mathematics as a science of patterns sep 1997

[...]... INTRODUCTION 9 'Mathematics as a Science of Patterns' , Here I expound a theory of patterns (structures) and argue that we can resolve a number of issues in the ontology of mathematics by construing mathematical objects as positions in patterns I also argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides... both necessary and sufficient Classifying a philosophical view as realist is like diagnosing an illness as a ease of arthritis, lupus, or schizophrenia For these and a number of other diseases there are no fixed diagnostic criteria Instead the appropriate medical associations, such as the American Rheumatism Association, have established lists of criteria for use in diagnosing patients If the patient meets... ONE: PROBLEMS AND POSITIONS demands that we not settle for an account of mathematical knowledge based upon processes, such as a priori i0tuition, that do not seem to be capable of scientific investigation or explanation This has led many contemporary philosophers of mathematics to disdain realism about mathematical objects, and to not read mathematics at face value Hartry Field has embraced an ingenious... plan of the book Part One: 'Problems and Positions' begins by explaining my mathematical realism and the version of truth I use Next is a chapter offering a prima facie case for mathematical realism, which is based largely on an argument that the indispensability of mathematics to seieaee justifies a realist stance towards much mathematics This is followed by a critique of antirealist work by Charles... it was and has been relatively easy for subsequent mathematicians to enlarge and enrich the structures they knew, and to postulate entirely new ones Basing the epistemology of mathematical objects on positing has the advantage of appealing to an apparently natural process akin to making up a story However, it also generates the obvious problem of showing how positing mathematical objects can lead us... educated people regard mathematics as our most highly developed science, a paradigm for lesser sciences to emulate Indeed, the more mathematical a science is the more scientists seem to prize it, and traditionally mathematics has been regarded as the 'Queen of Sciences* Thus it is ironic that philosophical troubles surface as soon as we inquire about its subject-matter Mathematics itself says nothing about... much mathematics and beyond is not as firm as it is for analysis and number theory, and, as a result, the case for a realist stance toward it is weaker The ontological component of my realism is a form of structuralism Mathematical objects are featureless, abstract positions in structures (or more suggestively, patterns) ; my paradigm mathematical objects are geometric points, whose identities are fixed... committed to classical number theory and analysis, for less than this opens the way to antirealist, constructive accounts of mathematics Moreover, accepting classical analysis already suffices for making a convincing case that the mathematical realm is independent of us and our mental life, thereby raising epistemological problems for realists I am inclined to commit myself to standard set theory as weE,... Logic and Mathematics 112 114 118 121 124 130 8 The Local Conception of Mathematical Evidence: Proof, Computation, and Logic 1 Some Norms of Mathematical Practice 2, Computation and Mathematical Empiricism 3, Mathematical Proof, Logical Deduction and Apriority 4 Summary 137 138 148 155 172 9 Positing Mathematical Objects 1 Introduction 2 A Quasi-Historical Account 175 175 177 3 Mathematical Positing Naturalized?... 'fictionalism', "deflationism*; in this respect 'realism* fits My view also has much in common with other so-called realist views in other areas of philosophy So I am going to retain the label, and try to define it so that it characterizes the contemporary debate about mathematical objects as well as traditional debates between realists and anti-realists in other areas of philosophy.1 1, TO CHARACTERIZE .