Re: What is the origin of the terms 12 AM and 12 PM?

First off, the use of A.M.(ante meridiem) and P.M. (post meridiem) dates back to some of the earliest sundials, which is why the meridian is set at noon rather than midnight. Second, 12 A.M. means "the 12th (of 12) hour before the meridian," not "12 hours before the meridian;" otherwise, 11 A.M. would occur one hour after midnight. And, since sundials and timekeeping predated the concept of "zero" by more than a millenium, the meridian was given a numerical value, hence 12 noon instead of 0 noon.

Now to the real question of why noon was assigned P.M. and midnight, A.M. The most obvious answer is that this is some arbitrary assignment that simplifies digital clocks, which only have A.M. and P.M., and not "noon" or "midnight" options in their displays. However, there is actually a mathematical reason for these assignments, which is still based somewhat on the limitations of timekeeping. If we start with the definitions of A.M. as before noon and P.M. as after noon, we can obviously agree that 11:59 A.M. is indeed one minute before noon and that 12:01 PM is one minute after noon. But, what holds for a minute or an hour should, mathematically, also hold for a second, such that 12:00:01 P.M. is one second after noon. Ignoring for sake of argument any limits on the shortest length of time, we can continue working backward through smaller fractions of seconds and prove that they still occur after noon, just as we can approach noon from 11:59:59 A.M. through fractions of seconds that occur before noon. Ultimately, we find that "noon" is a dimensionless point in time, such that in the time it takes even the most accurate clock to switch its display from A.M. to P.M., noon has long since passed.