Laplace's Demon

Neo, the main character from The Matrix, didn’t like the idea of fate because he wanted to be able to make his own choices. Let’s see what Pierre-Simon Laplace had to say about such free will back in 1814.

At the turn of the 17th and 18th century, Isaac Newton published a work in three books called Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy) (Newton 1687) or simply Principia. There, he originally wrote:

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

Which we can translate into English as follows:

The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.

We can express this idea mathematically as \(p’ = F\), where \(p’\) is the time derivative of momentum \(p\) and \(F\) is the force which represents the impulse that Newton referred to. Since momentum \(p\) is defined as \(p = mv\), where \(m\) is the mass of the body and \(v\) is its velocity (the time derivative of position \(x\), i.e. \(v = \frac{dx}{dt}\)), we can take the derivative of momentum with respect to time: \(F = \frac{dp}{dt} = m\frac{dv}{dt} = ma\), where \(a\) is the acceleration of the body (the time derivative of velocity, i.e. \(a = \frac{dv}{dt}\)). This is known as Newton’s second law of motion. If we need to consider various directions of the force \(F\) and acceleration \(a\), we can use vectors \(\vec{F} = m\vec{a}\). Newton also proposed the concept of derivatives so that we can actually solve and use the equations above. These ideas give us a very strong apparatus for calculating the movements of physical objects. If we have an abstract system of physical objects, and we know the initial forces, positions, and velocities of all the objects, then we exactly know how the system will behave in the future. Moreover, we don’t even need to know the initial properties; if we measure them at any time, we can calculate the past as well as the future. In other words, there’s no place for any random accident in the moving process. That’s also what Newton’s second law tells us, and that’s what Laplace noticed approximately 100 years later.

We will now need the term determinism. When we have a deterministic system, it means that the current state exactly predicts the next one. If the Universe were deterministic, it would mean that our destiny would be exactly predicted. This is also what Einstein believed, and that is why he said that “God does not play dice” when he was trying to show that the apparent randomness in quantum mechanics must have hidden deterministic variables underneath.

As far as we know, our Universe consists of elementary particles, and everything started with the Big Bang 13.8 billion years ago. That means that the initial conditions were apparently given, and if the elementary particles behaved according to Newton’s laws, the Universe would probably be fully deterministic. This idea is known as Laplace’s demon, a concept he outlined in his work named A philosophical essay on probabilities (Laplace 1902) in 1814:

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. (Laplace 1902)

When meteorologists got their first computers in the early 1960s, they thought that they would be able to make exact weather forecasts for a very long time. Unfortunately, it turned out that this is a very difficult task. Even though the weather is a deterministic system without any random elements involved, it is not easily predictable. This fact was explored by Edward Lorenz in 1961 when he was simulating weather evolution on a computer. After a very slight change in the input values, the simulated evolution was completely different. This is known as the Butterfly effect, after a talk Lorenz gave titled Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? (Lorenz 1972) The same sensitivity to initial conditions applies every time you throw dice, and that’s why we are not able to predict the weather for more than a few days even with the most advanced technology we have today. However, the Butterfly effect doesn’t weaken Laplace’s demon; it only creates so-called deterministic chaos. Lorenz summarized it in one sentence: Chaos: When the present determines the future, but the approximate present does not approximately determine the future. (Danforth 2013)

Quantum theory is perhaps the strongest challenge to the demon. It claims that the subatomic world behaves according to the laws of quantum mechanics and not Newton’s laws. This means, inter alia, that certain properties of particles do not have definite values until they are measured — we can only calculate the probability of obtaining a given value. This is not due to poor technology. It is because it is impossible in principle. One of the reasons for this is Heisenberg’s uncertainty principle. It states that the more precisely we know the position of any particle, the less precisely we can know its momentum, and vice versa. This is not a limitation of measurement — it reflects a fundamental property of nature. Under the standard interpretation of quantum mechanics, conjugate properties like position and momentum simply cannot both be precisely defined at the same time. However, whether this means the details genuinely don’t exist (as the Copenhagen interpretation holds) or are merely inaccessible to any observer (as hidden-variable theories suggest) remains debated. Either way, quantum theory shows us that there cannot be any “intelligent entity” inside our Universe which would know the exact state of all particles and which would be able to calculate the future or the past.

Stephen Hawking also applied his mind to the question of whether God plays dice or not. (Hawking, n.d.) The concept of black holes had been known for decades, and any mass or energy which falls into a black hole is lost forever without any chance to escape, so information in the Universe is lost. However, Hawking proved that black holes are not entirely so black and that they do send out particles at a steady rate. These particles don’t come directly from the black hole, but are created in pairs near the event horizon. During this creation, a whole pair can escape or be absorbed, but also one particle from a pair can escape while the other is absorbed. This vaporization reduces the size of the black hole, and in the end, the black hole disappears. Crucially, there is no way to predict the speed and position of the escaped particle without knowing about the absorbed one. This seemed to be the obvious funeral of Laplace’s demon; but in 2005, Hawking conceded that black holes may after all preserve information about what fell into them — though this rescues information conservation, not determinism itself.

What would it mean if we thought about the Universe being deterministic? The philosophical consequences are surprisingly fascinating. There wouldn’t be any place for uncertainty or accident; everything would have its reason. That seems fine, everything is explainable, but when we think more deeply, it gets terrifying and emotionally frustrating. It would be worse than the Matrix for Neo because there wouldn’t be any difference between living beings and machines. Our every step or thought would have been strictly set 13.8 billion years ago. Ethics, good and evil, or responsibility would be just illusions. There wouldn’t be any difference between life and lifeless; everything would be just a soulless result of moving atoms. We would be just actors on the stage of a big theater with no influence or chance to change anything. On the other hand, we might feel comfortable because it would give us certainty that we can’t mess up anything. If you got arrested for drinking and driving, you could argue that it was not your responsibility. You could say that elementary particles were responsible and you couldn’t do anything. Also, when we fall in love with somebody and it seems like an accident, we can only be thankful that the tiny particles decided to move in the right direction 13.8 billion years ago.

What about God? What is interesting is that people who believe in God may argue against a deterministic Universe. If God did create this theater, we could assume that he gave us freedom of will. Otherwise it would have been boring for him to observe us during such an artificial performance. And even if God really does not play dice, let’s keep our free will and thoughts and let’s hope that they’re really not just an illusion. Science has given rise to the demon and hopefully it will also destroy it.

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