Clocks show the time that is _passed/past_, that's the point. Hence flooring is correct approach.
Even when you involve physics, you are seeing the nanosecond past. Not the actual time. (The time that takes light to travel from clock to your eyes, then your brain's processing delay via neurons etc.)
Even the thought of being late is the same way. If something starts at 13:00 and you are not already there when it is 13:00:00.000, you are -by definition- late.