Mathematics without Numbers

Mathematics without Numbers : Towards a Modal-Structural Interpretation

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This treatise presents an interpretation of mathematics as the investigation of "structural possibilities", as opposed to absolute Platonic objects. After treating the natural numbers and analysis, the approach is extended to set theory, where it is shown how to dispense with a fixed universe of sets. Finally, problems of application to the physical world are addressed.
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Product details

  • Hardback | 166 pages
  • 138 x 216 x 19.05mm | 344g
  • Clarendon Press
  • Oxford, United Kingdom
  • English
  • bibliography, index
  • 0198249349
  • 9780198249344

Table of contents

Part 1 The natural numbers and analysis: the modal-structural framework - the hypothetical component; the categorical component - an axiom of infinity and a derivation (inspired by Dedekind with help from Frege); justifying the translation scheme; justification from within; extensions; the question of nominalism. Part 2 Set theory: informal principles - many versus one; the relevant structures; unbounded sentences - Putnam semantics; axioms of infinity - looking back; axioms of infinity - climbing up. Part 3 Mathematics and physical reality: the leading ideas; carrying the mathematics of modern physics; global solutions; metaphysical realist commitments - "synthetic determination" relations; a role for representation theorems.
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