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Nicolas Gisin
University of Geneva, CH

Non-determinism in Newtonian mechanics and the classical “measurement” problem

Starting from the hypothesis that a finite volume of space can’t hold more than a finite amount of information, I argue that the mathematical real numbers are physically unreal. Indeed, almost all so-called real numbers contain an infinite amount of information, like, e.g., the answers to all questions one may formulate in any human language. Moreover, all real numbers, except a countable subset, are incomputable in the sense that their digits are random. Hence, a better name for them is random numbers. This name illustrates the fact that determinism can’t be based on the use of “real” numbers to represent initial conditions. Hence, Newtonian classical mechanics is not deterministic, contrary to standard claims and beliefs, except for stable systems like harmonic oscillators. However, the use of the mathematical real numbers is undoubtedly very useful as an idealization to allow for, e.g., differential equations. But one should not make the confusion of believing that this idealization implies that nature herself is deterministic: A deterministic theoretical model of physics doesn’t imply that nature is deterministic.

Consequently, in most physical dynamical systems, i.e. in chaotic systems, the initial conditions are random: after some determined initial digits, the next digits are undetermined (they don’t have any ontological existence). Pretty soon, these random digits drive the system. This raises the question as to when the undetermined digits get actualized, i.e. get determined. This is the classical analog of the well-known quantum measurement problem. I argue that such a problem arises in all non-deterministic models.

The non-determinism of classical physics, as well as the non-determinism of quantum physics, imply that time really passes (arXiv:1602.01497). Einstein identified time with classical clocks, i.e. with classical harmonic oscillators. This, as well as the quantum unitary evolution, leads to what I call the boring time, the time when nothing truly new happens, the time when things merely are, time when what matters is being, i.e. Parmenides-time. But intrinsic indeterminism implies that there is another time, at least as real as Parmenides boring time, which I like to call creative time, or Heraclitus-time, when what matters is change.
 
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