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Ideas for 'works', 'Letters to Antoine Arnauld' and 'Letter to Weber'

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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

18244	I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
	Full Idea: Of my theory of irrationals you say that the irrational number is nothing else than the cut itself, whereas I prefer to create something new (different from the cut), which corresponds to the cut. We have the right to claim such a creative power.
	From: Richard Dedekind (Letter to Weber [1888], 1888 Jan), quoted by Stewart Shapiro - Philosophy of Mathematics 5.4
	A reaction: Clearly a cut will not locate a unique irrational number, so something more needs to be done. Shapiro remarks here that for Dedekind numbers are objects.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units

12920	There is no multiplicity without true units [Leibniz]
	Full Idea: There is no multiplicity without true units.
	From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30)
	A reaction: Hence real numbers do not embody 'multiplicity'. So either they don't 'embody' anything, or they embody 'magnitudes'. Does this give two entirely different notions, of measure of multiplicity and measures of magnitude?