newtonian mechanics - Does causality alone resolve the mathematical ambiguity of expressing physical systems?

Newton's 2nd law of motion is most often written in the differential form

$\sum F = {dp \over dt} $

but can also be expressed in an integral form

$ p = \int\sum F dt $

Each form of expressing the connection between force and momentum is mathematically correct, but if we adhere to causality we are limited to the second expression (Forces lead to momentum (movement) - not vice versa).

This limitation is further exemplified when one attempts to simulate the equations in a digital computer. If feedback is involved the first expression can lead to algebraic loops, while the second gracefully evolves simulated motion. Algebraic loops do not represent physical reality (even the fastest of all physical systems are limited by the speed of light!)

So other than what is defined to be the independent/dependent variables, and/or the operation of either differentiation vs integration, there is nothing in the mathematics that tells us which is the 'correct' formulation of the laws of motion. The presence of differentiation in the mathematics implies prediction - knowing the future and therefore expresses the physical system in a non-causal manner. Rewriting the system in an integral form expresses the system in a causal way. The integral form in a sense reveals the roles of each factor in cause and effect.

Are there perhaps other ways (missing equation(s), other types of mathematics) that can help to resolve the ambiguity that calculus can cause? Is $F = ma$ incomplete? Perhaps some expression involving entropy is required for completeness. Physical reality favors memory and shuns prediction.

BTW Newton's law just served as a simple example towards my question. The ambiguity can be realized in any other mathematics involving calculus that model physical systems (including Maxwell's equations).