What do free will and mathematics have in common? One may consider that no serious link can be done between a language, mathematics, and a metaphysical doctrine, determinism.
Nevertheless, such a question became a prominent one in the xix century, and concerned economists, social scientists, mathematician and physicists.
The common ground of all those debates was a vision of determinism as a consequence of a mechanistic view of nature that was prominent between physicists and was imported into the human sciences through the use of statistic by Adolphe Quetelet.
Quetelet, an astronomer, considered statistical regularities as the expression of hidden social laws, and was forced to explain the compatibility of those laws with free will. His widespread influence on the human sciences may explain Léon Walras’s rejection of mathematical economics by Pierre Emile Levasseur.
In order to save freedom of the will, Quetelet had to deny any link between the macroscopic regularities of the averages and individual behaviour. Walras’s general equilibrium, with his stress on the individual agent was understood as a reestablishment of this same link and thus rejected by Levasseur.
At the same time, Quetelet exerted a major influence on Maxwel’s kinetic theory of gases. Inspired by Quetelet views on statistics, Maxwell developed a kinetic theory of gases that treated atoms in the same way that Quetelet considered individuals. Maxwell was also concerned with the question of the compatibility between the determinism of physical laws and free will. Inspired by Quetelet answer on the independence between individual behaviour and statistical regularities, he showed that there are no reasons to consider atoms as deterministic systems in view of the macroscopic regularities of gas laws.
Still lacking a convincing source of indeterminism for the atomic behaviour, he became aware of a strange mathematical results by Joseph Boussinesq. Boussinesq, a French mathematician, had found a class of differential equations with more than one solution. Those equations, that describe very special kind of unstable equilibria, were interpreted by Boussinesq as the final proof that Laplacian determinism was a mistake. The debate that followed Boussinesq mémoire mixed very different kind of arguments, some of them of a physical nature, some indebted with Quetelet, and others of a speculative, philosophical nature.
The historical roots that we have reconstructed explain this unusual entanglement of very heterogeneous perspectives, spreading over different disciplines such as statistics, physics, mathematics and the social sciences, and helps us in understanding what free will and mathematics had in common.