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From a physicist's perspective I think that the latter part Ryan's answer really goes to the heart of the matter. The point is that the VAST majority of physical phenomena are purely local. Consider for example General Relativity. An observer existing for a finite time will probe a finite patch of spacetime. To describe what he sees he solves Einstein's equation:

$R_{\mu \nu}-\frac{1}{2}Rg_{\mu \nu}\sim T_{\mu \nu}$

where in the above $g$ is the metric, $R_{\mu \nu}$ its Ricci curvature and $T$ is a tensor field describing the distribution of energy and matter in spacetime. This is a local differential equation, and since the observer sees a small patch, for the most part he could care less whether the global structure of spacetime is $\mathbb{R}^{4}$ or any other smooth four-manifold.

A crucial point is that unlike derived physical equations, like say the heat equation, equations of fundamental physics (General Relativity, Electrodynamics, Quantum Field Theory, String Theory) are invariant under the Lorentz group of symmetries. This means that, for reasonable physical matter and energy distributions, there is a $finite$ signal propagation speed (the speed of light) and thus far away properties of the differentiable structure of spacetime take a very long time to have local consequences for any fixed observer.

2 Edited to account for Willie's comment.; deleted 5 characters in body

From a physicist's perspective I think that the latter part Ryan's answer really goes to the heart of the matter. The point is that the VAST majority of physical phenomena are purely local. Consider for example General Relativity. An observer existing for a finite time will probe a finite patch of spacetime. To describe what he sees he solves Einstein's equation:

$R_{\mu \nu}-\frac{1}{2}Rg_{\mu \nu}\sim T_{\mu \nu}$

where in the above $g$ is the metric, $R_{\mu \nu}$ its Ricci curvature and $T$ is a tensor field describing the distribution of energy and matter in spacetime. This is a local differential equation, and since the observer sees a small patch, for the most part he could care less whether the global structure of spacetime is $\mathbb{R}^{4}$ or any other smooth four-manifold.

A crucial point is that unlike derived physical equations, like say the heat equation, equations of fundamental physics (General Relativity, Electrodynamics, Quantum Field Theory, String Theory) are invariant under the Lorentz group of symmetries. This means that there is a $finite$ signal propagation speed (the speed of light) and thus far away properties of the differentiable structure of spacetime take a very long time to have local consequences for any fixed observer.

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From a physicist's perspective I think that the latter part Ryan's answer really goes to the heart of the matter. The point is that the VAST majority of physical phenomena are purely local. Consider for example General Relativity. An observer existing for a finite time will probe a finite patch of spacetime. To describe what he sees he solves Einstein's equation:

$R_{\mu \nu}-\frac{1}{2}Rg_{\mu \nu}\sim T_{\mu \nu}$

where in the above $g$ is the metric, $R_{\mu \nu}$ its Ricci curvature and $T$ is a tensor field describing the distribution of energy and matter in spacetime. This is a local differential equation, and since the observer sees a small patch, for the most part he could care less whether the global structure of spacetime is $\mathbb{R}^{4}$ or any other smooth four-manifold.