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But it also seems to me that one might, in principle, conduct an experiment that could distinguish between the symmetry of a mathematical equality and the causal implication, and that is to look for the inevitable time delay that must exist between a cause and its effect. Admittedly, for close encounters detecting this time delay may be beyond our current technological capability, but certainly if one thinks about it for a while it may be possible to come up with a suitable experiment. If so, and if it does demonstrate a causal relationship, then that would imply that we have been writing NSL wrong all these many centuries.
But that impulse to infer causality is very strong in humans, for better or for worse.