Ideas from 'Mathematics without Foundations' by Hilary Putnam [1967], by Theme Structure
[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,052129648x]].
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
9944

We understand some statements about all sets

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
9942

Gödel proved the classical relative consistency of the axiom V = L

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
9937

I do not believe mathematics either has or needs 'foundations'

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
9939

It is conceivable that the axioms of arithmetic or propositional logic might be changed

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
9940

Maybe mathematics is empirical in that we could try to change it

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
9941

Science requires more than consistency of mathematics

7. Existence / D. Theories of Reality / 3. Antirealism
9943

You can't deny a hypothesis a truthvalue simply because we may never know it!
