Continuity, causality and determinism in mathematical physics : from the late 18th until the early 20th century

It is commonly thought that before the introduction of quantum mechanics, determinism was a straightforward consequence of the laws of mechanics. However, around the nineteenth century, many physicists, for various reasons, did not regard determinism as a provable feature of physics. This is not to say that physicists in this period were not committed to determinism; there were some physicists who argued for fundamental indeterminism, but most were committed to determinism in some sense. However, for them, determinism was often not a provable feature of physical theory, but rather an a priori principle or a methodological presupposition. Determinism was strongly connected with principles of causality and continuity and the principle of sufficient reason; this thesis examines the relevance of these principles in the history of physics. Moreover, the history of determinism in this period shows that there were essential changes in the relation between mathematics and physics: whereas in the eighteenth century, there were metaphysical arguments which lent support to differential calculus, by the early twentieth century the development of rigorous foundations of differential calculus led to concerns about its applicability in physics. The thesis consists of six papers. In the first paper, "On the origins and foundations of Laplacian determinism", I argue that Laplace, who is usually pointed out as the first major proponent of scientific determinism, did not derive his statement of determinism directly from the laws of mechanics; rather, his determinism has a background in eighteenth century Leibnizian metaphysics, and is ultimately based on the law of continuity and the principle of sufficient reason. These principles also provided a basis for the idea that one can find laws of nature in the form of differential equations which uniquely determine natural processes. In "The Norton dome and the nineteenth century foundations of determinism", I argue that an example of indeterminism in classical physics which has attracted attention in philosophy of physics in recent years, namely the Norton come, was already discussed during the nineteenth century. However, the significance which this type of indeterminism had back then is very different from the significance which the Norton dome currently has in philosophy of physics. This is explained by the fact that determinism was conceived of in an essentially different way: in particular, the nineteenth century authors who wrote about this type of indeterminism regarded determinism as an a priori principle rather than as a property of the equations of physics. In "Vital instability: life and free will in physics and physiology, 1860-1880", I show how Maxwell, Cournot, Stewart and Boussinesq used the possibility of unstable or indeterministic mechanical systems to argue that the will or a vital principle can intervene in organic processes without violating the laws of physics, so that a strictly dualist account of life and the mind is possible. Moreover, I show that their ideas can be understood as a reaction to the law of conservation of energy and to the way it was used in physiology to exclude vital and mental causes. In "The nineteenth century conflict between mechanism and irreversibility", I show that in the late nineteenth century, there was a widespread conflict between the aim of reducing physical processes to mechanics and the recognition that certain processes are irreversible. Whereas the so-called reversibility objection is known as an objection that was made to the kinetic theory of gases, it in fact appeared in a wide range of arguments, and was susceptible to very different interpretations. It was only when the project of reducing all of physics to mechanics lost favor, in the late nineteenth century, that the reversibility objection came to be used as an argument against mechanism and against the kinetic theory of gases. In "Continuity in nature and in mathematics: Boltzmann and Poincaré", I show that the development of rigorous foundations of differential calculus in the nineteenth century led to concerns about its applicability in physics: through this development, differential calculus was made independent of empirical and intuitive notions of continuity and was instead based on mathematical continuity conditions, and for Boltzmann and Poincaré, the applicability of differential calculus in physics depended on whether these continuity conditions could be given a foundation in intuition or experience. In the final paper, "Determinism around 1900", I briefly discuss the implications of the developments described in the previous two papers for the history of determinism in physics, through a discussion of determinism in Mach, Poincaré and Boltzmann. I show that neither of them regards determinism as a property of the laws of mechanics; rather, for them, determinism is a precondition for science, which can be verified to the extent that science is successful

