Ideas from 'The Philosophy of Logic' by Hilary Putnam [1971], by Theme Structure
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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18200
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Very large sets should be studied in an 'if-then' spirit
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Full Idea:
Sets of a very high type or very high cardinality (higher than the continuum, for example), should today be investigated in an 'if-then' spirit.
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From:
Hilary Putnam (The Philosophy of Logic [1971], p.347), quoted by Penelope Maddy - Naturalism in Mathematics
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A reaction:
Quine says the large sets should be regarded as 'uninterpreted'.
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
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18199
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Indispensability strongly supports predicative sets, and somewhat supports impredicative sets
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Full Idea:
We may say that indispensability is a pretty strong argument for the existence of at least predicative sets, and a pretty strong, but not as strong, argument for the existence of impredicative sets.
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From:
Hilary Putnam (The Philosophy of Logic [1971], p.346), quoted by Penelope Maddy - Naturalism in Mathematics II.2
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8857
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We must quantify over numbers for science; but that commits us to their existence
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Full Idea:
Quantification over mathematical entities is indispensable for science..., therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.
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From:
Hilary Putnam (The Philosophy of Logic [1971], p.57), quoted by Stephen Yablo - Apriority and Existence
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A reaction:
I'm not surprised that Hartry Field launched his Fictionalist view of mathematics in response to such a counterintuitive claim. I take it we use numbers to slice up reality the way we use latitude to slice up the globe. No commitment to lines!
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