I have always been wondering 



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> why the term "model" is used by mathematicians (especially in mathematical logic) in a conceptually different (even opposite) way than it is used by other scientists, e.g. physicists, biologists, chemists, economists etc.
>  And when has this terminology first arisen?
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> 
> 


In essentially all of natural sciences we have a "world" (our physical world, a living being, a market...) which hosts specific instances of certain "phenomena" (that involve "objects" of some kind, and "relations" between them), and the goal of the theory is to provide a *model* that describes that phenomena. In this case, a *model* is understood to be some kind of conceptual construction that *abstracts* the common relevent properties of several instances of the phenomenon in question and puts them in a rational (mathematical or not) framework that enables us to draw consequences and predictions about the phenomenon itself. Think e.g. of the "Standard Model" of particle physics.


Also mathematics has a "world", populated by objects (e.g. groups or ordered fields), between which some relations hold. The goal of a mathematical "theory" (at least from the perspective of formal theories in the sense of mathematical logic) is to provide a simple "model" (mind the use of quotation marks) that describes all the instances of certain "phenomena", and it accomplishes this task by a list of *axioms* (e.g. the axioms of group theory) that incapsulate the relevant properties of the objects in question (e.g. being a group). 


It is possible that new "species" of the same kind of objects are found, that is, they verify the axioms, i.e. they fall under the description by the same "model".
A natural scientist would call such an object an incarnation (manifestation, realization, example, explicitation, instance...) of the "model" provided by the axioms. Mathematicians, on the contrary, call it a *model* (here in the tecnical sense, hence without quotation marks) for the axioms. 


Isn't it strange?


(Edit: there is also another way the word "model" is used in maths, as in "Weierstrass model" or "Néron model", in which cases it is essentially sinonimous with "normal form". This latter use of the word seems to me more consistent with the general natural sciences use)



The use of the word "model" in Mathematical Logic vs the same word in Natural Sciences
* yes, it is strange, because one expects words to have similar meaning even in different contexts (but as others have noted, whether this is strange or not, there are all sorts of words that have multiple meanings, and even contradictory ones).
* yes, I agree in your assessment of how model has two, not contradictory, just oppositely directed meanings. In common language, natural sciences, and statistics, a model is an abstraction, where incidental details are removed, leaving just the bare bones abstract thing of what you care about (a model airplane, a model of population dispersal, a regression model predicting a value from a small set of variables). In mathematical logic, it is the other way around; a model is an -example- of the syntactically presented axioms (the permutations of 3 items is a model of the group axioms, a valuation (an assignment of boolean values) is a model for a propositional statement.


In short, in mathematical logic, there is the abstract theory that has a model as an example, the model fits the theory. In the rest, the model -is- the theory and the phenomena of the world are the examples (which the theory/model are trying to fit).


When this divergence of use started and by who, I can't say. But the logical use is certainly later. Who first started using the concept (but not necessarily the usage) in logic? Was it Goedel? How about Hilbert in 'Foundations of Geometry' where he shows independence of individual axioms by giving different models satisfying the rest (i don't know what terms he uses in the original or if they were common usage at that time for the concept).


