Abstract
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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