How to reconcile physical theories with the idea of free will: From analysis of a simple model to interval and fuzzy approaches

Most modern physical theories are formulated in terms of differential equations. As a result, if we know exactly the current state of the world, then this state uniquely determines all the future events - including our own future behavior. This determination seems to contradict the intuitive notion of a free will, according to which we are free to make decisions - decisions which cannot be determined based on the past locations and velocities of the elementary particles. In quantum physics, the situation is somewhat better in the sense that we cannot determine the exact behavior, but we can still determine the quantum state, and thus, we can determine the probabilities of different behaviors - which is still inconsistent with our intuition. This inconsistency does not mean, of course, that we can practically predict our future behavior; however, in view of many physicists and philosophers, even the theoretical inconsistency is somewhat troubling. Some of these researchers feel that it is desirable to modify physical equations in such a way that such a counter-intuitive determination would no longer be possible. In this paper, we analyze the foundations for such possible theories, and show that on the level of simple mechanics, the formalization of a free will requires triple interactions - while traditional physics is based on pairwise interactions between the particles.