A quantitative and objective notion of emergence is proposed, using algorithmic information theory as a basis for an objective framework in which a bit string encodes observational data. A plurality of drops in the Kolmogorov structure function of such a string is seen as the hallmark of emergence. The definition offers some theoretical results and extends the notions of coarse-graining and boundary conditions.

Sophistication and logical depth are two quantitative measures of the non-trivial organization of an object. Although apparently different, these measures have been proven equivalent (by Pr. André Souto and collaborators), when logical depth is renormalized by the busy beaver function. However, if the measures are relativized to auxiliary information y, it turns out that the ability of y to solve the halting problem introduces a distortion between Soph(x|y) and Depth(x|y).

Finally, I share my latest thoughts about epistemology (the philosophy of knowledge), more precisely, about the problem of making it rigorous. I explain the gap between Bayesian and Popperian epistemology and speculate on some possible resolutions through algorithmic information theory. Note that this problem is not 'merely' philosophical – it is the key problem that needs to be solved to have proper thinking machines.