I was reading [a recent article][1] on Mach's Principle. In it, the author talks about inertia in an empty universe. I'll quote some lines from the article:

> Imagine a single body in an otherwise empty universe. In the absence
> of any forces, (Newton's second law gives): $$m\mathbf{a} = 0$$ What
> does this equation imply? Following Newton we would conclude from that
> $\mathbf{a} = 0$, that is, the body moves with uniform velocity. But
> we now no longer have a background against which to measure
> velocities. Thus $\mathbf{a} = 0$ has no operational significance.
> Rather, the lack of any tangible background for measuring motion
> suggests that $\mathbf{a}$ should be completely indeterminate. And it
> is not difficult to see that such a conclusion follows naturally,
> provided we come to the remarkable conclusion that
> $$m = 0$$ In other words, the measure of inertia depends on the
> existence of the background in such a way that in the absence of the
> background the measure vanishes!

I don't see how the argument is complete. For example, in an empty universe, how is it possible to assign a value of 0 to a force? And wouldn't the existence of mathematics in such an empty universe be questionable?

  [1]: http://www.ias.ac.in/resonance/April2011/p310-321.pdf