Interesting article... The derivative problem is particularly enlightening I think. A derivative is a limit. Thus the answer you get for a derivative is also a limit.
For example, if you have the real speed of a car at every point in time as well as the position, then nominally, in infinite world, the speed is just the derivative of the position curve.
But if you are in finite world, then the derivative curve in infinite world is the limit of the speed curve (and note that limit does not mean upper bound).. it just is a useful tool for reasoning what will happen if you take a more fine grained set of measurements.
For some fineness of measurement, there will be a point at which your observed speed curve is always within epsilon of the ideal one.
For example, if you have the real speed of a car at every point in time as well as the position, then nominally, in infinite world, the speed is just the derivative of the position curve.
But if you are in finite world, then the derivative curve in infinite world is the limit of the speed curve (and note that limit does not mean upper bound).. it just is a useful tool for reasoning what will happen if you take a more fine grained set of measurements.
For some fineness of measurement, there will be a point at which your observed speed curve is always within epsilon of the ideal one.
Good article