I had always assumed that Peano was an ancient Greek, but apparently he published his axioms in 1889 ( :smack: ). I had also always assumed that axioms are layed down first, and then things are built on top of them. However, I can’t believe that Peano arithmetic hadn’t been used before Peano published his axioms.
Was mathematics well formed before Peano?
Oh boy. There’s books and books going on about this topic. I’ll try to condense this as much as I can.
Before the late 19th century, mathematics wasn’t anywhere near as rigorous as it is today. There were a lot of things that were assumed to be true because no one had seen a counterexample, and a lot of ideas that were used just because they worked and not because they were actually consistent.
There were a few events that kinda kicked this viewpoint in the teeth. First, Weierstrass published a proof that a particular function is continuous over R, but not differentiable anywhere. That flew in the face of what everybody “knew” as true and caused considerable discomfort.
Second, non-Euclidean geometry, which had originally begun as a project to find an inconsistency in geometries that rejected the parallel postulate, was completely failing to do so.
Third, one of Dedekind’s students asked him what a real number is and he had no answer. That led him to define real numbers in terms of rational numbers, which led to the question of what rational numbers are.
Fourth, Cantor published his theory of infinite sets, which ruffled a few feathers.
Fifth, Russell came up with his famous paradox, which showed that the naive concept of a set was incorrect.
So the classical notion of mathematics was looking like it was in trouble. Meanwhile, the logicians were having a grand old time of publishing very rigorous stuff, which gave Russell the idea to try to reduce math to logic. Peano’s axioms were one of the starting points for that project.
Here’s a question back at you: what does well-formed mean?
Well, was everyone talking about the same thing, for a start. Presumably the axioms cemented what the natural numbers actually were, but before then, did mathematicians have differing opinions on what they were and how to define them?
Sometimes these things seem so obvious it’s hard to believe someone takes credit for their definition.
Uhm, we still have disagreements to the present day. You seem to be assuming that people are talking about the same thing now, which is far from clear.
For one thing, the axioms do not cement what the natural numbers actually are. There are other mathematical objects, besides the natural numbers, that satisfy the Peano axioms but which do not satisfy other properties that the natural numbers have. In other words, Peano’s axioms are “incomplete” in the sense of Godel.
That said, I expect that if you took a time machine to anytime when mathematicians have lived, and asked them whether they agreed that the objects they called numbers satisfied the properties encoded by Peano’s axioms, they would have said “yes”. The genius of Peano was not in observing that every natural numbers has a successor, etc. His contribution was to find a small collection of axioms that sufficed to deduce rigorously a very large portion of the properties we attribute to natural numbers.
To elaborate on this, any set that contains N satisfies the Peano axioms. For technical reasons, you can’t call N the intersection of all sets that satisfy the Peano axioms, but you can call it the smallest such set.
I think you’re missing something. What you’re saying “satisfies the Peano axioms” is really an inductive set. The last Peano axiom is that N is a universal inductive set, and this does determine N up to isomorphism. Z contains N, but fails to be universal.
The problem is really much deeper than Peano axioms. As I just said, within the topos of ZFC sets there’s a unique “natural numbers object” determined by Peano. The problem is that there are many different topoi out there, some having a unique NNO, some having all but the uniqueness (thus they fail the analogue of the last Peano axiom), some not even having that much.
I’ve never seen an axiom of universality as part of the Peano axioms. Given that even Mathworld’s presentation doesn’t contain it, I’ve got to wonder how standard it is.
That presentation indeed leaves out exactly what you said, something to the effect that N is the “smallest” such inductive set. Really, logicians (and this is really a logic/foundations question) have been using something like the following for a long time.
A “natural numbers object” in a cartesian closed category A consists of an initial object
1 --0–> N --S–> N
in the category of all diagrams
1 --a–> A --f–> A
in A. This means that for all such diagrams there is a unique arrow
N --h–> A
such that h0 = a, and hS = fh.
I suppose I should also mention that 1 is the terminal object in A.