Mathematics As a Science of Patterns

by Resnik, Michael D.

• ISBN13: 9780198236085
• ISBN10: 0198236085
• Format: Hardcover
• Copyright: 1997-09-11
• Publisher: Clarendon Press

Summary

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist.

Table of Contents

PART ONE: PROBLEMS AND POSITIONS	3	(96)
1. Introduction	3	(7)
2. What is Mathematical Realism?	10	(31)
1. To Characterize Realism	10	(4)
2. Immanent Truth	14	(16)
3. Realism and Immanent Truth	30	(9)
4. Some Concluding Remarks	39	(2)
3. The Case for Mathematical Realism	41	(11)
1. The Prima Facie Case for Realism	41	(2)
2. The Quine-Putnam View of Applied Mathematics	43	(1)
3. Indispensability Arguments for Mathematical Realism	44	(5)
4. Indispensability and Fictionalism about Science	49	(1)
5. Conclusion	50	(2)
4. Recent Attempts at Blunting the Indispensability Thesis	52	(30)
1. Synthetic Science: Field	53	(6)
2. Saving the Mathematical Formalism while Changing its Interpretation: Chihara and Kitcher	59	(8)
3. An Intermediate Approach: Hellman's Modal-Structuralism	67	(8)
4. What Has Introducing Modalities Gained?	75	(6)
5. Conclusion	81	(1)
5. Doubts about Realism	82	(17)
1. How Can We Know Mathematical Objects?	82	(5)
2. How Can We Refer to Mathematical Objects?	87	(2)
3. The Incompleteness of Mathematical Objects	89	(3)
4. Some Morals for Realists	92	(1)
5. An Aside: Penelope Maddy's Perceivable Sets	93	(6)
PART TWO: NEUTRAL EPISTEMOLOGY	99	(100)
Introduction to Part Two	99	(2)
6. The Elusive Distinction between Mathematics and Natural Science	101	(11)
1. How Physics Blurs the Mathematical Physical Distinction	102	(5)
2. Some Other Attempts to Distinguish Mathematical from Physical Objects	107	(1)
3. Our Epistemic Access to Space-Time Points	108	(2)
4. Morals for the Epistemology of Mathematics	110	(2)
7. Holism: Evidence in Science and Mathematics	112	(25)
1. The Initial Case for Holism	114	(4)
2. Objections to Holism	118	(3)
3. Testing Scientific and Mathematical Models	121	(3)
4. Global and Local Theories	124	(6)
5. Revising Logic and Mathematics	130	(7)
8. The Local Conception of Mathematical Evidence: Proof, Computation, and Logic	137	(38)
1. Some Norms of Mathematical Practice	138	(10)
2. Computation and Mathematical Empiricism	148	(7)
3. Mathematical Proof, Logical Deduction and Apriority	155	(17)
4. Summary	172	(3)
9. Positing Mathematical Objects	175	(24)
1. Introduction	175	(2)
2. A Quasi-Historical Account	177	(5)
3. Mathematical Positing Naturalized?	182	(2)
4. Positing and Knowledge	184	(4)
5. Postulational Epistemologies and Realism	188	(11)
PART THREE: MATHEMATICS AS A SCIENCE OF PATTERNS	199	(76)
Introduction to Part Three	199	(2)
10. Mathematical Objects as Positions in Patterns	201	(23)
1. Introduction	201	(1)
2. Patterns and their Relationships	202	(7)
3. Patterns and Positions: Entity and Identity	209	(4)
4. Composite and Unified Mathematical Objects	213	(3)
5. Mathematical Reductions	216	(4)
6. Reference to Positions in Patterns	220	(2)
7. Concluding Remarks on Reference and Reduction	222	(2)
11. Patterns and Mathematical Knowledge	224	(19)
1. Introduction	224	(2)
2. From Templates to Patterns	226	(6)
3. From Proofs to Truth	232	(8)
4. From Old Patterns to New Patterns	240	(3)
12. What is Structuralism? And Other Questions	243	(32)
1. Introduction	243	(1)
2. On `Facts of the Matter'	243	(3)
3. Patterns as Mathematical Objects	246	(4)
4. Structural Relativity	250	(4)
5. Structuralist Formulations of Mathematical Theories?	254	(3)
6. The Status of Structuralism	257	(4)
7. Structuralism, Realism, and Disquotationalism	261	(4)
8. Epistemic vs. Ontic Structuralism: Structuralism All the Way Down	265	(5)
9. A Concluding Summary	270	(5)
Bibliography	275	(8)
Index	283