A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics

By John P. Burgess; Gideon Rosen | Go to book overview

Preface

'A preface is not, in my book, an introduction.' So a prominent philosopher of mathematics once wrote. To judge from his practice, he took the function of an introduction to be to begin the exposition of his subject, and that of a preface to place it in its intellectual context. We will follow his example.

For all its wealth of results, and for all the power of its applications, mathematics as of about 1800 dealt with only a handful of mathematical structures, all closely connected with the models of time and space used in classical physics: the natural, rational, real, and complex number systems; the Euclidean spaces of dimensions one, two, and three. Indeed, mathematics was widely held to deal directly with the structure of physical space and time, and to provide an example of pure thought arriving at substantive information about the natural world. The central question in philosophy of mathematics at that period was how this could be possible. All that changed completely during the nineteenth century with the introduction of more and more novel mathematical structures, beginning with the first non-Euclidean spaces.

Among other consequences, the proliferation of novel structures that then ensued made it seem desirable to insist on a closer adherence than had become customary to the ancient ideal of rigour, according to which all new results in mathematics are to be logically deduced from previous results, and ultimately from a list of explicit axioms. For in the absence of rigour, intuitions derived from familiarity with more traditional mathematical structures might easily be unconsciously transferred to novel structures where they are no longer appropriate.

The introduction of so many new structures naturally tended to lead to increased specialization among mathematicians. However, the tendency to fragmentation was powerfully counteracted by the emergence of new cross-connections among its various branches, arithmetic, algebra, analysis, and geometry: a broadening of the notion of algebra, for instance, allowed mathematicians to recognize algebraic structures connected with geometric objects; a broadening of the notion of geometry allowed them to recognize geometric structures connected with analytic objects; and so on. Because of these cross-connections, rigorous axiomatic treatments of the several branches of mathematics separately would not suffice: a unified, general,

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A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics
Table of contents

Table of contents

  • A Subject with No Object iii
  • Preface v
  • Acknowledgements ix
  • Contents xi
  • Part I Philosophical and Technical Background 1
  • Part II Three Major Strategies 95
  • Part III Further Strategies and a Provisional Assessment 171
  • References 245
  • Section Index 253
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