Ideas from 'Models and Reality' by Hilary Putnam [1977], by Theme Structure

[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,0-521-29648-x]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
V = L just says all sets are constructible
                        Full Idea: V = L just says all sets are constructible. L is the class of all constructible sets, and V is the universe of all sets.
                        From: Hilary Putnam (Models and Reality [1977], p.425)
The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate
                        Full Idea: Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible.
                        From: report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism
                        A reaction: [He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal.
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language
                        Full Idea: The Löwenheim-Skolem Theorem says that a satisfiable first-order theory (in a countable language) has a countable model. ..I argue that this is not a logical antinomy, but close to one in philosophy of language.
                        From: Hilary Putnam (Models and Reality [1977], p.421)
                        A reaction: See the rest of this paper for where he takes us on this.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
It is unfashionable, but most mathematical intuitions come from nature
                        Full Idea: Experience with nature is undoubtedly the source of our most basic 'mathematical intuitions', even if it is unfashionable to say so.
                        From: Hilary Putnam (Models and Reality [1977], p.424)
                        A reaction: Correct. I find it quite bewildering how Frege has managed to so discredit all empirical and psychological approaches to mathematics that it has become a heresy to say such things.