Consider a Newtonian physics with gravity and point particles with non-zero mass. Take component forces and masses as primitive quantities. Then we can reduce the distance at time *t* between distinct particles *a* and *b* as (*m*_{a}*m*_{b}/*F*_{ab})^{1/2}, where *F*_{ab} is the magnitude of the gravitational force of *a* on *b* at *t*, and *m*_{a} and *m*_{b} are the masses at *t* of *a* and *b* respectively (I am taking the units to be ones where the gravitational constant is 1); we can define the distance between *a* and *a* to be zero. For every *t*, we may suppose that by law that the forces are such as to define a metric structure on the point particles.

If we want to extend this to a spatiotemporal structure, rather than just a momentary temporal structure, we need to stitch the metric structure into a whole. One way to do that is to abstract a little further. Let *S* be a three-dimensional Euclidean space. Let *P* be the set of all particles. Let *T* be the real line. For each object *a* in *P*, let *T*_{a} be the set of times at which *a* exists, and let *m*_{a}(*t*) be the mass of *a* at *t*. For any pair of objects *a* and *b* and time *t* in both *T*_{a} and *T*_{b}, let *F*_{ab}(*t*) be the magnitude of the gravitational force of *a* on *b* at *t*. Let *Q* be the set of all pairs (*a*,*t*) such that *t* is a member of *T*_{a}. Say that a function *f* from *Q* to *S* is an *admissible position function* provided that:

- If
*t*is a member of both*T*_{a}and*T*_{b}, then*F*_{ab}(*t*)=*m*_{a}(*t*)*m*_{b}(*t*)/|*f*(*b*,*t*)−*f*(*a*,*t*)|^{2}. -
*f*''(*a*,*t*) is equal to the sum over all particles*b*distinct from*a*of (*f*(*b*,*t*)−*f*(*a*,*t*))*F*_{ba}(*t*)/(*m*_{a}(*t*)|*f*(*b*,*t*)−*f*(*a*,*t*)|).

*a*relative to

*f*at

*t*is just

*f*(

*a*,

*t*).

The above account generalizes to allow for other forces in the equations.

So, instead of taking spatial structure to be primitive, we can derive it from component forces, masses and objects, taking the latter trio as primitive.

I don't know how to generalize this to work in terms of a spatiotemporal position function instead of just a spatial position function.

Of course, component forces are hairy.

Perhaps the method generalizes to less out-of-date physics. Perhaps not. But at least it's a nice illustration of how spatial relations might be non-fundamental, as in Leibniz (though Leibniz wouldn't like this particular proposal).

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