> ...and ask if the Peano axioms define a system that is isomorphic to the real "natural numbers". Or you could ask if the Peano axioms accurately define the artificial system...
This is exactly what I meant. You can define anything you want. You can even do so in a sophisticated way such that your defined system is internally coherent. But internal coherence alone can't be the standard of measure of truth -- we need to ask "does our internally coherent system correspond with reality?" To the questioner on MathOverflow, math will answer the coherence question, philosophy of math will answer the correspondence question.
> To me, this seems like a non-nonsensical question. 0 is defined to be a natural number.
Is it merely definitional though? They're called "natural" numbers for a reason! These are the set of numbers that seem most obvious to us, the kind that (I don't think) we can teach. For example, you can tell a child "this is 1 apple", "these are 2 apples", etc., but other than the name, you really can't teach a person what numbers are. They just know. Why is that? It's clearly is more than merely definitional. And I think 0 falls into this category unlike, say, complex numbers. But notice, whatever your response -- even if you disagree -- we're deep into philosophy territory here with the simple axiom "0 is a natural number".
> For example, you can tell a child "this is 1 apple", "these are 2 apples", etc., but other than the name, you really can't teach a person what numbers are. They just know. Why is that?
I would argue that this is a question for psychology, with input from neuroscience, biology, and likely numerous other fields of science. What you are asking is not a question about the universe, but rather a question about the human mind: why is it that the natural numbers seem innate to humans. In the same way we can ask why language is innate in humans, or the skills to walk.
Regarding 0, historically, 0 has been much more controversial as a number. To the best of my knowledge, we have no evidence of it existing as a number prior to around 400AD India. Also, for the record, 0 being a natural number is denominational. So much so that many (most?) mathematicians (myself included) do not consider 0 to be a natural number. Math has not imploded because of this, we just waste a little bit of effort here and there to clarify what we mean when it is an important distinction.
>And I think 0 falls into this category unlike, say, complex numbers.
Funny you should bring up complex numbers. If we look at physics, we find that the complex numbers seem far more "natural" than the natural numbers do. In fact, I cannot think of a single physics theory defined over the natural numbers. In contrast, the complex numbers show up all the time. Even quantum mechanics, which (being quantized) would seem to be an ideal candidate for a natural number theory, ends up being defined heavily in terms of complex numbers.
> You can even do so in a sophisticated way such that your defined system is internally coherent
Rest assured, any mathematical or logical system capable of expressing arithmetic is incoherent, if only trying to prove so using it's own rules. (You can shift the burden of proof to an external system you trust, but that external system would suffer from the same problem if a similar proof was attempted)
This is exactly what I meant. You can define anything you want. You can even do so in a sophisticated way such that your defined system is internally coherent. But internal coherence alone can't be the standard of measure of truth -- we need to ask "does our internally coherent system correspond with reality?" To the questioner on MathOverflow, math will answer the coherence question, philosophy of math will answer the correspondence question.
> To me, this seems like a non-nonsensical question. 0 is defined to be a natural number.
Is it merely definitional though? They're called "natural" numbers for a reason! These are the set of numbers that seem most obvious to us, the kind that (I don't think) we can teach. For example, you can tell a child "this is 1 apple", "these are 2 apples", etc., but other than the name, you really can't teach a person what numbers are. They just know. Why is that? It's clearly is more than merely definitional. And I think 0 falls into this category unlike, say, complex numbers. But notice, whatever your response -- even if you disagree -- we're deep into philosophy territory here with the simple axiom "0 is a natural number".
Regardless, thanks your your thoughtful response.