Demons and other curiosities
Pierre-Simon Laplace was a French scientist, perhaps one of the greatest ever, French or otherwise. His work affected the way we look at both mathematics and physics, among other areas of science. He may be least known for his discussion of what we now call Laplace’s demon.
Today I want to talk about his demon, and whether predicting the future is possible.
Can we predict the past? Can we predict the present? Can we predict the future? Predicting the past and predicting the present sound a bit silly. The usual question is: Can we predict the future? Although I think predicting the past—if taken to mean “what happened in the past?”—is not so easy.
So can we see the future? I would argue that many can and do very well every day predicting the future. The huge profits of options traders and hedge funds must say something about prediction. They make a lot of money by knowing—at least with some reasonable probability—what the future price will likely be of a stock or commodity.
There are many other predictions that we make that are often correct. We can predict that the sun will rise at tomorrow in Atlanta at 6:54 am. This prediction works very well. The Weather Channel does a reasonable job of predicting the weather for later today, a less good job on tomorrow, and not so good on predicting the weather this calendar day next year.
The issue is not these predictions, but whether it is possible to predict the future exactly. Laplace in 1814 claimed that given the exact position and speed of all objects in the universe at some time a “demon” would be able to use the laws of physics to predict their positions at an arbitrary time in the future. This is now called Laplace’s demon. Translated, he said:
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.
No Laplace Demon?
Since Laplace’s work there has been much discussion against the idea that even in principle there could be such a demon. Many attacks are now possible. Most are based on ideas and concepts that Laplace did not have available to him in 1814. Chaos theory has been advanced as one way to show hid demon is impossible, the random nature of quantum mechanics is another, and even the natural of computational complexity.
Laplace of course had no idea of the quantum nature of the world. So it is a bit unfair for us to attack him in this way. We could extend Laplace’s intent to a quantum world, noting that quantum mechanics is a deterministic theory even as it describes branching-off worlds. Not all conceivable branches are possible, and we could ask the demon to identify the excluded ones in advance.
We would assume the demon has perfect knowledge of the initial conditions of the universe or of any local Big-Bang event, thus distinguishing our setting from the more human-relevant one in this in-depth essay by Scott Aaronson. Still, it is easier to address Laplace’s argument in the kind of world where Newtonian mechanics holds sway, and where the demon could solve N-body problems exactly even with collisions.
Accordingly, some researchers have looked at the problem of prediction of the future in a Laplacian type world, where the future is deterministic. Not long ago, in 2008, David Wolpert used a riff on Cantor’s diagonalization argument to show that prediction machines could not exist. The latter is one of the reasons that I find the question relevant to theory. His theorem is here and is summarized here:
The theorem’s proof, similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?”
A Short Disproof
Recently a short note, called a “Mathbit,” was published by in the Math Monthly by Josef Rukavicka. A Mathbit is always at most a single page and is set in a gray font style.
He claims an even shorter proof that Laplace’s demon is impossible—David’s is more formal and has precise definitions. Here is the main part of Rukavicka’s argument:
Suppose that there is a device that can predict the future. Ask that device what you will do in the evening. Without loss of generality, consider that there are only two options: (1) watch TV or (2) listen to the radio. After the device gives a response, for example, (1) watch TV, you instead listen to the radio on purpose. The device would, therefore, be wrong. No matter what the device says, we are free to choose the other option. This implies that Laplace’s demon cannot exist.
Comments On This Proof
I have several reservations about this proof that there is no Laplace demon. For starters, it assumes a complexity type assumption: that the prediction of the future is fast. What if the prediction of time one day into the future took more than one day? Then of course the argument would fail. Of course this raises an interesting issue. Suppose to predict the future days into the future takes more that , then this is clearly not useful. However, even if the predictor takes only to do the prediction, the that is needed to get a useful prediction could be immense. What if the prediction took
This would clearly not allow Rukavicka’s argument to be meaningful.
Another basic issue that struck me is the choice of watching TV vs. radio. Rukavicka assumes implicitly in his argument that we have the free will to decide what to do. But this seems to be the essence of the whole issue. What if we cannot make this choice? We might listen to the predictor say “TV” and tomorrow we forget our contrarian intent to listen to the radio and watch TV anyway. What if we really do not have a choice? This seems to devolve into a circular argument—or am I missing something?
Well these two issues do take the argument back into the realm of Scott’s long essay.
What do you think? In a deterministic world could there be complexity results about predictions? Are these questions related to P=NP in some manner?