- STATISTICS 527
involves an abandonment of strict dynamical principles and an
adoption of the mathematical methods belonging to the theory
of probability... if the scientific doctrines most familiar to us had
been those which must be expressed in this way, it is possible
that we might have considered the existence of a certain kind of
contingency a self-evident truth, and treated the doctrine of philo-
sophical necessity as a mere sophism. [Quoted by Porter, (1986),
pp. 201-202]2.3. The metaphysics of probability and statistics. The statistical point of
view required an adjustment in thinking. Maxwell appeared to like the indetermi-
nacy that it introduced; Einstein was temperamentally opposed to it. Although
deterministic and probabilistic models might both produce the same predictions
because of the law of large numbers, there was a theoretical difference that could
be seen clearly in thermodynamics. If the laws of Newtonian mechanics applied
to the point-particles that theoretically made up, say, an ideal gas, the state of
the gas should evolve equally well in either direction, since those mechanical laws
are time-symmetric. Imagine, then, two identical containers containing the same
number of molecules of an ideal gas that, at a given instant, are in exactly the same
positions relative to the boundaries of the containers and such that each particle
is moving with equal speed but in exactly opposite direction, to the particle in the
corresponding place in the other container. By the laws of mechanics, the past
states of each container must be the future states of the other. But then one of the
two must be evolving in a direction that decreases entropy, in contradiction to the
second law of thermodynamics.
The explanation of that "must be" is statistical. It is not absolutely impossible
for the mechanical system to be in a state that would cause it to evolve, following the
deterministic laws of mechanics, in a direction of decreasing entropy. But the initial
conditions that lead to this evolution are extremely unlikely, so unlikely that no one
ever expects to observe such a system. As an illustration, Newtonian mechanics
can perfectly well explain water flowing uphill given that the initial velocities of all
the water molecules are uphill. But no one ever expects these initial conditions to
be satisfied in practice, in the absence of a tsunami. An additional consideration,
which Maxwell regarded as relevant, was that in some cases initial-value problems
do not have a unique solution. For example, the equation | · ^JL — y^1 /^4 — ï with
initial condition y = 0 when t = 0 is satisfied for t > 0 by both relations y = 0 and
y = t.s. Shortly before his death, Maxwell wrote to Francis Galton:
There are certain cases in which a material system, when it comes
to a phase in which the particular path which it is describing co-
incides with the envelope of all such paths may either continue in
the particular parth or take to the envelope (which in these cases
is also a possible path) and which course it takes is not determined
by the forces of the system (which are the same for both cases) but
when the bifurcation of the path occurs, the system, ipso facto, in-
vokes some determining principle which is extra physical (but not
extra natural) to determine which of the two paths it is to follow.
[Quoted in Porter, 1986, p. 206]