Ideas from 'Review of Tait 'Provenance of Pure Reason'' by Charles Parsons [2009], by Theme Structure

[found in 'Philosophia Mathematica' (ed/tr -) [- ,]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The old problems with the axiom of choice are probably better ascribed to the law of excluded middle
                        Full Idea: The difficulties historically attributed to the axiom of choice are probably better ascribed to the law of excluded middle.
                        From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], 2)
                        A reaction: The law of excluded middle was a target for the intuitionists, so presumably the debate went off in that direction.
6. Mathematics / C. Sources of Mathematics / 8. Finitism
If functions are transfinite objects, finitists can have no conception of them
                        Full Idea: The finitist may have no conception of function, because functions are transfinite objects.
                        From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], 4)
                        A reaction: He is offering a view of Tait's. Above my pay scale, but it sounds like a powerful objection to the finitist view. Maybe there is a finitist account of functions that could be given?
7. Existence / D. Theories of Reality / 10. Ontological Commitment / e. Ontological commitment problems
If a mathematical structure is rejected from a physical theory, it retains its mathematical status
                        Full Idea: If experience shows that some aspect of the physical world fails to instantiate a certain mathematical structure, one will modify the theory by sustituting a different structure, while the original structure doesn't lose its status as part of mathematics.
                        From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], 2)
                        A reaction: This seems to be a beautifully simple and powerful objection to the Quinean idea that mathematics somehow only gets its authority from physics. It looked like a daft view to begin with, of course.