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I am not sure if I understand this correctly. Please correct me.

In a formal system,

an interpretation is a mapping from its formal language to one of its structures ie models.

an formalization is a mapping from one of its models to its formal language?

So are they inverse processes to each other, in the sense that their domains and codomains are exchanged?

Thanks!

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A formalization of, say, Group Theory cannot reasonably be viewed as a mapping from a specific group, say the $1$-element group, to a formal language. You may be attempting to express everything in terms of the language of sets and functions prematurely. It is useful to first get a lot of concrete experience with concrete cases. –  André Nicolas Feb 16 '12 at 16:45
    
I don't understand what you mean by a formalization. –  Qiaochu Yuan Feb 16 '12 at 17:56
    
@QiaochuYuan: I heard of that word often, such as formalize this thing and that. It sounds like some serious concept but I cant find its definition. –  Tim Feb 16 '12 at 18:01
    
Not all serious concepts have an honest definition... –  Zhen Lin Feb 16 '12 at 21:47

1 Answer 1

I suspect that the structure that you seek is the Galois connection between theories and models. A web search on such should turn up many expositions, e.g. in a slightly more general context, see Section 2.2 in Goguen and Burstall: INSTITUTIONS: Abstract Model Theory for Specification and Programming.

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