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Perhaps, the most insightful analysis (possibly to this day) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equations that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to Popular Science Monthly (1882):"It then reaches the apex with a velocity of zero, and remains there till it pleases some directing principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again".

Boussinesq saw these singular "bifurcations" (his word) as creating gaps in causal chains, and suggesting an additional "directing principle" operating in living organisms. This answered the objection, raised earlier by Helmholtz, Du Bois-Reymond and others against such a principle, that conservation of energy precludes its operation. Maxwell praised the idea in a letter to Galton (1879):"it may at any instant, at its own sweet will, without exerting any force or spending any energy, go off along that one of the particular paths which happens to coincide with the actual condition of the system at that instant. In most of the former methods... there was a certain small but finite amount of... trigger-work for the Will to do. Boussinesq has managed to reduce this to mathematical zero... I think Boussinesq’s method is a very powerful one against metaphysical arguments about cause and effect and much better than the insinuation that there is something loose about the laws of nature..."

It was against this backdrop that Boussinesq advanced his idea. Earlier free will issues were in the domain of philosophers, who largely assumed, along with the common public, that mechanical laws did not apply to the mental. Bertrand, in his 1878 critique of Boussinesq gave a mathematician's version of this position (which incidentally rejects Laplace's extrapolation): "the results of equations could not attain absolute precision”, and “the certainty of equations cannot be greater than the certainty of principles from which they stem”, so it is naive to expect that solutions will be "loyally followed" even along regular segments. Bertrand's criticism was off the mark, Boussinesq did not suggest that mechanical description was precise, he explicitly described bifurcations as only structural toy models for what might be happening in the living organisms, but toy models with the benefit of mathematical precision. In 1880 Du Bois-Reymond acknowledged Boussinesq, and his precursors Saint-Venant and Cournot, but deemed their position unsatisfactory. Soon attention turned to new physics, and Boussinesq's programme of exploring the nature of "directing principle" scientifically fell into oblivion until recent times. Although some of his ideas about the role of causal gaps and directing principle in the mind-body problem were re-explored since 1950s by Heisenberg and Wigner (later Penrose and others) with quantum mechanics in place of classical.

Perhaps, the most insightful analysis (possibly to this day) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equations that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to Popular Science Monthly (1882):"It then reaches the apex with a velocity of zero, and remains there till it pleases some guiding principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again".

Boussinesq saw these singular "bifurcations" (his word) as creating gaps in causal chains, and suggesting an additional "guiding principle" operating in living organisms. This answered the objection, raised earlier by Helmholtz, Du Bois-Reymond and others against such a principle, that conservation of energy precludes its operation. Maxwell praised the idea in a letter to Galton (1879):"it may at any instant, at its own sweet will, without exerting any force or spending any energy, go off along that one of the particular paths which happens to coincide with the actual condition of the system at that instant. In most of the former methods... there was a certain small but finite amount of... trigger-work for the Will to do. Boussinesq has managed to reduce this to mathematical zero... I think Boussinesq’s method is a very powerful one against metaphysical arguments about cause and effect and much better than the insinuation that there is something loose about the laws of nature..."

It was against this backdrop that Boussinesq advanced his idea. Earlier free will issues were in the domain of philosophers, who largely assumed, along with the common public, that mechanical laws did not apply to the mental. Bertrand, in his 1878 critique of Boussinesq gave a mathematician's version of this position (which incidentally rejects Laplace's extrapolation): "the results of equations could not attain absolute precision”, and “the certainty of equations cannot be greater than the certainty of principles from which they stem”, so it is naive to expect that solutions will be "loyally followed" even along regular segments. Bertrand's criticism was off the mark, Boussinesq did not suggest that mechanical description was precise, he explicitly described bifurcations as only structural toy models for what might be happening in the living organisms, but toy models with the benefit of mathematical precision. In 1880 Du Bois-Reymond acknowledged Boussinesq, and his precursors Saint-Venant and Cournot, but deemed their position unsatisfactory. Soon attention turned to new physics, and Boussinesq's programme of exploring the nature of "guiding principle" scientifically fell into oblivion until recent times. Although some of his ideas about the role of causal gaps and the guiding principle in the mind-body problem were re-explored since 1950s by Heisenberg and Wigner (later Penrose and others) with quantum mechanics in place of classical.

