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.
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.
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
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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