> This is also meaningless, scientifically: there is nothing you can measure which is outside the universe. Things like a holographic universe are inherently untestable.
This is incorrect. Are you familiar with the inverse-square law? In 3-dimensional Euclidean space, the rate at which any force decays over a distance is inversely proportional to the square of the distance. This generalizes to n dimensions; but instead of the drop off rate being inversely proportional to the square of the distance, it becomes inversely proportional to the n - 1th power of the distance.
Suppose our universe is a 3-dimensional embedding of a n-dimensional manifold. Every local force we could empirically test would adhere to the inverse-square law, i.e. have drop off over distance k of ~ 1/(k^2). But theoretically speaking we could empirically test the dimensions of the manifold we reside in by identifying which power of distance is proportional to the drop off rate of extremely small forces where the compactified dimension can be detected.
For practical purposes this would require us to increase the precision with which we can empirically test (and reason about) forces at the subatomic level.
This is incorrect. Are you familiar with the inverse-square law? In 3-dimensional Euclidean space, the rate at which any force decays over a distance is inversely proportional to the square of the distance. This generalizes to n dimensions; but instead of the drop off rate being inversely proportional to the square of the distance, it becomes inversely proportional to the n - 1th power of the distance.
Suppose our universe is a 3-dimensional embedding of a n-dimensional manifold. Every local force we could empirically test would adhere to the inverse-square law, i.e. have drop off over distance k of ~ 1/(k^2). But theoretically speaking we could empirically test the dimensions of the manifold we reside in by identifying which power of distance is proportional to the drop off rate of extremely small forces where the compactified dimension can be detected.
For practical purposes this would require us to increase the precision with which we can empirically test (and reason about) forces at the subatomic level.