Abstract
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 that physics, mathematics and computing science have proposed. Computing science allows to discuss unpredictability in an abstract, yet very expressive way, which yields useful hierarchies of randomness and may help to relate its various forms in natural sciences. Finally we discuss biological randomness—its peculiar nature and role in ontogenesis and in evolutionary dynamics (phylogenesis). Randomness in biology has a positive character as it contributes to the organisms’ and populations’ structural stability by adaptation and diversity.
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Notes
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....
Non-analyticity is stronger than the presence of positive Lyapunov exponents in a non-linear function. These exponents may appear in the solution of a non-linear system or directly in a function describing a dynamics; they quantify how a minor difference in the initial conditions may be amplified along the trajectory. In this case, one has a form of “controlled” randomness, as the divergence of the trajectories starting within the same best interval of measurement will not exceed an exponentially increasing, yet pre-given value. In the absence of (analytic) solutions, bifurcations and homoclinic orbits may lead to sudden, “uncontrolled”, divergence.
A macroscopic cause cannot have more elements of symmetry than the effect it produces. Its informational equivalent—called data processing inequality—states that no manipulation of the data can improve the inferences that can be made from the data (Cover and Thomas 1991).
Also Laplace was aware of this, but Lagrange, Laplace, Fourier firmly believed that any (interesting or “Cauchy”) system of equations possesses a linear approximation (Marinucci 2011).
A correlation between random events and symmetry breakings is discussed in Longo and Montévil (2015). In this case, measurement produces a value (up or down), which breaks the in-determined or in-differentiated (thus, symmetric) situation before measurement.
The model does not assess the ability to make statistical predictions—as probabilistic models might—but rather the ability to predict precise measurement outcomes.
Eagle has argued that a physical process is random if it is “maximally unpredictable” (Eagle 2005).
Some molecular types are present in a few tenth or hundreds molecules. Brownian motion may suffice to split them in slightly but non-irrelevantly different numbers.
An organism is an ecosystem, inhabited, for example, by about \(10^{14}\) bacteria and by an immune system which is, per se, an ecosystem (Flajnik and Kasahara 2010). Yet, an ecosystem is not an organism: it does not have the relative metric stability (distance of the components) nor global regulating organs, such as the neural system in animals.
Some may prefer to consider viruses as the least form of life. The issue is controversial, but it would not change at all Gould’s and our perspective: we only need a minimum biological complexity which differs from inert matter.
This was Borel’s definition of randomness (Borel 1909).
The British mathematician and logician Frank P. Ramsey studied conditions under which order must appear.
The adjective “large” has precise definitions for both finite and infinite sets.
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).
It is not unreasonable to hypothesise that pseudo-randomness rather 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.
Incidentally, the conference where Gödel presented his famous incompleteness theorem.
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Acknowledgments
The authors have been supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant and benefitted from discussions and collaboration with A. Abbott, S. Galatolo, M. Hoyrup, T. Paul and K. Svozil. We also thank the referees for their excellent comments and suggestions.
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Calude, C.S., Longo, G. Classical, quantum and biological randomness as relative unpredictability. Nat Comput 15, 263–278 (2016). https://doi.org/10.1007/s11047-015-9533-2
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DOI: https://doi.org/10.1007/s11047-015-9533-2