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4 ideas
9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
Full Idea: I do not believe mathematics either has or needs 'foundations'. | |
From: Hilary Putnam (Mathematics without Foundations [1967]) | |
A reaction: Agreed that mathematics can function well without foundations (given that the enterprise got started with no thought for such things), the ontology of the subject still strikes me as a major question, though maybe not for mathematicians. |
9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
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. | |
From: Hilary Putnam (Mathematics without Foundations [1967], 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! |
9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
Full Idea: Mathematics might be 'empirical' in the sense that one is allowed to try to put alternatives into the field. | |
From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
A reaction: He admits that change is highly unlikely. It take hardcore Millian arithmetic to be only changeable if pebbles start behaving very differently with regard to their quantities, which appears to be almost inconceivable. |
9941 | Science requires more than consistency of mathematics [Putnam] |
Full Idea: Science demands much more of a mathematical theory than that it should merely be consistent, as the example of the various alternative systems of geometry dramatizes. | |
From: Hilary Putnam (Mathematics without Foundations [1967]) | |
A reaction: Well said. I don't agree with Putnam's Indispensability claims, but if an apparent system of numbers or lines has no application to the world then I don't consider it to be mathematics. It is a new game, like chess. |