# Examples of application of theories to completely unexpected structures?

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.

• – Bart Michels May 27 '18 at 14:11

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".

• This is the type of answer I was looking for – user56834 May 27 '18 at 15:25

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.

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.

• That was not really what I was thinking of. Please see my edit – user56834 May 27 '18 at 14:06

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.

(Not sure this counts.)