A new Equation for Intelligence F = T ∇ Sτ - a Force that Maximises the Future Freedom of Action
Intelligence is a Force with the Power to Change the World
Describing intelligence as a physical force that maximises the future freedom of action, adds a new aspect to intelligence that is often forgotten: the power to change the world. This, I think, was the biggest revelation for me, when I started thinking about the the new equation for intelligence. The second revelation was, that intelligent systems are survival engines, that increase their chances of survival by maximising a single quantity: the freedom of action. Both insights may sound trivial or obvious, but I don't think they are.
A few days ago a saw the TED talk "A new equation for intelligence" by Alex Wissner-Gross. He presents an equation he published in April 2013 in a physics journal. It may not be the most impressive talk I have ever seen. And I had to watch it twice to fully understand it. But the message excites me so much, that I don't sleep well since a few days. I thought everybody must be excited about this equation. But, it seems that this is not the case. Either I am not understanding it correctly or others don't get it. Or maybe it resonates with me, because I am physicist, with a strong background in computing, who has done research in computational biology. To find this out, let me explain my understanding of the equation. Please tell what your think and what's wrong with my excitement (I need sleep)....
So, why did the equation blow me away? Because this very simple physical equation can guide us in our decisions and it makes intelligent behaviour measurable and observable. It adds a new real physical force to the world, the force of intelligence. From the equation we can deduce algorithms to act intelligently, as individuals, as societies and as mankind. And we can build intelligent machines using the equation. Yes, I know, you may ask: "How can the simple equation F = T ∇ Sτ do all of that?"
Intelligence is a Force that Maximises the Future Freedom of Action
Before we look at the equation in more detail, let me describe its essence in every day terms. Like many physical laws or equations the idea behind it is simple:- Intelligence is a force that maximises the future freedom of action.
- It is a force to keeps options open.
- Intelligence doesn't like to be trapped.
The new Equation for Intelligence F = T ∇ Sτ
Note: skip this section, if you are not interested in understanding the mathematics of the equation!
This is the equation:
This is the equation:
F = T ∇ Sτ
Where F is the force, a directed force (therefore it is bold), T is a system temperature, Sτ is the entropy field of all states reachable in the time horizon τ (tau). Finally, ∇ is the nabla operator. This is the gradient operator that "points" into the direction of the state with the most freedom of action. If you are not a physicist this might sound like nonsense. Before I try to explain the equation in more detail, let's look at a another physical equation of force.
The intelligence equation very similar to the equation for potential energy F = ∇ Wpot. Wpot is the potential energy at each point is space. The force F pulls into the direction of lower energy. This is why gravitation pulls us in direction of the center of the earth. Or think of a landscape. At each point the force points downhill. The direction is the direction a ball would roll starting at that point. The strength of the force is determined by the steepness of the slope. The steeper the slope, the stronger the force. Like the ball is pulled downhill by the gravitational force to reach the state with the lowest energy, an intelligent system is pulled by the force of intelligence into a future with lowest number of limitations. In physics we use the ∇ Nabla operator or gradient to turn a "landscape" into a directed force (a force field).
Where F is the force, a directed force (therefore it is bold), T is a system temperature, Sτ is the entropy field of all states reachable in the time horizon τ (tau). Finally, ∇ is the nabla operator. This is the gradient operator that "points" into the direction of the state with the most freedom of action. If you are not a physicist this might sound like nonsense. Before I try to explain the equation in more detail, let's look at a another physical equation of force.
The intelligence equation very similar to the equation for potential energy F = ∇ Wpot. Wpot is the potential energy at each point is space. The force F pulls into the direction of lower energy. This is why gravitation pulls us in direction of the center of the earth. Or think of a landscape. At each point the force points downhill. The direction is the direction a ball would roll starting at that point. The strength of the force is determined by the steepness of the slope. The steeper the slope, the stronger the force. Like the ball is pulled downhill by the gravitational force to reach the state with the lowest energy, an intelligent system is pulled by the force of intelligence into a future with lowest number of limitations. In physics we use the ∇ Nabla operator or gradient to turn a "landscape" into a directed force (a force field).
Back to our equation F = T ∇ Sτ. What it says is that intelligence is a directed force F that pulls into the direction of states with more freedom of action. T is a kind of temperature, that defines the overall strength (available resources) the intelligent system has (heat can do work, think of a steam engine: the more heat the more power). Sτ is the "freedom of action" of each state that can be reached by the intelligence within a time horizon τ (tau). The time horizon is how far into future the intelligence can predict. Alex Wissner-Gross uses the notion of entropy S to express the freedom of action in the future. The force of intelligence is pointing into that direction. As we have seen, in physics the direction of the force at each state is calculated by a gradient operation ∇ (think of the direction the ball is pulled). The Nabla operator ∇ is used to assign a directional vector (the direction of the force of intelligence) to each state (in our case: all possible future states). The more freedom of action a state provides the stronger the force is pulling in that direction. So, ∇Sτ is the pointing into the direction with the most freedom of action. The multiplication with T means the more power we have to act, the stronger the force can be.
Note: the optimal future state is the optimal state form the viewpoint of the intelligent system. It might not the optimal state for other systems or for the entire system.
If you want to understand the equation in more detail read the original paper 'Causal Entropic Forces - by A. D. Wissner-Gross and C. E. Freer'.
Understanding the Laplace operator conceptually
The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it?
Any good essays (combining both history and conceptual understanding) on the Laplace operator, and its subsequent variations (e.g. Laplace-Bertrami) that you would highly recommend? | |||||||||||||
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The Laplacian
and integrate: The integrals where The Laplace-Beltrami operator is essentially the same thing in the more general Riemannian setting - all the nasty curvy terms will be higher order, so the same formula should hold. | |||||||||||||
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I think the most important property of the Laplace operator
Some good books on the subject:
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To gain some (very rough) intuition for the Laplacian, I think it's helpful to think of the Laplacian on
Just as Anthony's answer discusses, the second derivative at Generally, a function is harmonic if and only if it satisfies the mean value property. In The maximum principle states roughly that if Finally, let me go up one dimension and mention some of my intuition for harmonic functions | |||
Another view along the lines of the answers above:
Suppose you have some region in the plane What does "as smooth as possible" mean? Well, one measure of the smoothness of How do we minimize
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Here is some intuition:
I think the most basic thing to know about the Laplacian Notice that the integration by parts formula can be interpreted as telling us that |