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Ideas for 'Issues of Pragmaticism', 'Reply to Foucher' and 'works'

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5 ideas

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

9147	Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz]
	Full Idea: Leibniz's talk of the addition of ones cannot define number, since it cannot be specified how often they are added without using the number itself. Number must be an organic unity of ones, achieved by a single act of abstraction.
	From: comment on Gottfried Leibniz (works [1690]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §1
	A reaction: I doubt whether 'abstraction' is the right word for this part of the process. It seems more like a 'gestalt'. The first point is clearly right, that it is the wrong way round if you try to define number by means of addition.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite

19406	I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz]
	Full Idea: I am so much for the actual infinite that instead of admitting that nature abhors it, as is commonly said, I hold that it affects nature everywhere in order to indicate the perfections of its author.
	From: Gottfried Leibniz (Reply to Foucher [1693], p.99)
	A reaction: I would have thought that, for Leibniz, while infinities indicate the perfections of their author, that is not the reason why they exist. God wasn't, presumably, showing off. Leibniz does not think we can actually know these infinities.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility

19375	The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R]
	Full Idea: Leibniz said the division of the continuum should not be conceived 'to be like the division of sand into grains, but like that of a tunic or a sheet of paper into folds'.
	From: report of Gottfried Leibniz (works [1690], A VI iii 555) by Richard T.W. Arthur - Leibniz
	A reaction: This from the man who invented calculus. This thought might apply well to the modern physicist's concept of a 'field'.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals

18080	A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz]
	Full Idea: We have only to keep in mind that to find a tangent means to draw a line that connects two points of a curve at an infinitely small distance.
	From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1
	A reaction: [The quote can be tracked through Kitcher's footnote]

18081	Nature uses the infinite everywhere [Leibniz]
	Full Idea: Nature uses the infinite in everything it does.
	From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1
	A reaction: [The quote can be tracked through Kitcher's footnote] He seems to have had in mind the infinitely small.