Perhaps, the most insightful analysis (possibly to date) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equations that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to Popular Science Monthly (1882):"It then reaches the apex with a velocity of zero, and remains there till it pleases some directing principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again".

Perhaps, the most insightful analysis (possibly to this day) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equations that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to Popular Science Monthly (1882):"It then reaches the apex with a velocity of zero, and remains there till it pleases some directing principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again".

Perhaps, the most insightful analysis (possibly to date) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equation that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to Popular Science Monthly (1882):"It then reaches the apex with a velocity of zero, and remains there till it pleases some directing principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again".

Boussinesq saw these singular "bifurcations" (his word) as creating gaps in causal chains, and suggesting an additional "guiding principle" operating in living organisms. This answered the objection, raised earlier by Helmholtz, Du Bois-Reymond and others against such a principle, that conservation of energy precludes its operation. Maxwell praised the idea in a letter to Galton (1879):"it may at any instant, at its own sweet will, without exerting any force or spending any energy, go off along that one of the particular paths which happens to coincide with the actual condition of the system at that instant. In most of the former methods... there was a certain small but finite amount of... trigger-work for the Will to do. Boussinesq has managed to reduce this to mathematical zero... I think Boussinesq’s method is a very powerful one against metaphysical arguments about cause and effect and much better than the insinuation that there is something loose about the laws of nature..."

To put this into perspective note that non-uniqueness of solutions to equations of motion is neither necessary nor sufficient for indeterminism. When two identical rigid balls collide moving along a line, conservation of energy and momentum are not enough to determine their motion after the collision, unless we also assume that they remain on that line. That extra assumptions are needed to resolve collisions was known even before Newton to Huygens, Wren and Wallis, but it was seen as reflection of the model being too idealized and not fully describing the physics involved, rather than indeterminism. Poisson took it the same way when he considered medium resistance laws of the form $\dot{v}=-av^n$, and discovered that for $0<n<1$ there are non-trivial solutions when the initial velocity is zero (in modern terms, because $v^n$ in non-Lipschitz). This is the viewpoint reflected in the 19th century textbooks of Duhamel and Cournot.

The term determinism did not even come into common usage until 1860s (Kant mentions it as unwelcome neologism in 1893, and so does Mill in 1865), and Laplace's famous quote was likely just a flowery metaphor. Only in 1865 Bernard in his influential 1865 Introduction to the Study of Experimental Medicine publicized the word "determinism" as reducing "the features of living beings to physical-chemical features”, but implied that the reduction may not be complete. And only in 1872 Du Bois-Reymond fully articulated modern (mechanical) determinism of “reducing all transformations taking place in the material world to atomic motions” ruled by “mechanical necessity”, turning Laplace's metaphor into a doctrine. According to Bordoni, "it seems reasonable to think that the mythology of Laplacian determinism was a late reconstruction, and the physiologist Emile Du Bois-Reymond played an important role in the emergence of that mythology". The subsequent philosophical discussions involved Renouvier, Peirce, Boutroux and James. James is usually credited with the first two-stage model of free will, outlined in his famous Dilemma of Determinism (1884), but Boussinesq's 1878 model is also two-stage, although his determined/free stages are reversed compared to James's.

It was against this backdrop that Boussinesq advanced his idea. Earlier free will issues were in the domain of philosophers, who largely assumed, along with the common public, that mechanical laws did not apply to the mental. Bertrand, in his 1878 critique of Boussinesq gave a mathematician's version of this position (which incidentally rejects Laplace's extrapolation): "the results of equations could not attain absolute precision”, and “the certainty of equations cannot be greater than the certainty of principles from which they stem”, so it is naive to expect that solutions will be "loyally followed" even along regular segments. Bertrand's criticism was off the mark, Boussinesq did not suggest that mechanical description was precise, he explicitly described bifurcations as only structural toy models for what might be happening in the living organisms, but toy models with the benefit of mathematical precision. In 1880 Du Bois-Reymond acknowledged Boussinesq, and his precursors Saint-Venant and Cournot, but deemed their position unsatisfactory. Soon attention turned to new physics, and Boussinesq's programme of exploring the nature of "guiding principles" scientifically fell into oblivion until recent times. Although some of his ideas about the role of causal gaps and guiding principles in the mind-body problem were re-explored since 1960s by Heisenberg and Wigner (later Penrose and others) with quantum mechanics in place of classical.

