In the science of Galileo and Newton there is a precise correspondence between our mental procedures and the world events, since M extracts the relevant things neglecting the “secondary qualities”, that is, those nuances which are useful for poetry but not scientific predictions. Within this framework, certitude (i.e. syntactical correctness) and truth (i.e. adequacy to reality) seem to coincide.
The complexity of the dynamical model of a many particle system can be seen as the ambiguity in predicting the state reached at a given time starting from a definite initial condition. Let us introduce the notion of “stability of a trajectory”. Suppose we go by bicycle on a road laying in the bottom of a valley. Let such a road be unique. However a change of landscape may transform the valley into a ridge with two lateral valleys; if the rule is that the road has to be in the valley, then we have two roads. This duplication is called “bifurcation”. The bicycle on the road is a metaphor of the system dynamics. The system is mathematically represented by a point in an n-dimensional space, where n is the number of different measurements by which we characterize the system. In the case of a single point-like particle, we need three coordinates for the position and three for the velocity, hence n=6; for a system of N particles, n=6N.
The time evolution of the system is the collection of successive state points, that is, a trajectory in the n dimensional space. The bifurcation is the transition from a trajectory with a single branch to a trajectory with two possible different branches.
The topography of the valley roads looks like a fork. In the “qualitative dynamics” of Poincaré, we plot the position of the stable dynamical states as a function of a control parameter. For all the control parameters which admit a single valley we have a continuous line; as the control parameter implies two valleys, that line bifurcates into two separate lines. This topography is called the bifurcation diagram.
The only knowledge of the topography does not allow to establish whether the system is taking one branch or the other. If each one of the separate branches bifurcates on its turn, and thus on, we have a tree whose branch number grows exponentially, that is, two after the first bifurcation, four after the second, eight after the third and so on. If we call “syntax” the assignment of the topography, we see that to resolve the ambiguity and decide on the precise branch, we must add some extra information at each bifurcation. This means that the “semantics” enters not only in the preparation of the model, as we choose the degrees of freedom, but also in the course of the evolution, since we must also decide how the environment breaks the symmetry of each bifurcation imposing a specific choice.
This is the main message of nonlinear dynamics. Before people realized this bifurcation explosion, it was considered safe to limit the investigation to a range of control parameters allowing for a single branch. However only in laboratory demonstration it is up to us to fix the control parameters; in real life, the piece of world that we aim to model is embedded in an environment which provides parameters not controllable by us. We should then be ready to face plenty of possible bifurcations.
Thus, complexity in science denotes two different things.
If we refer to a closed piece of world, removed from external influences, once the essential connotations have been captured into a model, then the deductive machine provides a tree of possible solutions, with many branches all equally likely. In such a case the complexity is syntactical, intrinsic to the formal language, in so far as it is due to in the presence of alternative paths. We can take as indicator of complexity the amount of computational resources necessary to solve the problem. This complexity, considered in computer science, will be rather called by us as complication.
More realistically, we must realize that a model of closed world is not very convenient. It is instead more useful to build the model upon some salient features on which we have detailed information, and consider the system under investigation as “open” to the rest of the world. This openness manifests itself as suitable “boundary conditions”, that we must apply at each bifurcation to decide which side to continue.
