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I recall reading that it's important to separate mathematics and metamathematics. What exactly does this mean, and why is it so?

I understand that this question may make no sense without more context, but I've been curious about this for a while and can't seem to find where I originally read that.

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For example, it is usually important to distinguish between the meta-natural numbers $0, 1, 2, \ldots$ and the elements in a model of first-order arithmetic you happen to be analysing... – Zhen Lin Sep 2 '12 at 14:21
@Blue: Can you mention the context in which you read this? – Kartik Audhkhasi Sep 2 '12 at 14:24
@KartikAudhkhasi: Unfortunately, I can't. I read this a couple years ago, so I forgot the source. It didn't turn up anything in a Google search, either. – Blue Sep 2 '12 at 14:58
up vote 2 down vote accepted

Metamathematics is a branch of mathematics, so the assertion does not really make sense. It is certainly important not to confuse a formal system with the mathematical tools and ideas used to look into properties of the formal system, or formal systems in general.

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Thanks. While I'll probably never know for sure, that seems like a plausible explanation of what the author of whatever I was reading may have meant – Blue Sep 3 '12 at 8:34

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