Perhaps, the most insightful analysis (possibly to date) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equations that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to Popular Science Monthly (1882):"It then reaches the apex with a velocity of zero, and remains there till it pleases some directing principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again".

Boussinesq saw these singular "bifurcations" (his word) as creating gaps in causal chains, and suggesting an additional "directing principle" operating in living organisms. This answered the objection, raised earlier by Helmholtz, Du Bois-Reymond and others against such a principle, that conservation of energy precludes its operation. Maxwell praised the idea in a letter to Galton (1879):"it may at any instant, at its own sweet will, without exerting any force or spending any energy, go off along that one of the particular paths which happens to coincide with the actual condition of the system at that instant. In most of the former methods... there was a certain small but finite amount of... trigger-work for the Will to do. Boussinesq has managed to reduce this to mathematical zero... I think Boussinesq’s method is a very powerful one against metaphysical arguments about cause and effect and much better than the insinuation that there is something loose about the laws of nature..."

To put this into perspective note that non-uniqueness of solutions to equations of motion is neither necessary nor sufficient for indeterminism. When two identical rigid balls collide moving along a line, conservation of energy and momentum are not enough to determine their motion after the collision, unless we also assume that they remain on the same line. That extra assumptions are needed to resolve collisions was known even before Newton to Huygens, Wren and Wallis, but it was seen as reflection of the model being too idealized and not fully describing the physics involved, rather than indeterminism. Poisson took it the same way in 1806, when he considered medium resistance laws of the form $m\dot{v}=-av^n$, and discovered that for $0<n<1$ there are non-trivial solutions when the initial velocity is zero (in modern terms, because $v^n$ in non-Lipschitz). This is the viewpoint reflected in the 19th century textbooks of Duhamel and Cournot.

The term determinism did not even come into common usage until 1860s (Kant mentions it as unwelcome neologism in 1793, and so does Mill in 1865), and Laplace's famous quote was likely little more than a flowery metaphor. Only in 1865 Bernard in his influential 1865 Introduction to the Study of Experimental Medicine publicized the word "determinism" as reducing "the features of living beings to physical-chemical features”, but implied that the reduction may not be complete. And only in 1872 Du Bois-Reymond fully articulated modern (mechanical) determinism of “reducing all transformations taking place in the material world to atomic motions” ruled by “mechanical necessity”, turning Laplace's metaphor into a doctrine. According to Bordoni, "it seems reasonable to think that the mythology of Laplacian determinism was a late reconstruction, and the physiologist Emile Du Bois-Reymond played an important role in the emergence of that mythology". The subsequent philosophical discussions involved Renouvier, Peirce, Boutroux and James. James is usually credited with the first two-stage model of free will, outlined in his famous Dilemma of Determinism (1884), but Boussinesq's 1878 model is also two-stage, although his determined/free stages are reversed compared to James's.

It was against this backdrop that Boussinesq advanced his idea. Earlier free will issues were in the domain of philosophers, who largely assumed, along with the common public, that mechanical laws did not apply to the mental. Bertrand, in his 1878 critique of Boussinesq gave a mathematician's version of this position (which incidentally rejects Laplace's extrapolation): "the results of equations could not attain absolute precision”, and “the certainty of equations cannot be greater than the certainty of principles from which they stem”, so it is naive to expect that solutions will be "loyally followed" even along regular segments. Bertrand's criticism was off the mark, Boussinesq did not suggest that mechanical description was precise, he explicitly described bifurcations as only structural toy models for what might be happening in the living organisms, but toy models with the benefit of mathematical precision. In 1880 Du Bois-Reymond acknowledged Boussinesq, and his precursors Saint-Venant and Cournot, but deemed their position unsatisfactory. Soon attention turned to new physics, and Boussinesq's programme of exploring the nature of "directing principle" scientifically fell into oblivion until recent times. Although some of his ideas about the role of causal gaps and directing principle in the mind-body problem were re-explored since 1950s by Heisenberg and Wigner (later Penrose and others) with quantum mechanics in place of classical.