Abstract
There is a persistent confusion about determinism and predictability. In spite of the opinions of some eminent philosophers (e.g., Popper), it is possible to understand that the two concepts are completely unrelated. In few words we can say that determinism is ontic and has to do with how Nature behaves, while predictability is epistemic and is related to what the human beings are able to compute. An analysis of the Lyapunov exponents and the Kolmogorov-Sinai entropy shows how deterministic chaos, although with an epistemic character, is non subjective at all. This should clarify the role and content of stochastic models in the description of the physical world.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Cited by Dyson (2009).
- 2.
We shall see how determinism refers to ontic descriptions, while predictability (and, in some sense, chaos) has an epistemic nature.
- 3.
In brief, van Kampens argument is the following. Suppose the existence of a world A which is not deterministic and consider a second world B obtained from the first using the following deterministic rule: every event in B is the copy of an event occurred one million years earlier in A. Therefore, all the observers in B and their prototypes live the same experiences despite the different natures of the two worlds.
- 4.
Shannon (1948) showed that, once the probabilities \( P(C_{m} ) \) are known, the entropy (6) is the unique quantity which measures, under natural conditions, the surprise or information carried by \( \{ C_{m} \} \).
- 5.
The precise definition of mixing in dynamical systems requires several specifications and technicalities. To have an idea, imagine to put flour and sugar, in a given proportion (say 40 and 60 %, respectively) and initially separated, in a jar with a lid. After shaking the jar for a sufficiently long time, we expect the two components to be mixed, i.e., the probability to find flour or sugar in every part of the jar matches the initial proportion of the two components: a teaspoonful of the mixture taken at random will contain 40 % of flour and 60 % of sugar.
- 6.
A very broad definition of an ergodic system relies on the identification of time averages and averages computed with the invariant probability density (7). Said in other words, a system is ergodic if its trajectory in phase space, during its time evolution, visits (and densely explores) all the accessible regions of phase space, so that the time spent in each region is proportional to the invariant probability density assigned to that region. Therefore, if a system is ergodic, one can understand its statistical features looking at the time evolution for a sufficient long time; the conceptual and technical relevance of ergodicity is quite clear.
- 7.
There are several definitions of a Model, but to our purposes the following is a reasonable one: Given any system \( {\mathcal{S}} \), by which we mean a set of objects connected by certain relations, the system \( {\mathcal{M}} \) is said a model of \( {\mathcal{S}} \) if a correspondence can be established between the elements (and the relations) of \( {\mathcal{M}} \) and the elements (and the relations) of \( {\mathcal{S}} \), by means of which the study of \( {\mathcal{S}} \) is reduced to the study of \( {\mathcal{M}} \), within certain limitations to be specified or determined.
- 8.
Experiments are usually carried out under controlled conditions, meaning that every possible care is taken in order to exclude external influences and focus on specific aspects of the physical world. In his “Dialogues concerning two new sciences”, Galilei (English translation 1914) describes the special care to be taken in order to keep the accidents under control: “… I have attempted in the following manner to assure myself that the acceleration actually experienced by falling bodies is that above described. A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball …” (the italicized emphases are ours).
- 9.
The attractor of a dynamical system is a manifold in phase space toward which the system tends to evolve, regardless of the initial conditions. Once close enough to the attractor, the trajectory remains close to it even in the presence of small perturbations.
- 10.
Typically \( h(\epsilon ){ \sim }\epsilon^{ - \alpha } \) where the value of \( \alpha \) depends on the process under investigation (Cencini et al. 2009).
References
Atmanspacher, H.: Determinism is ontic, determinability is epistemic’. In: Atmanspacher, H., Bishop, R. (eds.) Between Chance and Choice. Imprint Academic, Thorverton (2002)
Boyd, R.: Determinism, laws and predictability in principle. Phylosophy Sci. 39, 43 (1972)
Campbell, L., Garnett, W.: The Life of James Clerk Maxwell. MacMillan and Co., London (1882)
Cencini, M., Cecconi, F., Vulpiani, A.: Chaos: From Simple Models to Complex Systems. World Scientific, Singapore (2009)
Chibbaro, S., Rondoni, L., Vulpiani, A.: Reductionism, Emergence and Levels of Reality. Springer-Verlag, Berlin (2014)
Dyson, F.: Birds and frogs. Not. AMS 56, 212 (2009)
Franceschelli, S.: Some remarks on the compatibility between determinism and unpredictability. Prog. Biophys. Mol. Biol. 110, 61 (2012)
Galilei, G.: Dialogues Concerning Two New Sciences. MacMillan, New York (1914)
Koertge, N.: Galileo and the problem of accidents. J. Hist Ideas 38, 389 (1977)
Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963)
Ma, S.K.: Statistical Mechanics. World Scientific, Singapore (1985)
Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505 (1953)
Pais, A.: Subtle is the Lord: The science and the life of Albert Einstein. Oxford University Press, Oxford (2005)
Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892)
Popper, K.R.: The open universe: An argument for indeterminism. From the Postscript to the Logic of Scientific Discovery. Routledge, London (1992)
Popper, K.R.: The Logic of Scientific Discovery. Routledge, London (2002)
Prigogine, I., Stengers, I.: Les Lois du Chaos. Flammarion, Paris (1994)
Primas, H.: Hidden determinism, probability, and times arrow. In: Bishop, R., Atmanspacher, H. (eds.) Between Chance and Choice, p. 89. Imprint Academic, Exeter (2002)
Shannon, C.E.: A note on the concept of entropy. Bell Syst. Tech. J. 27, 379 (1948)
Stone, M.A.: Chaos, prediction and Laplacean determinism. Am. Phylos. Q. 26, 123 (1989)
Takens, F.: Detecting strange attractors in turbulence. In: Rand, D., Young, L.S. (eds.) Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898. Springer-Verlag, New York (1981)
van Kampen, N.G.: Determinism and predictability. Synthese 89, 273 (1991)
von Plato, J.: Creating Modern Probability. Cambridge University Press, Cambridge (1994)
Vauclair, S.: Elémentes de Physique Statistique. Interéditions, Paris (1993)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Caprara, S., Vulpiani, A. (2016). Chaos and Stochastic Models in Physics: Ontic and Epistemic Aspects. In: Ippoliti, E., Sterpetti, F., Nickles, T. (eds) Models and Inferences in Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-28163-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-28163-6_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28162-9
Online ISBN: 978-3-319-28163-6
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)