I’m assuming that gravity following an inverse square law is just a fact which couldn’t be deduced from first principles.
It may actually be derivable anthropically: exponents other than 2 or 1 prohibit stable orbits, and an exponent of 1, as Zack says, implies 2-dimensional space, which might be too simple for observers.
Though it should be noted that even if we allow for anthropic arguments, it is impossible to ascertain whether the inverse-square law is fundamentally true, or just a very good approximation of some far more complex actual law. Therefore, the truly fundamental laws are impenetrable to such reasoning: at maximum, we can ascertain that the fundamental laws, whatever they are, must have approximations with anthropically relevant properties to the extent that we are influenced by them. And indeed, when it comes to gravity, the inverse-square law is highly accurate for our practical purposes, but it’s only a good approximation of the predictions of the more complicated general relativity—itself likely just an approximation of the more accurate and complicated quantum gravity—that happens to hold in the conditions that prevail in the part of spacetime we inhabit.
I suppose the only way out of this would be to devise an anthropic argument where our existence hinges on the lack of arbitrarily small deviations from the law we wish to derive anthropically. I don’t know if perhaps some sound arguments along those lines could be derived from reasoning about the very early universe.
It may actually be derivable anthropically: exponents other than 2 or 1 prohibit stable orbits, and an exponent of 1, as Zack says, implies 2-dimensional space, which might be too simple for observers.
Though it should be noted that even if we allow for anthropic arguments, it is impossible to ascertain whether the inverse-square law is fundamentally true, or just a very good approximation of some far more complex actual law. Therefore, the truly fundamental laws are impenetrable to such reasoning: at maximum, we can ascertain that the fundamental laws, whatever they are, must have approximations with anthropically relevant properties to the extent that we are influenced by them. And indeed, when it comes to gravity, the inverse-square law is highly accurate for our practical purposes, but it’s only a good approximation of the predictions of the more complicated general relativity—itself likely just an approximation of the more accurate and complicated quantum gravity—that happens to hold in the conditions that prevail in the part of spacetime we inhabit.
I suppose the only way out of this would be to devise an anthropic argument where our existence hinges on the lack of arbitrarily small deviations from the law we wish to derive anthropically. I don’t know if perhaps some sound arguments along those lines could be derived from reasoning about the very early universe.