### Single Idea 9800

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic]

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 Reference

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).