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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. |
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). |