2007-03-23

Several days ago Steinn wrote an interesting post about

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 microstate, but couples via gravity to the brain it tries to measure, it may be impossible

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.

2007-03-29

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 prediction impossible.

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 interaction.

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 any cause.

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 physicist.

(*) 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.

2007-08-25

I would like to add to this topic a somewhat related quote from the book 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 effect')."

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 physical laws of our universe. The impossibility results establish that Laplace was wrong to claim that even in a classical, non-chaotic universe the future can be unerringly predicted. Alternatively, they can be viewed as a non-quantum 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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