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Full Idea
I believe that under certain circumstances revisions in the axioms of arithmetic, or even of the propositional calculus (e.g. the adoption of a modular logic as a way out of the difficulties in quantum mechanics), is fully conceivable.
Gist of Idea
It is conceivable that the axioms of arithmetic or propositional logic might be changed
Source
Hilary Putnam (Mathematics without Foundations [1967], p.303)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.303
A Reaction
One can change the axioms of a system without necessarily changing the system (by swapping an axiom and a theorem). Especially if platonism is true, since the eternal objects reside calmly above our attempts to axiomatise them!
| 9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
| 9941 | Science requires more than consistency of mathematics [Putnam] |
| 9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
| 9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
| 9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
| 9944 | We understand some statements about all sets [Putnam] |