It takes my breath away how consistent the mean length of a day is and how accurately we can measure it. So accurately that if it deviates by mere tens of milliseconds, and it does, we can notice it.

It boggles my mind that a hunk of rock made up of differing densities, with so much of salty water sloshing around, Moon exerting her brakes, bombardments by extraterrestrial matter, Earth still remains so consistent. Of course, this is not surprising if one does the math.

Yes math and physics are different. It's the the fact that math can model the physics so accurately at all that it's breathtaking.

Even without venturing into the realms of Quantam Mechanics, all measurements are, by physical definition, upto the resolution of the measuring instrument. So it always come with inherent +/- error bars. Sometimes these error bars are made explicit, at other times they are elided, often with the assumption that it's obvious.

Models are always simplifications. That is precisely why they are useful. ( A 1:1 scale map is not very useful, especially when we already have one). Models are obtained by ignoring the effect of many known disturbances, but whose effect one deems not to exceed a tolerance bound.