> 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.
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.