Abstract
It can be shown that certain kinds of classical deterministic and indeterministic descriptions are observationally equivalent. Then the question arises: which description is preferable relative to evidence? This paper looks at the main argument in the literature for the deterministic description by Winnie (The cosmos of science—essays of exploration. Pittsburgh University Press, Pittsburgh, pp 299–324, 1998). It is shown that this argument yields the desired conclusion relative to in principle possible observations where there are no limits, in principle, on observational accuracy. Yet relative to the currently possible observations (of relevance in practice), relative to the actual observations, or relative to in principle observations where there are limits, in principle, on observational accuracy the argument fails. Then Winnie’s analogy between his argument for the deterministic description and his argument against the prevalence of Bernoulli randomness in deterministic descriptions is considered. It is argued that while the arguments are indeed analogous, it is also important to see they are disanalogous in another sense.
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Notes
There are various possibilities of interpreting this probability measure. For instance, according to the popular time-average interpretation, the probability of A is defined as the (long-run) proportion of time that a solution spends in A (see Werndl 2009b).
Here the paper follows the extant literature Suppes (1993), Suppes and de Barros (1996) and Winnie (1998) in assuming that measure-theoretic deterministic and stochastic descriptions are tested and confirmed by deriving probabilistic predictions from them. That nonlinear models can be confirmed has been questioned (see Bishop 2008). This is just to mention the controversy about confirmation of nonlinear models; a thorough treatment would require another paper.
The baker’s transformation involves strong idealisations—in particular, that there is no friction. Consequently, scientists do not derive probabilistic predictions from it to compare it with the data. Because the example is so easy to understand, instead of referring to another example, the reader is asked to pretend that scientists indeed use it for this purpose.
This is not hard to see. As shown in Werndl (2009a, Section 4.1), the baker’s transformation (X, f t , P) is isomorphic via ϕ to the Bernoulli shift (Y, h t , Q) with values 0 and 1 (the states are bi-infinite sequences \(y\!=\!\ldots y_{-2}y_{-1}y_{0}y_{1}y_{2}\ldots, y_{i}\in\{0,1\}\)). For (Y, h t , Q) define the observation function \(\Upxi_{16}\) as follows. It takes the value \((\frac{1}{8},\frac{1}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0000, (\frac{3}{8},\frac{1}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0001, (\frac{5}{8},\frac{1}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0010, (\frac{7}{8},\frac{1}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0011, (\frac{1}{8},\frac{3}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1000, (\frac{3}{8},\frac{3}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1001, (\frac{5}{8},\frac{3}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1010, (\frac{7}{8},\frac{3}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1011, (\frac{1}{8},\frac{5}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0100, (\frac{3}{8},\frac{5}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0101, (\frac{5}{8},\frac{5}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0110, (\frac{7}{8},\frac{5}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!0111, (\frac{1}{8},\frac{7}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1100, (\frac{3}{8},\frac{7}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1101, (\frac{5}{8},\frac{7}{8})\) if \(y_{-1}y_{0}y_{1}y_{2}\!=\!1110, (\frac{7}{8},\frac{7}{8})\) if y −1 y 0 y 1 y 2 = 1111. Since (Y, h t , Q) is a Bernoulli shift, \(\{\Upxi_{16}(h_{t})\}\) is a Markov description, which is easily seen to be identical to {W t }. Because (X, f t , P) is isomorphic via ϕ to (Y, h t , Q) and \(\Upphi_{16}(x)=\Upxi_{16}(\phi(x)), \{\Upphi_{16}(f_{t})\}\!=\!\{\Upxi_{16}(\phi(f_{t}))\}\!=\!\{\Upxi_{16}(h_{t}(\phi)\}; \) thus \(\{\Upphi_{16}(f_{t})\}\) is also identical to {W t }.
A nontrivial finite-valued observation function has at least two outcomes with positive probability.
