[P&N] Chapter 10: Determinism

Citation: Edward A. Lee, 2017: Plato and the Nerd - the creative partnership of humans and technology. MIT Press, Cambridge, MA

Laplace’s Demon

According to to the Austrian computer scientist Hermann Kopetz, the determinism is a property of models and not a property of the physical world.

A model is determinism if given an initial state of the model, and given all the inputs that are provided to the model, the model defines exactly one possible behavior.

I mentioned in the previous notes of this book that:”whenever you find something very hard to get through, think about the territory and map.” Now, again, we are confronted with this very subtle trap - “Is the physical world deterministic?”.

Before answer this question, let’s consider the following statement, which is called “Laplace’s Demon” first.

In the early 1800s, the French scientist Pierre-Simon Laplace made an argument for determinism in the universe. Laplace argued that if someone(a demon) were to know the precise location and velocity of every particle in the universe, then the past and future locations and velocities fore each particle would be completely determined and could be calculated from the laws of classical mechanics(Laplace, 1901).

Well, I personally once thought about something really similar to this topic when I was washing my foot, only end up finding the questions itself meaningless because there is no such thing that can remember all this information of the whole universe. But let’s get serious.

Credits go to the modern physics, the following facts can undermine the “Laplace’s Demon” statement:

1. The classical mechanics, specifically, “Newton’s Laws”, only apply to modest speed and scales. They need to be adjusted using Einstein’s relativity to be precise.

2. The notion of “location” and “velocity” can not be known simultaneously according to modern quantum mechanics.

But does this implies that there’s no such “Laplace’s Demon”? Considering the probabilistic nature of the wave function in quantum mechanics, does this undermine the idea of “deterministic universe”? Stephen Hawking argues that it does not:

At first, it seemed that these hopes for a complete determinism would be dashed by the discovery early in the 20th century that events like the decay of radioactive atoms seemed to take place at random. It was as if God was playing dice, in Einstein’s phrase. But science snatched victory from the jaws of defeat by moving the goal posts and redefining what is meant by a complete knowledge of the universe.

Humanity rules all! After all, we can redefine what is meant by victory. But let’s get serious again, “In quantum theory, it turns out one doesn’t need to know both the positions and the velocities[of the particles], (it is enough to know how the wave function(Schrodinger’s Equations) evolved in time).” Just think about the famous “Schrodinger’s Cat”, which can be both alive and dead.

I would like to invite Lee again to summarize for me:

So Laplace’s question still stands, except that now we have to update it to consider the “state” of the system to be represented by its wave function not by the positions and velocities of its particles, and we have to account for the curvature of space time no matter how small. If we make these adjustments, is the resulting model of the universe deterministic? The question of whether the physical world itself is deterministic is probably unanswerable. However, we can answer the question of whether any particular model of the universe is deterministic. We need to keep distinct the map and territory.

But Lee further states that “deterministic models my also be foiled by complexity(and chaos)”, which will be examined next.

The Butterfly Effect

Everybody knows “the Butterfly Effect” these day, which comes from a metaphor saying that a tornado could be caused by the wing of a butterfly.

And we can say that pseudorandomness is a form of chaos, or butterfly effect. Another examples of chaos is the famous game created by the British mathematician John Horton Conway, which is called “Conway's Game of Life”.

The game has a rectangular grid of cells that are either alive or dead (shown as black or dead squares). An initial state has some cells alive and some dead. At each step of the game, the cells are updated following only 4 rules:

1. Underpopulation: Any live cell with fewer than 2 live neighbors dies.
2. Stabling: Any live cell with 2 or 3 live neighbors lives on to the next step.
3. Overpopulation: Any live cell with more than 3 live neighbors dies.
4. Reproduction: Any dead cell with exactly 3 live neighbors becomes a live cell.

This is a perfect example illustrating the idea of chaos. Google it!

Incompleteness of Determinism

From the last section, we know that models my exhibit chaos, which can limit the utility of the models as predictors, making Laplace’s demon a difficult concept to accept. But why don’t we come up with a “hybrid” system, which combines the “continuous and discrete behaviors”.

Let consider an example where 2 elastic balls bouncing directly into another still ball simultaneously on a frictionless ground.

Suppose the 2 balls’ masses are not the same, how do we construct our models? Basically, we have the following 2 options:

1. Treat the very instant when the collision occurs as a discrete event, which does not take time. And construct the model using Newton’s laws where both momentum and energy converse.

2. Treat the whole process as a continuous process and construct the model using either classical mechanics (Newton’s laws) or quantum mechanics (using the Schrodinger equation).

In the first option, we lose predictability to nondeterminism(the behavior at the collision cannot be determined for that the results are different if we pick different ball to analyze first). In the second option, we lose predictability to chaos(the model is extremely sensitive to initial conditions such as masses, positions and velocities). In both cases, we lose predictability. But the first one is much simpler than the second one, so it seems that the hybrid one that admits nondeterminism is the better one. But after all, they are incomplete.

The Hard and the Soft of Determinism

Determinism is extremely valuable, but if a deterministic model is too complex to analyze, a nondeterministic model may be more valuable.

Lee asserts that any modeling framework rich enough to include Newton’s laws of motion, if it also admits discrete behaviors, is incomplete. And because our ability to predict is limited by both chaos and nondeterminism, which are unavoidable, our vision of the future will always remain uncertain.

It seems that Lee has just beat many children’s dream, to predict the future, to death. Sad. But come on, our future are not doomed from this perspective, right? You can build your future by yourself, just go for it!