Combining Philosophers

Ideas for Anaxarchus, Hilary Putnam and Paolo Mancosu

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2 ideas

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities

18959	Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam]
	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.
	From: Hilary Putnam (Philosophy of Logic [1971], Ch.7)
	A reaction: This attitude goes back to Hilbert, but it fits with Quine's view of what is indispensable for science. It is hard to see a reason for the cut-off, just looking at the logic of expanding sets.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity

18200	Very large sets should be studied in an 'if-then' spirit [Putnam]
	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.
	From: Hilary Putnam (The Philosophy of Logic [1971], p.347), quoted by Penelope Maddy - Naturalism in Mathematics
	A reaction: Quine says the large sets should be regarded as 'uninterpreted'.