We can precisely predict all future states of the system

Watching some lectures on differential equations, I hear the lecturer say, "If we can explicitly solve the equation, we can precisely predict all future states of the system."

The use of "precisely predict" here is very odd. If we are speaking of a pure mathematical system, then we are not "predicting" anything: all states of the system are called into being at once with the creation of the system of differential equations, which, even though we may label a variable 't', do not have a future or a past.

On the other hand, if he is speaking of an actual physical system (say, a mass-spring system), then "precisely" is grossly inaccurate: what he ought to have said is "We can predict future states of the physical system to whatever extent it does resemble and continues to resemble the abstract system of equations." Our predictions of the actual mass-spring's behavior based on the mass-spring equation will go seriously awry if someone walks in the room and grabs the spring, or an American drone blows up the building containing it, suspecting that an Afghan wedding party might be inside.


  1. Even then he is on shaky ground. No measurement can be precise enough to predict accurately forever. He can say the mathematical model is determinate, but that does not imply the world is, or that any actual observable application of the model to the world is. Any prediction must come with error bars, and eventually they get large.


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