I have always been wondering

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?

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)

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