the particle’s instantaneous velocity v at t codes a lot of information about what its velocity and location are at nearby times—but not ‘‘categorical information’’. The information is conditional or hypothetical information about average velocities (and consequently, locations). [...] for nearby times the collection of average velocities must be so ‘‘well-behaved’’ as to have a single limit, v, as the times get closer to t; or in spacetime terms, by saying that the nearby history of locations (the local segment of the worldline) must be smooth enough to have at t a tangent vector (a 4-velocity determined by v).

Also note that QM doesn’t quantize space-time. Indeed QM doesn’t really have a concept of space-time per se. Now various attempts at GUT do have quantized space time with varying degrees of failure.

Therefore, it is only reasonable to speak of a particle’s average velocity over some finite distance and time, and then only within a particular range of uncertainty.

On the philosophical subject of “burden of proof”, however, I believe that each party in an argument has a responsiblity to define, at the beginning of the discussion, just what *level* of certainty he or she is willing to accept as “proof”. Only in mathematics is there absolute proof; everywhere else there is only evidence.

For example, Richard Dawkins asserts that he considers the existence of God to be so unlikely, that literally all the evidence in the Universe could not convince him that God exists– that there would still be, in his mind, a simpler and more likely explanation. Specifically, he argues that God is more unlikely, statistically speaking, than the entire Universe.

Quantum mechanics does not define the limits of our mechanical abilty to measure quantities, but the theoretical universal limits of our ability to measure the universe.