Mathematics as a Science of PatternsMathematics as a Science of Patterns is the definitive exposition of 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 defence of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology. |
Contents
Introduction | 3 |
What is Mathematical Realism? | 13 |
The Case for Mathematical Realism | 41 |
Recent Attempts at Blunting the Indispensability Thesis | 52 |
Doubts about Realism | 83 |
Introduction to Part | 99 |
Some Other Attempts to Distinguish Mathematical | 107 |
Proof | 137 |
Concluding Remarks on Reference and Reduction | 222 |
Patterns and Mathematical Knowledge | 224 |
From Templates to Patterns | 226 |
From Proofs to Truth | 232 |
From Old Patterns to New Patterns | 240 |
What is Structuralism? And Other Questions | 243 |
Patterns as Mathematical Objects | 246 |
Structural Relativity | 250 |
Computation and Mathematical Empiricism | 148 |
Mathematical Proof Logical Deduction and Apriority | 155 |
Summary | 172 |
Introduction to Part Three | 199 |
Patterns and their Relationships | 202 |
Entity and Identity | 209 |
Composite and Unified Mathematical Objects | 213 |
Mathematical Reductions | 216 |
Reference to Positions in Patterns | 220 |
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Common terms and phrases
abstract anti-realists argue axioms beliefs biconditionals branches of mathematics causal Chapter Chihara commit complex numbers computations conception concerning concrete consistency proof count deduction defined describe develop disquotational ematical empirical entities epistemic epistemology Euclidean example existence fact first-order formal Furthermore geometric Hartry Field Hellman holism hypotheses ical identity immanent infinite intuitions isomorphic iterative hierarchy language math mathematical claims mathematical evidence mathematical knowledge mathematical practice mathematical proof mathematical realism mathematical structures mathematical theories mathematicians matter modal models natural number sequence notion number theory omega sequence ontology philosophical philosophy of mathematics physical objects positing mathematical objects positions in patterns postulates predicate premisses presuppose proof quantum question Quine Quine's real numbers reason recognize reference reflective equilibrium relations scientific second-order second-order logic sense sentences set theory set-theoretic simply space-time struc structuralist sub-pattern templates theorems theory of patterns things tion true truth-theory tures