Synopses & Reviews
An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge.
Synopsis
Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.
About the Author
James Franklin is Professor of Mathematics at the University of New South Wales, Australia and founder of the 'Sydney School' in the philosophy of mathematics. His books include The Science of Conjecture: Evidence and Probability Before Pascal; Corrupting the Youth: A History of Philosophy in Australia; and What Science Knows.
Table of Contents
Preface
PART I: THE SCIENCE OF QUANTITY AND STRUCTURE
1. The Aristotelian Realist Point of View
2. Uninstantiated Universals and 'Semi-Platonist' Aristotelianism
3. Elementary Mathematics: Quantity and Number
4. Higher Mathematics: Science of the Purely Structural
5. Necessary Truths about Reality
6. The Formal Sciences Discover the Philosophers' Stone
7. Comparisons and Objections
8. Infinity
9. Geometry: Mathematics or Empirical Science?
PART II: KNOWING MATHEMATICAL REALITY
10. Knowing Mathematics: Pattern Recognition and Perception of Quantity and Structure
11. Knowing Mathematics: Visualization
12. Knowing Mathematics: Understanding and Proof
13. Explanation in Mathematics
14. Idealization: An Aristotelian View
15. Non-Deductive Logic in Mathematics
Epilogue: Mathematics, Last Bastion of Reason
Notes
Select Bibliography
Index