Abstract
Recently some results have been presented which show that certain kinds of deterministic descriptions and indeterministic descriptions are observationally equivalent (Werndl 2009a, 2011). These results prompt interesting philosophical questions, such as what exactly they show or whether the deterministic or indeterministic description is preferable. There is hardly any philosophical discussion about these questions, and this paper contributes to filling this gap. More specifically, first, I discuss the philosophical comments made by mathematicians about observational equivalence, in particular Ornstein and Weiss (1991). Their comments are vague, and I argue that, according to a reasonable interpretation, they are misguided. Second, the results on observational equivalence raise the question of whether the deterministic or indeterministic description is preferable relative to evidence. If the deterministic and indeterministic description are equally well supported by evidence, there is underdetermination. I criticise Winnie’s (1998) argument that, by appealing to different observations, one finds that the deterministic description is preferable. In particular, I clarify a confusion in this argument. Furthermore, I show that the argument delivers the desired conclusion relative to in principle possible observations, but that the argument fails relative to currently possible observations.
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.
For discrete-time descriptions, see Werndl (2009a).
- 2.
There are various interpretations of this probability measure. For instance, according to the time-average interpretation, the probability of A is the long-run average of the fraction of time a solution spends in A (Werndl 2009b).
- 3.
The text in braces is in a footnote.
- 4.
If the descriptions are about different levels of reality, the situation seems different. For instance, in certain cases one might argue that at one level of reality the deterministic description, and at another level of reality the stochastic description is preferable.
- 5.
To give an example, on \(X=[0,1] \times [0,1]\) consider:
\(\textrm{T}((x,y)) = (2x,y/2)\) if \(0 \le x < 1/2\), \((2x - 1,(y + 1)/2)\) if \(1/2 \le x \le 1\).
For the Lebesgue probability measure p one obtains the discrete-time deterministic description \((X,\;{\textrm{T}}_{\textrm{t}} ,\;{\textrm{p}})\), called the baker’s transformation. Consider \(\Phi ((x,y)) = {\textrm{o}}_1 \chi _{{{\upalpha }}1} ((x,y)) + {\textrm{o}}_2 \chi _{{{\upalpha 2}}} ((x,y))\), where \({{\upalpha 1}} = [0,1] \times [0,1/2]\), \({{\upalpha 2}} = [0,1] \times [1/2,1]\) and \(\Psi ((x,y)) = {\textrm{q}}_1 \chi _{{{\upbeta 1}}} ((x,y)) + {\textrm{q}}_2 \chi _{{{\upbeta }}2} ((x,y)) + {\textrm{q}}_3 \chi _{{{\upbeta 3}}} ((x,y)) + {\textrm{q}}_4 \chi _{{{\upbeta 4}}} ((x,y))\), where \({{\upbeta 1}} = [0,1/2] \times [0,1/2]\), \({{\upbeta }}2 = [1/2,1] \times [0,1/2]\), \({{\upbeta }}3 = [0,1/2] \times [1/2,1]\), \({{\upbeta 4}} = [1/2,1] \times [1/2,1]\) (\(\chi _A (z):\,\, = 1\) for \(z \in A\); 0 otherwise). Clearly, if one observes q1 (with Ψ), the probability that one next observes o1 (with Φ) is 1; if one observes q2, the probability that one next observes o2 is 1; if one observes q3, the probability that one next observes o1 is 1; and if one observes q4, the probability that one next observes o2 is 1.
- 6.
For example, consider the baker’s transformation \((X,{\textrm{T}}_{\textrm{t}} ,{\textrm{p}})\). Let \(\Psi ((x,y)) = {\textrm{q}}_1 \chi _{{{\upbeta 1}}} ((x,y)) + {\textrm{q}}_2 {{\upchi }}_{{{\upbeta 2}}} ((x,y)) + {\textrm{q}}_3 {{\upchi }}_{{{\upbeta 3}}} ((x,y)) + {\textrm{q}}_4 {{\upchi }}_{{{\upbeta 4}}} ((x,y))\) be as in the previous footnote and let \(\Phi ((x,y)) = {\textrm{o}}_1 {{\upchi }}_{{{\upgamma 1}}} ((x,y)) + {\textrm{o}}_2 {{\upchi }}_{{{\upgamma 2}}} ((x,y))\), \({{\upgamma 1}} = [0,1/2] \times [0,1]\), \({{\upgamma 2}} = [1/2,1] \times [0,1]\). Clearly, for all i, \(1 \le {\textrm{i}} \le {\textrm{4}}\), and all j, \(1 \le {\textrm{j}} \le {\textrm{2}}\), the probability that qi is followed by oj 1/2. Still Φ is coarser than Ψ, and for the observation Ψ at the finer level one obtains a stochastic description at a smaller scale.
