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Ideas for 'fragments/reports', 'System of Logic' and 'The Correspondence Theory of Truth'

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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers

8742	The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
	Full Idea: Mill says arithmetic has two axioms, that 'things which are equal to the same thing are equal to each other', and 'equals added to equals make equal sums', plus a definition for each numeral as 'formed by the addition of a unit to the previous number'.
	From: report of John Stuart Mill (System of Logic [1843], p.610?) by Stewart Shapiro - Thinking About Mathematics 4.3
	A reaction: The difficulty here seems to be the definition of 1, and (even worse for an empiricist), of 0. Then he may have a little trouble when he reaches infinity.

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic

9800	Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill]
	Full Idea: The inductions of arithmetic are based on so-called definitions (such as '2 and 1 are three'), and on two axioms: The sums of equals are equal, The differences of equals are equal.
	From: John Stuart Mill (System of Logic [1843], 2.6.3)
	A reaction: These are axioms for arithmetical operations, rather than for numbers themselves (which, for Mill, do not require axioms as they are empirically derived).