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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
Source
Richard Dedekind (Continuity and Irrational Numbers [1872], §1)
Book Ref
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]
17611 | We want the essence of continuity, by showing its origin in arithmetic [Dedekind] |
17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
10572 | A cut between rational numbers creates and defines an irrational number [Dedekind] |
18087 | If x changes by less and less, it must approach a limit [Dedekind] |