# On the observational equivalence of continuous-time deterministic and indeterministic descriptions

## Abstract

This paper presents and philosophically assesses three types of results on the observational equivalence of continuous-time measure-theoretic deterministic and indeterministic descriptions. The first results establish observational equivalence to abstract mathematical descriptions. The second results are stronger because they show observational equivalence between deterministic and indeterministic descriptions found in science. Here I also discuss Kolmogorov’s contribution. For the third results I introduce two new meanings of ‘observational equivalence at every observation level’. Then I show the even stronger result of observational equivalence at every (and not just some) observation level between deterministic and indeterministic descriptions found in science. These results imply the following. Suppose one wants to find out whether a phenomenon is best modeled as deterministic or indeterministic. Then one cannot appeal to differences in the probability distributions of deterministic and indeterministic descriptions found in science to argue that one of the descriptions is preferable because there is no such difference. Finally, I criticise the extant claims of philosophers and mathematicians on observational equivalence.

## Keywords

Observational equivalence Determinism Indeterminism Continuous-time descriptions Classical physics Stochastic processes Ergodic theory## Notes

### Acknowledgements

I am indebted to Jeremy Butterfield and Jos Uffink for valuable suggestions and stimulating discussion on previous versions of this manuscript. For useful comments I also want to thank Franz Huber, James Ladyman, Miklós Rédei, and the audiences at the Philosophy of Physics Seminar, Cambridge University, the Philosophy of Physics Research Seminar, Oxford University, and the Philosophy and History of Science Seminar, Bristol University. Many thanks also to two anonymous referees and the editor Carl Hoefer for valuable suggestions. I am grateful to the Queen’s College, Oxford University, for supporting me with a junior research fellowship.

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