more from Richard Dedekind

Single Idea 17612

[catalogued under 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic]

Full Idea

I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers.

Gist of Idea

Arithmetic is just the consequence of counting, which is the successor operation


Richard Dedekind (Continuity and Irrational Numbers [1872], 1)

Book Reference

Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.4

A Reaction

Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting?

Related Idea

Idea 17614 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]