Theorem 1 is equivalent to Proposition 1 in Werndl (2009a), but there are two minor differences in the statement of the theorems. First, unlike Werndl (2009a), it is not required that (X, f t , P) is ergodic because this follows automatically from the premises (ergodicity is equivalent to the condition that there exists no \(G\subseteq X, 0<P(G)<1, \) such that f 1(G) = G; cf. Petersen 1983, 42). Second, it is stated here that 0 < P(Z t+n = e h given Z t = e l ) < 1 for any arbitrary \({n\in\mathbb{{N}}}\) and not only for n = 1 as in Werndl (2009a). This stronger conclusion follows immediately from the weaker one because if the premises hold for (X, f t , P), they also hold for (X, f n t , P) where f n t is the n-th iterate of f t .
There are other results in the literature which are based on quite different assumptions but which could also be interpreted as results about observational equivalence between deterministic and stochastic descriptions. An example is Judd and Smith (2001, 2004); they present results under the ideal condition of infinite past observations of a deterministic system and the assumption that there is observational error modeled by a probability distribution around the true value. While these results are interesting, this paper will not discuss them. They are based on different assumptions and hence are of limited relevance for assessing the extant philosophical literature of concern here (i.e., Suppes 1993, Suppes and de Barros 1996, Winnie 1998).
Still, the nesting argument seems a bit complicated because the reason why there is no underdetermination is simply that only the deterministic description agrees with the in principle possible observations.
For in principle possible observations where there are no limits, in principle, on observational accuracy, the nesting argument succeeds in giving sufficient conditions under which the deterministic description is preferable. Here it is also true that if the premises of the nesting argument are not true, a stochastic description will be preferable. Then there is an observation function \(\Upphi\) such that the values corresponding to \(\Upphi\) can be observed, but where observations show that there are no other states apart from these values. Hence only the stochastic description \(\{Z_{t}\}=\{\Upphi(f_{t})\}\) agrees with the observations and is thus preferable (cf. Sect. 4).
Namely, A confirms A. Given any arbitrary statement B, both A and B are derivable from A∧ B. From this it follows that A confirms B (see Okasha 1998).
For example, consider the choice between a stochastic description in statistical mechanics of the evolution of the macrostates of a gas and the deterministic representation (see Werndl 2009a) of this stochastic description. Then, at the level of reality of the macrostates, the stochastic description can be preferable because of indirect evidence.
There are observation functions which lead to Bernoulli descriptions with equal probabilities (fair dice), and also observation functions which lead to Bernoulli description with non-equal probabilities (biased dice).
It is questionable that the literature which Winnie criticises really claims this. But we can set this issue aside because it will not be relevant in what follows.
This point, Winnie argues, also underlies a wrong interpretation of Brudno’s theorem. There is no need to go here into Brudno’s theorem because the basic conceptual point is exactly the same: Brudno’s theorem does not tell us that all coarse-grainings lead to Bernoulli random sequences; all it tells us is that there are some coarse-grainings which lead to Bernoulli random sequences.
The basic idea of Werndl’s (2009c) proof is as follows. By assumption, there is an observation function \(\Upphi:X\rightarrow Q\) of the deterministic description such that \(\{\Upphi(f_{t})\}\) is no Bernoulli description. Now suppose that there exists an observation function \(\Uppsi:X\rightarrow O\) finer than \(\Upphi\) such that \(\{\Uppsi(f_{t})\}\) is a Bernoulli description. Because \(\Uppsi\) is finer than \(\Upphi, \) there is a function \(\Upgamma:O\rightarrow Q\) such that \(\Upphi=\Upgamma(\Uppsi). \) But it is not hard to see that if \(\{\Uppsi(f_{t})\}\) is a Bernoulli description, then \(\{\Upphi(f_{t})\}=\{\Upgamma(\Uppsi(f_{t}))\}\) is also a Bernoulli description. But this contradicts the assumption that \(\{\Upphi(f_{t})\}\) is no Bernoulli description.
The text in square brackets has been replaced by the terminology used in this paper. I will use this convention throughout.
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Werndl, C. Evidence for the Deterministic or the Indeterministic Description? A Critique of the Literature About Classical Dynamical Systems. J Gen Philos Sci 43, 295–312 (2012). https://doi.org/10.1007/s10838-012-9199-8
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DOI: https://doi.org/10.1007/s10838-012-9199-8