Abstract Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as “physically real” and classical mechanics, like quantum physics, is indeterministic.
Abstract Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as “physically real” and classical mechanics, like quantum physics, is indeterministic.
Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as ``physically real" and classical mechanics, like quantum physics, is indeterministic.
Gisin and Del Santo have argued, in a series of influential papers (2019–2023), that real numbers are the “hidden variables of classical mechanics,” on the grounds that a typical real number contains an infinite amount of information that a finite volume of space cannot store. We show that this argument rests on a fundamental confusion between a physical quantity and the number that results from its numerization. The infinite sequence of decimals of a real number is not a property of the world but the structural product of the measurement procedure: the iterated application of a standard to an independent magnitude. The generic incommensurability of any two magnitudes of the same kind suffices to generate this infinite sequence. We identify and name three nested confusions: the word–thing confusion (the word “magnitude” taken for the real quantity), the quantity–number confusion (the real quantity taken for its numerical value), and their product, the numerization confusion, through which the real disappears from both sides at once. We show that these confusions symmetrically vitiate both classical determinism and Gisin’s informational indeterminism, and sketch the consequences for the determinism/indeterminism debate.
Abstract It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.
The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.
It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers.However, almost all real numbers contain an infinite amount of information.I argue that a finite volume of space can't contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant.Moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random.I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers.This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory.Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic.Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.
Abstract Del Santo and Gisin have recently argued that classical mechanics exhibits indeterminacy and that by treating the observables of classical mechanics with real number precision we introduce hidden variables that restore determinacy. In this article we introduce the conceptual machinery required to critically evaluate these claims. We present a characterization of indeterminacy which can capture both quantum indeterminacy and the classical indeterminacy Del Santo and Gisin propose. This allows us to show that there is an important difference between the two: their classical indeterminacy can be resolved with hidden variables in a manner which is not possible for quantum indeterminacy.