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      1 1 Such as the loan, in 1332, to the King of Britain Edward III who never returned it to the Bank of Bardi and Peruzzi – as all high school kids in Italy and our colleague Alberto Peruzzi in Florence know very well.

      2 2 The non-analyticity is stronger than the presence of positive Lyapunov exponents for a nonlinear function. These exponents can appear in the solution of a nonlinear system or directly in a function describing a dynamic. They quantify how a minor difference in initial conditions can be magnified along a trajectory. In this case, we can have a form of “controlled” randomness because the divergence of the trajectories starting within the same best measurement interval will never exceed a pre-assumed, exponentially increasing value. In the absence of (analytical) solutions, bifurcations and homoclinic orbits can lead to sudden and “uncontrolled” divergence.

      3 3 A macroscopic cause cannot have more elements of symmetry than the effects it produces. Its informational equivalent, called data processing inequality, asserts that no manipulation of information can improve the conclusions drawn from such data (Cover and Thomas 1991).

      4 4 Laplace was also aware of this, but Lagrange, Laplace and Fourier firmly believed that any system of Cauchy equations possessed a linear approximation (Marinucci 2011).

      5 5 A correlation between random events and symmetry breakings is discussed in Longo et al. (2015). In this case, measurement produces a value (up or down), which breaks the in-determined or in-differentiated (thus, symmetric) situation before measurement.

      6 6 The model does not assess the ability to make statistical predictions – as probabilistic models might – but rather the ability to predict precise measurement outcomes.

      7 7 Eagle argued that a physical process is random if it is “maximally unpredictable” (Eagle 2005).

      8 8 Some molecular types are present in a few tenths or hundreds of molecules. Brownian motion may suffice to split them in slightly but non-irrelevantly different numbers.

      9 9 An organism is an ecosystem inhabited by about 1014 bacteria, for example, and by an immune system, which in itself is an ecosystem (Flajnik and Kasahara 2010). Yet an ecosystem is not an organism: it has no relative metric stability (distance from its constituents), nor general organs of regulation and action, such as the nervous system found in animals.

      10 10 Some may prefer to consider viruses as the least form of life. The issue is controversial, but it would not change Gould’s and our perspective at all: we only need a minimum biological complexity which differs from inert matter.

      11 11 This was the definition of chance informally proposed by Borel (1909): the proportion of times that a finite sequence is in an infinite (even very long) sequence corresponds (approximately) to the probability that such a sequence occurs during a particular test.

      12 12 The British mathematician and logician Frank P. Ramsey studied conditions under which order must appear.

      13 13 The adjective “large” has precise definitions for both finite and infinite sets.

      14 14 Consider a gas particle and its momentum: the average value of the momentum over time (the time integral) is asymptotically assumed to coincide with the average momenta of all particles in the given, sufficiently large, volume (the space integral).

      15 15 It is not unreasonable to hypothesize that pseudo-randomness reflects its creators’ subjective “understanding” and “projection” of randomness. Psychologists have known for a long time that people tend to distrust streaks in a series of random bits; hence, they imagine a coin flipping sequence alternates between heads and tails much too often for its own sake of “randomness”. As we said, the gambler’s fallacy is an example.

      16 16 Incidentally, the same conference at which Gödel presented his famous incompleteness theorem.

      Chapter written by Cristian S. CALUDE and Giuseppe LONGO.