Mathematics without Numbers

Mathematics without Numbers : Towards a Modal-Structural Interpretation

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Geoffrey Hellman presents a detailed interpretation of mathematics as the investigation of structural possibilities, as opposed to absolute, Platonic objects. After dealing with the natural numbers and analysis, he extends his approach to set theory, and shows how to dispense with a fixed universe of sets. Finally, he addresses problems of application to the physical world.
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Product details

  • Paperback | 168 pages
  • 138 x 216 x 13mm | 241g
  • Clarendon Press
  • Oxford, United Kingdom
  • English
  • Revised ed.
  • 0198240341
  • 9780198240341

Table of contents

Introduction; Chapter 1: The natural numbers and analysis; Introduction; 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; Chapter 2: Set theory; Introduction; Informal principles: Many vs. one; The relevant structures; Unbounded
sentences: Putnam semantics; Axioms of infinity: Looking back; Axioms of infinity: Climbing up; Appendix; Chapter 3: Mathematics and physical reality; Introduction; The leading ideas; Carrying the mathematics of modern physics: RA2 as a framework; Global solutions; Metaphysical realist commitments?
`Synthetic Determination' relations; A role for representation theorems
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Review quote

rich, subtle, and insightful * Stewart Shapiro, Nous * Without doubt this volume is a major contribution to recent philosophy of mathematics and should be read by anyone interested in this difficult but philosophically central field. * Peter Clark, Times Higher Education Supplement *
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