If in the consideration of a simply infinite system N set in order by a transformation φ we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation φ, then are these elements called natural numbers or ordinal numbers or simply numbers, and the base-element 1 is called the base-number of the number-series N . With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relations or laws which are […] always the same in all ordered simply infinite systems, whatever names may happen to be given to the individual elements […], form the first object of the science of numbers or arithmetic.
Richard Dedekind, Was sind und was sollen die Zahlen? (1888), tr. Wooster Woodruff Beman as The Nature and Meaning of Numbers in Essays on the Theory of Numbers (1901). Peano derived his 1889 axiomatization (now standard) from Dedekind’s.
What’s interesting to me about this is it’s an early set-theoretic definition of the natural numbers but it’s essentially structuralist. In fact it’s so early that Dedekind has to spend the bulk of the paper laying out his set theory and justifying it on both mathematical and philosophical grounds. It predates the foundational controversies that established “pure” set theories, so he defines “thing” as “object of our thought” and “system” (by which he means “set”)* as “different things […] considered from a common point of view, […] associated in the mind”, and justifies the existence of an infinite set by the claim that every possible thought has a thought-about-it, which is a bijection with a proper subset of thoughts.
This is very different from Frege’s definition of the natural numbers: they’re both taking place in naive set theories, but Dedekind identifies the natural numbers with the relational positions in an ordinal sequence whereas Frege’s defining them absolutely in cardinal terms, for philosophical reasons in both cases. This fits with Dedekind’s definition of real numbers as cuts vs equivalence classes of Cauchy sequences or w/e – Cauchy sequences are more closely tied to the historical motivation for a formal definition of the real numbers (analysis of continuity), but cuts are an extremely neat and simple way of identifying a structural position. (If I’m reading this right, he does differ from modern structuralism, which often positions itself against set-theoretic platonism, by using set theory itself as his theory of structures. But then he’s not using a modern set theory either.)
My impression, about which I need to read more, is that the foundational controversies turned a lot of people off this sort of philosophizing, which led to the standard (at least popular) understanding of ‘foundations’ abandoning what i would call canonicity, which Frege and Dedekind’s definitions both strive to attain. I think this was a mistake – even without this philosophical shift, there was already plenty of demand for rigor which would have produced something that does the job of ZF(C), but could maybe have had a better understanding of what it is and isn’t. But even if it was a mistake, it’s hard to blame anyone.
*German “Systeme”. The paper and its translation both predate the standardization of set-theoretic terminology, both the English use of “set” rather than “class”, “aggregate”, and so on, and the German “Menge”. “system” sounds rather structuralist to me but I don’t actually know how it’d connote in an 1888 German math paper. in any case the mathematics are clearly a (naive) set theory.