The Norton Dome and the Nineteenth Century Foundations of Determinism

The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome

Why Bohm was never a determinist

Bohm’s interpretation of quantum mechanics has generally been understood as an attempt to restore the determinism of classical physics. However, although Bohm’s interpretation of quantum mechanics, as he initially proposed it in 1952, does indeed have the feature of being deterministic, for Bohm this was never the main point. In fact, in other texts which he published shortly before and after, as well as in correspondence from this period, he argued that the assumption that nature is deterministic is unjustified and should be abandoned. His aim was a different one: to develop an account of quantum mechanics that can be understood. Whereas it has been argued before that Bohm’s commitment to determinism can be linked to his interest in Marxism, I argue for the opposite: in Marxist philosophy, especially Friedrich Engels’ dialectical materialism, Bohm found resources for developing a notion of causality different from determinism. This non-deterministic causality is based on the idea of infinite complexity and an infinite number of levels of nature. From ca. 1954, Bohm’s conception of causality further weakened, as he developed the idea of a dialectical relation between causality and chance. From now on, Bohm argued that causality and chance are equally indispensable in science, and that wherever we find causality, we can assume that there is underlying chance and vice versa

Why Bohm was never a determinist

Bohm’s interpretation of quantum mechanics has generally been understood as an attempt to restore the determinism of classical physics. However, although Bohm’s interpretation of quantum mechanics, as he initially proposed it in 1952, does indeed have the feature of being deterministic, for Bohm this was never the main point. In fact, in other texts which he published shortly before and after, as well as in correspondence from this period, he argued that the assumption that nature is deterministic is unjustified and should be abandoned. His aim was a different one: to develop an account of quantum mechanics that can be understood. Whereas it has been argued before that Bohm’s commitment to determinism can be linked to his interest in Marxism, I argue for the opposite: in Marxist philosophy, especially Friedrich Engels’ dialectical materialism, Bohm found resources for developing a notion of causality different from determinism. This non-deterministic causality is based on the idea of infinite complexity and an infinite number of levels of nature. From ca. 1954, Bohm’s conception of causality further weakened, as he developed the idea of a dialectical relation between causality and chance. From now on, Bohm argued that causality and chance are equally indispensable in science, and that wherever we find causality, we can assume that there is underlying chance and vice versa

Why Bohm was never a determinist

Bohm’s interpretation of quantum mechanics has generally been received as an attempt to restore the determinism of classical physics. However, although this interpretation, as Bohm initially proposed it in 1952, does indeed have the feature of being deterministic, for Bohm this was never the main point. In fact, in other publications and in correspondence from this period, he argued that the assumption that nature is deterministic is unjustified and should be abandoned. Whereas it has been argued before that Bohm’s commitment to determinism was connected to his interest in Marxism, I argue for the opposite: Bohm found resources in Marxist philosophy for developing a nondeterministic notion of causality, which is based on the idea of infinite complexity and an infinite number of levels of nature. From ca. 1954 onwards, Bohm’s conception of causality further weakened, as he developed the idea of a dialectical relation between causality and chance

Vital instability: life and free will in physics and physiology, 1860-1880

During the period 1860-1880, a number of physicists and mathematicians, including Maxwell, Stewart, Cournot and Boussinesq, used theories formulated in terms of physics to argue that the mind, the soul or a vital principle could have an impact on the body. This paper shows that what was primarily at stake for these authors was a concern about the irreducibility of life and the mind to physics, and that their theories can be regarded primarily as reactions to the law of conservation of energy, which was used among others by Helmholtz and Du Bois-Reymond as an argument against the possibility of vital and mental causes in physiology. In light of this development, Maxwell, Stewart, Cournot and Boussinesq showed that it was still possible to argue for the irreducibility of life and the mind to physics, through an appeal to instability or indeterminism in physics: if the body is an unstable or physically indeterministic system, an immaterial principle can act through triggering or directing motions in the body, without violating the laws of physics

On the origins and foundations of Laplacian determinism

International audienceIn this paper I examine the foundations of Laplace's famous statement of determinism in 1814, and argue that rather than derived from his mechanics, this statement is based on general philosophical principles, namely the principle of sufficient reason and the law of continuity. It is usually supposed that Laplace's statement is based on the fact that each system in classical mechanics has an equation of motion which has a unique solution. But Laplace never proved this result, and in fact he could not have proven it, since it depends on a theorem about uniqueness of solutions to differential equations that was only developed later on. I show that the idea that is at the basis of Laplace's determinism was in fact widespread in enlightenment France, and is ultimately based on a re-interpretation of Leibnizian metaphysics, specifically the principle of sufficient reason and the law of continuity. Since the law of continuity also lies at the basis of the application of differential calculus in physics, one can say that Laplace's determinism and the idea that systems in physics can be described by differential equations with unique solutions have a common foundation