> Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.
Oh, by no means do I object to calling an updated and generalised version by the name of the person who originated the subject! Since you've brought up Galois, I hardly think that he'd recognize the modern theory of Galois connections, but I think that the name is wholly appropriate.
No, what I thought was a shame is if the original theory doesn't get discussed at all. If my only exposure to Peano's work was, for example, the axiom schema in Enderton, then I don't think I'd be able to appreciate why it's such a big deal. That would feel to me like teaching the theory of Galois connections without ever saying anything about field theory! Whereas, on the other hand, I did immediately understand as an undergraduate the magic of being able to define everything in terms of the pointed set using induction, and I think I'd appreciate even more having seen that first and then seeing how it is updated for modern mathematical logic.
In fact, at a casual glance, I still don't see why L1, L3, and the A, M, and E axioms can't be omitted in the presence of the axiom(s) on p. 269, which has been the whole substance of my objection. I believe that there's an answer, but, if I don't see it as a professional mathematician (though not a logician), then surely it can't be true that every undergraduate will appreciate it!