Several days ago Steinn
wrote an interesting post about free will. In the
comment thread I challenged his statement "Classically measuring
the state of the brain is not a problem, since it can in principle be
done with arbitrarily delicacy, and the same is the case with the
inputs, they can be tapped with infinitesimal perturbation." This is an
argument one encounters quite frequently in discussions about free
will vs the determinism of Newtonian mechanics and here is my
Every molecule in your brain interacts via gravity with the whole
planet; There is no way to shield this interaction. Since your brain
consists of many molecules we can assume sensitive dependence on
initial conditions and one would have to measure and then predict
positions and velocities with incredible high precision to forecast
just a few milliseconds. This means, once can probably not neglect the
interaction of the molecules in your brain with the rest of the planet,
if one wants to measure and predict its microstate. But in order to
know the microstate of the Earth you need to consider the whole solar
system of course.
The 2nd part of my argument considers that the system brain-earth-solar
system would necessarily include the computer and other machinery to
determine and predict the state of Steinn's brain.
Since this machinery and prediction system cannot know its own
but couples via gravity to the brain it tries to measure, it may be
impossible in principle to determine and predict the microstate
of a brain.
By the way, while this argument considers deterministic, Newtonian
mechanics only, I assume it would also hold if one replaces
'(Newtonian) microstate' with 'quantum state'. (More about that perhaps
in another post.)
Of course, I already wrote much earlier
about the fact that, strictly speaking, predictions are impossible in
physics. But I think the above argument goes beyond that.
As a follow-up to my post above, I would like to provide some
pointers for further reading and thinking.
Cris Moore showed that simple Newtonian systems (e.g. a particle moving
in a three-dimensional potential) can be equivalent to a
Turing machine (*). In order to predict the behavior of such a
system or 'machine' M1, one would have to duplicate it, so that the
evolution of system M2 would provide the desired forecast (but one
would have to set the parameters of M2 so that it evolves faster than
M1). However, the argument I made previously suggests that the
(gravitational) interaction between M1 and M2 would make such a
In general, one cannot turn off the interaction between M1 and M2,
because one needs to duplicate the initial state of M1 and be able to
read out the prediction made by M2 and this requires physical
In a much more general framework one can show the Impossibility
of Predicting the Behavior of Rational Agents (*). Dean P. Foster
and H. Peyton Young demonstrate that "there are games in which it is
impossible for perfectly rational players to learn to predict the
future behavior of their opponents (even approximately) no matter what
learning rule they use. The reason is that, in trying to predict the
next-period behavior of an opponent, a rational player must take an
action this period that the opponent can observe. This observation may
cause the opponent to alter his next-period behavior, thus invalidating
the first player’s prediction."
John D. Norton describes The Dome:
An Unexpectedly Simple Failure of Determinism (*). A fascinating
example of a non-deterministic system in Newtonian mechanics; A
particle at rest begins to move at an arbitrary point in time, without
Last but not least, let me mention the issue of non-collision
singularities in n-body systems. Their existence has been conjectured
for n>3 by Painlevé and Zhihong Xia was able to prove the
existence of such singularities for the 5-body system.
I would like to conclude this topic with the following conjecture(s):
#1: In general, a physicist cannot predict her own behavior (x).
#2: In general, a physicist cannot predict the behavior of another
(*) I thank Cosma Shalizi
who provided those links and I am also very grateful for an interesting
discussion of this topic.
(x) With 'behavior' I really mean microstate (or quantum state). I use
'in general', because in special situations (e.g. the physicist gets
killed) it might be possible to predict the behavior. I do not define
'physicist' but assume the equivalent of an information processing
state-machine, with many internal states.
I would like to add to this topic a somewhat related quote from the
Physical Basis of The Direction of Time, chap 3.1, p.56: "Borel
(1924) estimated the effect of a gravitational force that would arise
here on earth by the displacement of a mass of the order of a few grams
by a few centimeters at the distance of Sirius. He thereby pointed out
that this would lead to a completely different microscopic state for
the molecules forming a gas in a vessel under normal conditions within
seconds. Although distortions of the individual molecular trajectories
are extremely small, they would be amplified in each subsequent
collision by a factor of the order of l/R, the ratio of the mean free
path over the molecular radius.
This extreme sensitivity to the environment describes in effect a local microscopic indeterminism. In
many situations, the microscopic distortions may even co-determine
macroscopic effets (thus inducing an effective macroscopic
indeterminism), as discussed, in particular in the theory of chaos ('butterfly
David Wolpert discusses the physical
limits of inference in this
paper beyond such concrete examples and assumptions.The abstracts
states "We present existence and impossibility results
for inference devices. These results hold independent of the precise
laws of our universe. The impossibility results establish that Laplace
wrong to claim that even in a classical, non-chaotic universe the
future can be
unerringly predicted. Alternatively, they can be viewed as a
mechanical 'uncertainty principle'. "
However, he then states in the Introduction:
"The crucial property underlying our results is that inference devices
are embodied in the very physical system (namely the universe) about
which they are making inferences."
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