Well, here is an idea that propagates backwards from linguistics to differential geometry:
The higher order derivatives of position on a manifold (or at least an embedded manifold), can be used to find an object which "best fit"s the manifold at a point. The family of objects used for the fitting is restricted to entities having constant such derivatives after a specified order. For example, curves in 3D space can be fit with lines, osculating circles, and helices of constant radius and torsion. Surfaces in 3D can be fit with planes, spheres, and....
Singularities in a manifold are an opportunity to "best fit" with other families of entities besides the ones with constant higher order derivatives. That one can choose from among different such families is an idea - the best model - that is more evident in linguistics than in geometry - but there it is - a way to analyze singularities.
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