As a human, a finite mode as you can still insist and ask whether mathematically speaking it is final purpose or casuality? As a finite mode, you can still persist and ask whether, mathematically speaking, it's final purpose or casuality? Because if one option is a cheaper calculus of life, then that's important.
Mathematically there is no such thing as a final cause. Not hidden, not implicit, not “emergent.” Zero. Mathematics describes relations between states, not intentions toward states. A dynamical system is specified by an initial condition plus a rule of evolution. The rule does not point forward to a goal; it maps present state to next state. Even optimization problems do not smuggle in purpose but define a constraint and ask which configuration satisfies it. The “goal” exists only in the modeller’s head, not in the system.
What does exist is causality, sometimes deterministic, sometimes probabilistic. Deterministic causality means that given a state, the next state is fixed. Probabilistic causality means that given a state, there is a probability distribution over next states. In neither case is the future pulling the present toward itself. The arrow always runs from present to next, never backward from outcome to cause.