiro (Stewart) - Plato's Rationalism, and Aristotle"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 3


  1. The World of Being – 51
  2. Plato on Mathematics – 52
  3. Mathematics on Plato – 60
  4. Aristotle, the Worthy Opponent – 63
  5. Further Reading – 71


"Shapiro (Stewart) - Near Opposites: Kant and Mill"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 4


  1. Reorientation – 73
  2. Kant – 76
  3. Mill – 91
  4. Further Reading – 102


"Shapiro (Stewart) - Logicism: Is Mathematics (Just) Logic?"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 5


  1. Frege – 108
  2. Russell – 115
  3. Carnap and Logical Positivism – 124
  4. Contemporary Views – 133
  5. Further Reading – 138


"Shapiro (Stewart) - Formalism: Do Mathematical Statements Mean Anything?"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 6


  1. Basic Views; Frege's Onslaught – 141
    … 1.1. Terms – 142
    … 1.2. Games – 144
  2. Deductivism: Hilbert's Grundlagen Der Geometrie – 148
  3. Finitism: The Hilbert Programme – 158
  4. Incompleteness – 165
  5. Curry – 168
  6. Further Reading – 171


"Shapiro (Stewart) - Intuitionism: Is Something Wrong with Our Logic?"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 7


  1. Revising Classical Logic – 172
  2. The Teacher, Brouwer – 174
  3. The Student, Heyting – 185
  4. Dummett – 190
  5. Further Reading – 197


"Shapiro (Stewart) - Numbers Exist"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 8


  1. Godel – 202
  2. The Web of Belief – 212
  3. Set-Theoretic Realism – 220
  4. Further Reading – 225


"Shapiro (Stewart) - No They Don't"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 9


  1. Fictionalism1 – 227
  2. Modal2 Construction – 237
  3. What Should We Make of All This? – 243
  4. Addendum: Young Turks – 249
  5. Further Reading – 256


"Shapiro (Stewart) - Structuralism"

Source: Shapiro - Thinking about Mathematics - The Philosophy of Mathematics, 2000, Chapter 10


  1. The Underlying Idea – 257
  2. Ante Rem Structures, and Objects – 263
  3. Structuralism Without Structures – 270
  4. Knowledge of Structures – 275
    … 4.1. Pattern Recognition and Other Abstraction – 276
    … 4.2. Implicit Definition – 283
  5. Further Reading – 289


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  1. Blue: Text by me; © Theo Todman, 2023
  2. Mauve: Text by correspondent(s) or other author(s); © the author(s)


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