I was reading a recent article 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?
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