Yes, I know what Bertrand Russell thought of those axioms (that they don’t, in fact, characterize the natural numbers), but I also know that Russell was very taken with Peano at a conference in Paris some years later and saw him in a new light. It also seems that Peano did think the axioms characterise the natural numbers. I tend to agree with Russell, but I still find it slightly astonishing that the axioms were numbered. Talk about bootstraps!

¹NB. I don’t have an original copy of FDM to hand; I only have secondary sources that quote it. If they are misquoting, then this post is bunk and those secondary sources slide a little lower in my estimation. If you have a copy of FDM, why not photograph it and put it online for the world to see? These issues need clearing up; foundational maths is still a mess, and the history of foundational maths is even worse.