> Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point.
> 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!
Second addendum, chapter 4 is about second order logic and apparently I just forgot that exercise 1 is simply showing that you get all of the structure built up in Chapter 3 with Peano's original formulation in second-order logic. Seems that here I'm the one suffering from a lack of historical context!
I think from a logic standpoint this also makes sense -- getting to undecidability quickly makes taking the direct route through first-order logic more appealing.
If I'm being honest, I now do feel a little bit deprived, I probably would have enjoyed the categorical view when I was learning this too.
Oh, then yeah, I totally agree. In general I think it's a shame that so little emphasis is placed on the history of mathematics, though at the same time I appreciate that most of my peers just didn't care :(
> 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!