344
M. Svrˇ cek
Although Norton disproves very carefully the above mentioned objections, a curious situation arises: On one hand Norton is correct in his arguments against his
opponents, but on the other his opponents are right about the determinism of classical physics. Malament [122] has expressed this quandary in a very elegant way: “But
I am not sure that the full complexity and interest of the breakdown is adequately
captured by saying, either, “Newtonian particle mechanics is an indeterministic theory” (full stop) or “Norton’s example is not a well-defined Newtonian system” (full
stop). Indeed, I am not convinced we have clearly posed alternatives here—because
we do not have a sufficiently clear idea in the first place what should count as a “Newtonian system” (or count as falling within the “domain of application” of Newtonian
theory). My inclination is to avoid labels here and direct attention, instead, to a rich
set issues that the example raises.”
In order to gain a deeper insight into this problem one should look for analogies
such as the Norton dome. A historical review of comparable examples was written
by van Strien. In her article “The Norton Dome and the Nineteenth Century Foundations of Determinism” [123] she says: “Lipschitz-indeterministic systems played
an important role in the work of Boussinesq, who, in 1878, discussed several such
systems, one of which was similar to the Norton dome [124].” This paper contains
also an observation of an unexpected importance: “Though determinism as a general metaphysical principle was seldom doubted, it was not necessarily founded on a
physical theorem about the uniqueness of solutions to certain differential equations.”
It’s quite astonishing to notice the difference in the way 19th century scientists
were thinking! Maxwell, in his letter to Darwin’s half-cousin Galton (quoted in
Sect. 7), being fascinated by the works of Boussinesq and al., does not speak about
the problem of indeterminism in classical physics at all. Yet he mentions “some
determining principle which is extra physical (but not extra natural)” instead. It
indicates that Maxwell was well aware of a missing law of nature, when he continues:
“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, not of sensible magnitude but enough to bring her round in
time.”
It appears that Norton already used the correct word “acausal” in his first paper
dealing with the dome [52]. Using the word “acausal” as a proper opposite to the
word “causal” is to be preferred, because it leaves the door open for teleological phenomena. Any attempt to analyse indeterminism on the classical level is meaningless
since classical physics is unable to deal with it, perhaps at best can somehow “tolerate” it as in the case of Norton’s dome. We may not even need to discuss the problem
of Lipschitz continuity: the Mexican hat in the Ginzburg-Landau equations has no
such discontinuity at the apex and, after all, we observe the same type of double solutions for the Norton dome and ideal conductors (first solution) and superconductors
(second solution) the latter leading to the Meissner effect. Classical physics is able to
describe the Norton dome as well as the Meissner effect, but is unable to determine
what evoked the phenomenon.
M. Svrˇ cek
Although Norton disproves very carefully the above mentioned objections, a curious situation arises: On one hand Norton is correct in his arguments against his
opponents, but on the other his opponents are right about the determinism of classical physics. Malament [122] has expressed this quandary in a very elegant way: “But
I am not sure that the full complexity and interest of the breakdown is adequately
captured by saying, either, “Newtonian particle mechanics is an indeterministic theory” (full stop) or “Norton’s example is not a well-defined Newtonian system” (full
stop). Indeed, I am not convinced we have clearly posed alternatives here—because
we do not have a sufficiently clear idea in the first place what should count as a “Newtonian system” (or count as falling within the “domain of application” of Newtonian
theory). My inclination is to avoid labels here and direct attention, instead, to a rich
set issues that the example raises.”
In order to gain a deeper insight into this problem one should look for analogies
such as the Norton dome. A historical review of comparable examples was written
by van Strien. In her article “The Norton Dome and the Nineteenth Century Foundations of Determinism” [123] she says: “Lipschitz-indeterministic systems played
an important role in the work of Boussinesq, who, in 1878, discussed several such
systems, one of which was similar to the Norton dome [124].” This paper contains
also an observation of an unexpected importance: “Though determinism as a general metaphysical principle was seldom doubted, it was not necessarily founded on a
physical theorem about the uniqueness of solutions to certain differential equations.”
It’s quite astonishing to notice the difference in the way 19th century scientists
were thinking! Maxwell, in his letter to Darwin’s half-cousin Galton (quoted in
Sect. 7), being fascinated by the works of Boussinesq and al., does not speak about
the problem of indeterminism in classical physics at all. Yet he mentions “some
determining principle which is extra physical (but not extra natural)” instead. It
indicates that Maxwell was well aware of a missing law of nature, when he continues:
“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, not of sensible magnitude but enough to bring her round in
time.”
It appears that Norton already used the correct word “acausal” in his first paper
dealing with the dome [52]. Using the word “acausal” as a proper opposite to the
word “causal” is to be preferred, because it leaves the door open for teleological phenomena. Any attempt to analyse indeterminism on the classical level is meaningless
since classical physics is unable to deal with it, perhaps at best can somehow “tolerate” it as in the case of Norton’s dome. We may not even need to discuss the problem
of Lipschitz continuity: the Mexican hat in the Ginzburg-Landau equations has no
such discontinuity at the apex and, after all, we observe the same type of double solutions for the Norton dome and ideal conductors (first solution) and superconductors
(second solution) the latter leading to the Meissner effect. Classical physics is able to
describe the Norton dome as well as the Meissner effect, but is unable to determine
what evoked the phenomenon.