- 7.
Statement A confirms itself; A is derivable from A&B (B is any statement); B is derivable from A&B. Hence, A confirms B.
References
Chernov, Nikolai, and Roberto Markarian. 2006. Chaotic billiards. Providence: American Mathematical Society.
Janssen, Jacques, and Nikolaos Limnios. 1999. Semi-Markov models and applications. Dordrecht: Kluwer.
Laudan, Larry. 1995, Damn the consequences! Proceedings and Addresses of the American Philosophical Association 69: 27–34.
Laudan, Larry, and Jarrett Leplin. 1991. Empirical equivalence and underdetermination. The Journal of Philosophy 88: 449–472.
Luzzatto, Stefano, Ian Melbourne, and Frederic Paccaut. 2005. The Lorenz attractor is mixing. Communications in Mathematical Physics 260: 393–401.
Okasha, Samir. 2002. Underdetermination, holism and the theory/data distinction. The Philosophical Quarterly 208: 303–319.
Ornstein, D. 1970. Imbedding Bernoulli shifts in flows. In Contributions to ergodic theory and probability, eds. Albrecht Dold and Beno Eckmann, 178–218. Berlin: Springer.
Ornstein, Dan, and Benjamin Weiss. 1991. Statistical properties of chaotic systems. Bulletin of the American Mathematical Society 24: 11–116.
Simanyi, Nandor. 1999. Ergodicity of hard spheres in a box. Ergodic Theory and Dynamical Systems 19: 741–766.
Simanyi, Nandor. 2003. Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems. Inventiones Mathematicae 154: 123–178.
Suppes, Patrick. 1993. The transcendental character of determinism. Midwest Studies in Philosophy 18: 242–257.
Werndl, Charlotte. 2009a. Are deterministic descriptions and indeterministic descriptions observationally equivalent? Studies in History and Philosophy of Modern Physics 40: 232–242.
Werndl, Charlotte. 2009b. What are the new implications of chaos for unpredictability? The British Journal for the Philosophy of Science 60: 195–220.
Werndl, Charlotte. 2011a. On the observational equivalence of continuous-time deterministic and indeterministic descriptions. European Journal for the Philosophy of Science 1(2): 193–225.
Werndl, Charlotte. 2011b. On choosing between deterministic and indeterministic models: Underdetermination and Indirect Evidence. Synthese. doi: 10.1007/S1122901199669.
Winnie, J. 1998. Deterministic chaos and the nature of chance. In The cosmos of science – essays of exploration, eds. John Earman and John Norton, 299–324. Pittsburgh, PA: Pittsburgh University Press.
Acknowledgements
I am indebted to Jeremy Butterfield for valuable suggestions. I am grateful for comments to Franz Huber, James Ladyman, Miklos Redei, Jos Uffink, two anonymous referees, and the audiences at the Oxford Philosophy of Physics Research Seminar, the Bristol Philosophy of Science Research Seminar, and the EPSA conference 2009. This research has been supported by a Junior Research Fellowship from the Queen’s College, Oxford University.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media B.V.
About this paper
Cite this paper
Werndl, C. (2012). Observational Equivalence of Deterministic and Indeterministic Descriptions and the Role of Different Observations. In: de Regt, H., Hartmann, S., Okasha, S. (eds) EPSA Philosophy of Science: Amsterdam 2009. The European Philosophy of Science Association Proceedings, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2404-4_35
Download citation
DOI: https://doi.org/10.1007/978-94-007-2404-4_35
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2403-7
Online ISBN: 978-94-007-2404-4
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)