Saturday, July 18, 2009

Misunderstanding Laplace

I finished Leonard Mlodinow's "The Drunkard's Walk: How Randomness Rules Our Lives" this past week, and have a couple of thoughts related to it.

In Chapter 10 he quotes Laplace:
"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes."

and Mlodinow states that this is an expression of determinism. He then further states
But for Laplace's dream to hold true, several conditions must be met. First, the laws of nature must dictate a definite future, and we must know those laws. Second, we must have access to data that completely describe the system of interest, allowing no unforeseen influences. Finally, we must have sufficient intelligence or computing power to be able to decide what, given the data about the present, the laws say the future will hold.

He then criticizes it with the following three problems:

  1. society is not governed (as far as we know) by definite and fundamental laws in the way physics is
  2. like Lorenz, we cannot obtain the precise data necessary for making predictions
  3. human affairs are so complex that it is doubtful we'd be able to make the calculations anyway

He concludes "as a result, determinism is a poor model for the human experience." His point seems to be, in some ways, obvious and in other ways irrelevant.

Laplace was simply saying that "God" would not find anything random, because of complete knowledge. The connection between knowledge and inference, which probability theory affords, was worked out by Laplace in great detail and it known to use today as Bayesian inference. The structure of Bayesian inference describes randomness simply as the product of our ignorance of the model, the parameters, the initial conditions, the measurement details, etc... Laplace was simply saying that with perfect knowledge, there is no randomness. E.T. Jaynes would describe the "random process" as a mind-projection fallacy: you have ignorance of the system, so you attribute its unpredictable behavior as a product of the system itself. A rolled die is following Newton's Laws, deterministically, and detailed knowledge of the die and the roll and the surface should allow you to predict 100% of the time what it will do. We lack that knowledge, thus the behavior becomes unpredictable. We often then attribute that unpredictable behavior as a "random die", as if it were the die that contains the randomness and not our own ignorance.

Bringing in Lorenz, and chaos theory, is irrelevant here. Lorenz's systems were completely deterministic, and it is theoretically possible for a being to know the state of the system out to a sufficient number of decimal places to provide any particularly set level of uncertainty in the system. With the quantization of states, it then becomes possible to know *exactly* what state something is in. Of course, quantum mechanics is a two-edged sword in this example: it solves the chaos problem, but adds an inherent, physical, randomness to the system which is very peculiar.

The problem with Mlodinow, it seems, is that he hold human activity to be a bit too special. We are, after all, made up of atoms and would thus be governed by the laws of physics. Certainly it would be too complex to handle, for us, but Laplace was not talking about us in his quote, or at least not us right now or in the near future.

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