Randomness and Determination in the Interplay between the Continuum and the Discrete
Francis Bailly · Advances in computer science and engineering · 2011
This paper is a conceptual analysis of the role of the mathematical continuum vs. the discrete in the understanding of randomness, as a notion with a physical meaning or origin. The presentation is “informal”, as we will not write formulas; yet, we will refer to non-obvious technical results from various scientific domains. And we will propose a conceptual frame for understanding randomness (and predictability), which is essentially original, we believe. The idea we started from is that the mathematical structures, constructed for the intelligibility of physical phenomena, according to their continuous (mostly in Physics) or discrete nature (generally in Computing), may propose different understandings of Nature. In particular, as hinted in [Bailly, Longo, 2006; Longo, 2007], the causal relations, as structures of intelligibility (we “understand Nature ” by them), are mathematically related to the use of the continuum or the discrete. But … what discrete (mathematical) structures are we talking about? We believe that there is one clear mathematical definition of “discrete”, that we will use in this paper: a structure is discrete when the discrete topology on it is “natural”. Of course, this is not a formal definition, but in mathematics we all know what “natural ” means. For example, one