Single Idea 8742

[catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers]

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'.

Gist of Idea

The only axioms needed are for equality, addition, and successive numbers

Source

report of John Stuart Mill (System of Logic [1843], p.610?) by Stewart Shapiro - Thinking About Mathematics 4.3

Book Reference

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.95


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.