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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.
Gist of Idea
Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal
Source
John Stuart Mill (System of Logic [1843], 2.6.3)
Book Ref
Mill,John Stuart: 'System of Logic (9th ed, 2 vols)' [Longmans, Green etc 1875], p.297
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).
| 11041 | Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle] |
| 17843 | The idea of 'one' is the foundation of number [Aristotle] |
| 17850 | Each many is just ones, and is measured by the one [Aristotle] |
| 17851 | Number is plurality measured by unity [Aristotle] |
| 1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
| 17783 | A number is not a multitude, but a unified ratio between quantities [Newton] |
| 9800 | Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill] |
| 14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 19093 | Greek mathematics is wholly sensory, where ours is wholly inferential [Macbeth] |