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Mathematics without Numbers : Towards a Modal-Structural Interpretation - Geoffrey Hellman

Mathematics without Numbers

Towards a Modal-Structural Interpretation


Published: 25th November 1993
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Hellman here presents a detailed interpretation of mathematics as the investigation of "structural possibilities," as opposed to absolute, Platonic objects. After treating the natural numbers and analysis, he extends the approach to set theory, where he demonstrates how to dispense with a fixed universe of sets. Finally, he addresses problems of application to the physical world.

`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 `Several important insights are woven together and brought to bear on the nature of mathematics in this signal contribution to the philosophy of the field.' Daniel H. Cohen, Mathematical Reviews `rich, subtle, and insightful' Stewart Shapiro, Nous 'This is an exciting and intriguing book, rich in ideas and not short on technical ingenuity, that will doubtless repay careful study. A short review leaves no scope for the detailed critical assessment it merits.' Bob Hale, University of St Andrews, The Philosophical Review, Vol. 101, No. 4 (October 1992) 'the amount of mathematics developed within Hellman's modal framework is impressive ... Hellman's approach makes use of categorical modal-existence postulates, thus achieving truth-determinateness in a way not available to the traditional postulationist.' A.D. Irvine, University of British Columbia, History & Philosophy of Logic '...concise mathematical monograph...' Contemporary Physics, Vol. 38 No. 4, 1997 `Concise mathematical monograph.' Contemporary Physics

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

ISBN: 9780198240341
ISBN-10: 0198240341
Series: Clarendon Paperbacks
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 168
Published: 25th November 1993
Country of Publication: GB
Dimensions (cm): 21.59 x 13.89  x 1.32
Weight (kg): 0.17