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AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a specific structure, or a specific set of structures.

For example, when people think of a “field” they AFAIK usually think of $\mathbb R$, or $\mathbb C$. A topology started out as a concept defined on $\mathbb R^n$ if I’m not mistaken.

But I’ve also seen cases where a certain structure has a natural topological structure, such as certain sets of propositions in first order logic. As far as I know, the people who formulated the axioms of a topology had no idea of this application. And the topological structure of a set of FOL statements is certainly conceptually vastly different from one on $\mathbb R^n$, certainly not two structures I would have expected to have such a deep commonality.

I would like to make a list of examples of mathematical structures that

  1. Are interesting and well-behaved structures (e.g. not mere pathological counter examples)

  2. satisfy the axioms of some mathematical theory in an interesting and nontrivial way,

  3. But whose emergence is (conceptually/historically) very different from the structure of which those axioms were originally intended as a generalization.

user56834
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The useful Zariski topology in algebraic geometry satisfies the usual axioms for a topology in a context that doesn't really match "the structure of which those axioms were originally intended as a generalization".

Ethan Bolker
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Several significant examples from mathematical physics:

  • The usefulness of Hilbert space in the formalization of quantum mechanics.

  • Riemannian manifolds as the appropriate language for general relativity.

  • Calabi -Yau manifolds come up in string theory.

Calculus for Newtonian mechanics doesn't count because the wish to formalize mechanics was much of what led Newton to invent calculus.

Ethan Bolker
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The Peano's arithmetic was thought to axiomatize the structure of natural numbers. However, exists the structure of non standar naturals where exists a biggest natural number and satisfies that axioms.

YCB
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Number theory (the "structure of the integers") had no applications for years - a fact that particularly pleased G. H. Hardy.

Now it's central to cryptography: prime factorization, discrete logarithms, elliptic curves.

See https://crypto.stackexchange.com/questions/59537/how-come-public-key-cryptography-wasnt-discovered-earlier

(Not sure this counts.)

Ethan Bolker
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