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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12735
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Everything has a fixed power, as required by God, and by the possibility of reasoning [Leibniz]
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Full Idea:
It follows from the nature of God that there is a fixed power of a definite magnitude [non vagam] in anything whatsoever, otherwise there would be no reasonings about those things.
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From:
Gottfried Leibniz (De aequopollentia causae et effectus [1679], A6.4.1964), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 6
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A reaction:
This is double-edged. On the one hand there is the grand claim that the principle derives from divine nature, but on the other it derives from our capacity to reason and explain. No one doubts that powers are 'fixed'.
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