Law Without Law or "Just" Limit Theorems?: Some Reflections About a Proposal of Wheeler's

About 35 years ago Wheeler introduced the motto "law without law" to highlight the possibility that (at least a part of) Physics may be understood only following regularity principles and few relevant facts, rather than relying on a treatment in terms of fundamental theories. Such a proposal can be seen as part of a more general attempt (including the maximum entropy approach) summarized by the slogan "it from bit", which privileges the information as the basic ingredient. Apparently it seems that it is possible to obtain, without the use of physical laws, some important results in an easy way, for instance, the probability distribution of the canonical ensemble. In this paper we will present a general discussion on those ideas of Wheeler's that originated the motto "law without law". In particular we will show how the claimed simplicity is only apparent and it is rather easy to produce wrong results. We will show that it is possible to obtain some of the results treated by Wheeler in the realm of the statistical mechanics, using precise assumptions and nontrivial results of probability theory, mainly concerning ergodicity and limit theorems.