Of course, I meant F = G(m1/r)(m2/r). :smack:
… because there are no geometries with a non natural number of dimensions.
If the ship is half a million miles away, how are you measuring the length of the line on it, to compare with the stick in your hand?
You have an incredibly accurate (mythical) measuring device that can precisely (within about half a millimeter) measure the length of anything you can see with it. It is also able to compensate for any variation in the ship’s orientation.
A bit of a philosophical re-interpretation, if I may. I interpret the OP as asking the relatively simple question of why gravity obeys the inverse-square law in space as we commonly understand it, rather than some other function – perhaps one governed by some obscure non-integral constant. And the simple answer, as already indicated by several and supported by the informative diagram linked by Standingwave, is that it arises from the basic geometry of space.
The other points raised are interesting ones, but somewhat digressionary from that perspective. If, for instance, gravity operates differently when very weak, as proposed by Modified Newtonian Dynamics, then this is simply a refinement of the fundamental underlying inverse-square law. Likewise, if the inverse-square law must be modified because space is curved, it’s yet again a refinement of a fundamental truth brought about by a more detailed understanding.
I guess I’m just trying to get across the idea that the inverse-square law is an important fundamental truth determined by the nature of our universe the same way that Newtonian physics represents such fundamental truths, even though they turn out to be based on a simplified model of the universe that was overturned by relativity. The simplified model is still a very close and useful approximation of the practical world, Newtonian laws are fully self-consistent within it, and understanding why and how those laws hold true gives us a great deal of practical knowledge about the physical world, like how rocket propulsion works.
It’s really not viable, and certainly not simple at all. It’s tough to construct a model for MOND that’s even self-consistent, and so far as anyone can tell impossible to do so in a way that’s consistent with anything that even resembles General Relativity. It also accounts for only a few of the many observations which provide evidence for dark matter. And further, if you’re going to posit that dark matter does not exist, then you ought to give an explanation for that, since it would be quite remarkable if all matter in the Universe interacts electromagnetically.
There are, however, two possible ways in which the force might deviate from 1/r^2 which are taken seriously. For one, if the graviton has a nonzero mass, then its flux won’t be conserved, and so on very long length scales you would expect an exponential falloff, not an inverse square one (this is called the Yukawa potential, and is often used as a reasonable approximation of the Strong Nuclear Force).
For another possibility, the Universe in which we live might not actually have three spatial dimensions. It might have more, with the “extra” ones curled in on themselves on some small length scale. In this case, gravity would appear to fall off as 1/r^2 on long length scales, but as 1/r^3 (or 1/r^4 or some higher power) on sufficiently-small length scales.
Experiments have been performed, and will continue to be performed, to test both of these possibilities. So far, they have not detected any deviation from 1/r^2 of either form, but neither one can ever actually be ruled out entirely: All we can do is make statements like “If the graviton has a mass, it is less than X”, or “If there are extra dimensions, their characteristic scale is smaller than Y”, with better experiments just pushing down the values of X and Y.
You are blurring a lot of stuff. Newton’s law of universal gravitation is an empirical truth. No one disputes its importance and usefulness. But it’s a contradiction to claim a theory is both approximate and fundamental. Fundamental truth is usually taken to mean derived from first principles, not essential to our practical understanding of the world…
Then that measuring device is either:
Subject to the same distortion as anything else in the reference frame it is in when used, and therefore does not appear to do anything out of the ordinary.
or
Inhabiting a privileged/absolute reference frame, which is impossible.
I’m not so sure about that. https://www.youtube.com/watch?v=Oc8sWN_jNF4
That depends on how you define “geometries” and “dimensions”.
Mind… blown. If only Zeno of Elea had had YouTube in his day.
Some comments on this comment…
Of the different particle interactions we know about, one was first observed only in 1974.
Of the six quarks we know about, one was first observed only in 1995.
Of the three neutrinos we know about, one was first observed only in 2000.
The Higgs boson was first observed only in 2012.
We know of particles that interact via all the (known) fundamental forces of particle physics.
We know of particles that interact via only two of the forces.
We know of particles that interact via only one of the forces (and different forces at that).
It’s not that crazy at all to imagine a particle that we haven’t successfully observed yet (at least not terrestrially) that interacts via a force we haven’t successfully observed yet (or even one that we have). In fact, introducing just one such particle is almost too simple a model, since most other particles come in families of three. It’s even less crazy when introducing such a particle explains so many pieces of data.
Interesting, I just learned about the concept of Hausdorff dimension. I have to concede that point. 
Still, in physics (“space”), I believe there are no theories positing non integer dimension, even where n > 4.