In an equation like i(t) = CdV/dt, is "V" the cause and "i" the effect?
It's not so simple even in mechanics. Imagine you are a nail, with a hammer coming fast towards you. In order to stop the hammer in the space available, it has to be accelerated (decelerated with a change of sign). While undergoing that rapid acceleration, it applies a large force to the nail.

The problem is what you mean by 'cause'. 

Usually, we think of effect following cause, that cause never follows effect. The wayward football broke the neighbour's window. It is totally reasonable to argue that the broken window did not cause the football to be moving.

However, in electronics, and mechanics, where the behaviour of accelerations, forces, voltages and currents are conveniently described by equations of the form dI/dt = V/L, and f = ma, the two go hand in hand.

We can say the voltage across the coil caused the current to change, if that's convenient at the time, for instance when 'charging' the inductor of a boost DC-DC converter. We can also say that the changing current caused a large voltage across the coil, when 'discharging' the inductor of the same converter into the higher output voltage. Both descriptions are correct. It is not reasonable to argue that one did not cause the other. I've had long arguments about whether a BJT transistor is voltage controlled or current controlled. It is of course both, it's nonsense to argue that the current flow doesn't cause the voltage difference or vice versa, but concentrating on one or the other may be more useful choice at any given time.

Similarly, the large force applied to the hammer head by the nail caused it to slow down rapidly.