Dynamical systems, causality and derivatives order Talking about input/output representation of a dynamical system, the professor said that the equation(s) involved must satisfy this condition in order for the system to be qualified as "causal":

the greatest derivatives order of the output should be lower than the greatest derivatives order of the input

He explained the fact saying that if we dismiss the condition, we could imagine a system described by 
$$y(t) = \dot{u}(t)$$
that is  not feasible in our physical world where actions produce effects. He said "if we knew the derivative of input, we'd know the future".
At first, I can't fully understand this assertion. If it is true, then why does the very knowledge of u(t) not imply the knowledge of the future?
I think this misunderstanding is due to the fact that I have just started studying system, so maybe i'm not in the correct perspective.
However there's another problem yet: in the quoted condition there is not written " derivatives of input are not allowed". They can appear, given that their max order is lower than (..). Why this constraint implies causality?
Finally, I'd be very grateful if you could point out main concepts that link together systems, derivatives, future.
Thanks a lot
 A: consider the discrete time evolution of the function $$u_n = u(t_0 + n\Delta t)$$ and its derivative
$$ \dot u = \frac{u_{n+1} - u_n}{\Delta_t} $$, where n is the current time step, and the denominator is the infinitesimal time step. Clearly, it involves knowing the function u(t) at step n+1, which is in the future. At present, we know u(t) for all n from 0 to n
Edit:
To answer your 2nd question: Consider any law of physics. I don't remember who was the first to make this proof , it may be Landau or Noether, but the statement was that all laws of physics must be fixing the same derivative order. In the case of our universe, it seems to be the 2nd order. Short googling finds ample modern proofs, e.g.
http://www.academia.edu/2995322/Why_Physics_Uses_Second_Derivatives
Consider an apple falling on Newton's head. In the first case, we know the force of gravity, which fixes the 2nd time derivative, and thus gives us 2nd order ODE. We solve that ODE to find its velocity and position as function of time for some initial conditions. Given the above discrete representation of derivatives, we can model the falling of an apple as a step-wise process, where the laws of physics will always tell us acceleration at step n, and that in turn will tell us the velocity and position at step n+1, given velocity and position at step n.
Now, in the 2nd case, lets say that the laws of physics tell us what the velocity should be. We formulate the 1st order ODE, solve it, and find the position as function of time. Again we can formulate the step-by step process for velocity and position. However, now somebody comes and tells us that they know all velocities at steps 0 to n, and want to know the current acceleration at step n. We will fail to answer that question, as acceleration at step n depends on the future velocity at step n+1, which we do not know.
Hope this helps :)
A: Taking your professor's example:
$$y(t) = \dot{u}(t) \implies y = \lim_{\delta \to 0} \frac{u(t+\delta)-u(t)}{\delta}$$
If we assume we can't know $u(t+\delta)$ for $\delta > 0$ (can't know the future), then this limit cannot be calculated in an applied setting.
A: I don't agree with this, at least not with the specific example given. If $u$ is differentiable at $t$, then $$\lim_{\delta \to 0} \frac{u(t+\delta)-u(t)}{\delta} = \lim_{\delta \to 0^-} \frac{u(t+\delta)-u(t)}{\delta},$$
i.e. we are approaching the limit from left and hence only consider time up to $t$.
My definition of causality would be $u_1(t) = u_2(t)$ for $t < t_0$ $\Rightarrow$ $y_1(t) = y_2(t)$ for $t < t_0$. This is equivalent to $u(t) = 0$ for $t < t_0$ $\Rightarrow$ $y(t) = 0$ for $t < t_0.$ Clearly the system $y(t) = \dot{u}(t)$ has this property.
I will add however that the system is unphysical for a different reason, namely that it is unstable $\left( \right.$consider for example $u(t) = e^{-t^2}$ $\left)\right.$
