My question is pretty straightforward to be stated but I don't know whether the answer is as easily reported. It is remarkable and very interesting that Physics work in (almost) any number of spacetime dimensions, but is it possible to actually measure the dimensionality of spacetime, even indirectly? I have found elsewhere in this site an argument regarding measuring the power law of the distance-dependence of forces, for example, the magnitude of the electrostatic force exerted by a charged point particle with charge $q$ would be of the form, \begin{equation} F_{Coulomb} \sim\frac{q}{r^{d-1}} \end{equation} with $r$ the distance from the particle's position and $d$ the number of spatial dimensions.
I can accept this as long as the procedure of measuring distances is well defined. Fundamentally, an observer would hold a meter stick and measure distances along the spatial axes. But how can the observer be certain of the number of axes he is capable of laying the meter stick along. How can he be sure there aren't any more axes that he simply cannot realize?
