17697
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The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
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Full Idea:
The standpoint of pure experience seems to me to be refuted by the objection that the existence, possible or actual, of an arbitrarily large number can never be derived through experience, that is, through experiment.
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From:
David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.130)
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A reaction:
Alternatively, empiricism refutes infinite numbers! No modern mathematician will accept that, but you wonder in what sense the proposed entities qualify as 'numbers'.
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18810
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Aristotle's proofs give understanding, so it can't be otherwise, so consequence is necessary [Smiley, by Rumfitt]
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Full Idea:
The ingredient of necessity [in Aristotle's account of consequence] is required by his demand that proof should produce 'understanding' [episteme], coupled with his claim that understanding something involves seeing that it cannot be otherwise.
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From:
report of Timothy Smiley (Conceptions of Consequence [1998], p.599) by Ian Rumfitt - The Boundary Stones of Thought 3.2
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A reaction:
An intriguing reverse of the normal order. Not 'necessity in logic delivers understanding', but 'reaching understanding shows the logic was necessary'.
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