Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable.

What It Means

This is a placeholder explanation generated by the low-overhead model. It interprets "Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable." in the context of "Bertrand Russell,The Principles of Mathematics (1903) Vol. 1 p.248" to mean that wisdom is timeless.