The book aims at the identification of the organising concepts of some physical and
biological phenomena, by means of an analysis of the foundations of mathematics and of …

§ 1. Introduction. For most of its history, mathematics was algorithmic in nature. The
geometric claims in Euclid's Elements fall into two distinct categories:“problems,” which …

We present a list of open questions in reverse mathematics, including some relevant
background information for each question. We also mention some of the areas of reverse …

Science constructs tools for knowledge and, occasionally, this bold enterprise may let a few
believe in the “completeness” of a given theoretical frames: as for these phenomena, we can …

We propose the thesis that randomness is unpredictability with respect to an intended theory
and measurement. From this point of view we briefly discuss various forms of randomness …

We exhibit a close correspondence between $ L_1 $-computable functions and Schnorr
tests. Using this correspondence, we prove that a point $ x\in [0, 1]^ d $ is Schnorr random if …

We consider the question of computing invariant measures from an abstract point of view.
We work in a general framework (computable metric spaces, computable measures and …

We prove the effective version of Birkhoffʼs ergodic theorem for Martin-Löf random points
and effectively open sets, improving the results previously obtained in this direction (in …

We show that if a point in a computable probability space $ X $ satisfies the ergodic
recurrence property for a computable measure-preserving $ T\colon X\to X $ with respect to …

In biology, phenotypes' variability stems from stochastic gene expression as well as from
extrinsic fluctuations that are largely based on the contingency of developmental paths and …