Phase Transitions and Power Laws

This is a good introduction to power law distributions.

Most natural events and outcomes follow Gaussian and Poisson distributions. They are made up of random, independent factors. But there are certain events which follow power-law distributions.

This is connected with phase transitions and critical points. When ice reaches a critical point it undergoes a phase transition to water. Likewise, the evaporation of water into gas. This is measured through the heat in the system. Self-organized Criticality is responsible for some, but not all, phase transitions. The Sand pile model most often used to illustrate the idea.

 

Small Worlds

Ever hear of Six Degrees of Separation? It’s about Small World networks. Stanley Milgram ran the Small World experiment to find out if two randomly selected individuals knew each other. Usually, most knew each other through just a few intermediaries.

Any individual in a social network is just a few “hops” to any other individual.

 

Graph Theory and Kidneys

Here’s a really pragmatic use of graph theory. This is a better method of matching compatible kidney donors and recipients than the current inefficient method of arranging transplants.

 

Bayesian Reasoning

Once you learn about Bayesian Reasoning, you understand why humans are semi-retarded apes. But I’m being too generous.

Eliezer Yudkowsky gives an excellent primer to Bayes’ Theorem: An Intuitive Explanation of Bayesian Reasoning
And the advanced lesson: A Technical Explanation of a Technical Explanation
Remember this: “I usually take the moral that your strength as a rationalist is measured by your ability to be more confused by fiction than by reality. If you are equally good at explaining any story, you have zero knowledge.”

Keeping in spirit, I cannot supply a verbal explanation.

 

Misusing Quantum Mechanics

There’s a difference between understanding a verbal explanation of a scientific or mathematic concept and understanding the mathematical formulas involved.

Quantum Mechanics is almost purely mathematical without verbal explanations. It’s difficult to convey the conceptual idea. This, of course, allows charalatans to abuse the concept to spread mystical nonsense like Deepak Chopra’s Ayurvedic Medicine.

 

Unreasonable Effectiveness of Mathematics

Here’s the classic essay by Eugene Wigner.

 

The Real Nash Equilibrium

The movie Beautiful Mind managed to get Nash’s major discovery wrong.

There were a number of historical inaccuracies. The movie portrayed Nash’s schizophrenia as a bunch of right-wing fantasies. In reality, they were left-wing fantasies. Nash also spoke with space aliens.

Besides the historical inaccuracies, they got the game theory wrong too. They showed Nash “discovering” Nash Equilibrium in a bar discussion. Except what he described as the Equilibrium is not the Nash equilibrium.

 

Brownian Motion and Stochastic Drift

Brownian motion is one of Einstein’s less popularly known proofs.

When you look at a particle under the microscope, it moves. But it does not move in one direction or according to any seen force. It seems to move in completely random directions. It’s a form of stochastic drift.

 

Complex Adaptive Systems

So how does complexity work?

I’ll use John Holland’s categories. A Complex Adaptive System (cas) is made up of three mechanisms (Schema, Schemata, and Signals) and four properties (Aggregation, Non-Linearity, Flows and Diversity). Agents interact on the local level, self-organize, and produce “emergent” complex behavior.

The Agents are the adaptive actors. They make IF/THEN responses. If they see specific stimuli, then they respond with an action. They learn new information from the outcome, and continue making IF/THEN responses.

This is a bit of a generalization, so I’m not striving for extreme detail.

 

Prisoner’s Dilemma

The good old Prisoner’s Dilemma.

You and a friend have committed a crime and have been caught. You are being held in separate cells. You are both offered a deal but have to decide what to do. But you are not allowed to communicate with your partner and you will not be told what they have decided until you have made a decision.

 

Monty Hall Paradox

The Monty Hall “Paradox” isn’t actually logical paradox. It’s counterintuitive probability.

Game shows and casinos deliberately rig the probabilities to ensure the house wins. Sometimes, they rely on ignorance as did the gameshow Let’s Make a Deal.

 

Non-Euclidean Geometry

Euclidean Geometry was once considered a perfectly consistent and complete mathematical system.

The emergence of Non-Euclidean Geometries, like hyperbolic and elliptic geometry was an intellectual revolution. It turns out that Euclid’s Parallel Axiom is incomplete. It was an assumption about a line would far beyond our experience. What if we change the rule? What happens then?

 

Free Statistical Software

The Impoverished Social Scientist’s Guide to Free Statistical Software and Resources

 

Different Sized Infinities

Georg Cantor used set theory to prove that the infinity of real numbers is larger than the infinity of natural numbers.

No, not all infinities are equal in size. It’s a counterintuitive result, but it is logical.

 

Arrow’s Impossibility Theorem

There is no ideal voting system. I’ve noticed that many people dream up ‘perfect’ voting systems where everyone’s vote is equal and the results clearly reflect social preferences. What they’re asking for is mathematically impossible. Mathematics always beats utopian fantasies.

Elections are supposed aggregate individual preferences into social preferences. Arrow’s Impossibility Theorem demonstrates that there can be no perfect democratic voting system. The results are inconsistent with the basic concept of democracy. No matter which system you use, some votes will be worth more than others.

Each of Arrow’s axioms are obvious and vital to the idea of democracy. Yet he proved that it is mathematically impossible to be consistent with every single axiom at once. Something has to give. One of the tradeoffs is if a system has no dictatorship, it becomes intransitive. If it is not intransitive, it becomes a dictatorship.

 

Mathematics and Language

Mathematics offers precise logic. Functions accurately describe relationships between objects and sets. Language, consisting of words, lacks mathematical rigor and logic. It seems to be based upon estimations. It is less accurate – indeed, this sentence shows the lack of precise knowledge. Estimations are useful if we do not need precise information.

Artists deceives us because art lacks functions. It describes nothing. To truly understand reality, we must use mathematics. In the words of Galileo:

“Philosophy is written in this grand book–I mean the universe–which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.” – Galileo Galilei, Il Saggiatore (The Assayer, 1616)

 

Predators and Prey

The predator prey dynamic is a classic non-linear system. This creates the Feast and Famine Cycle. If the prey population grows, it causes the predator population to grow, which causes the prey population to fall, etc. The Lotka-Volterra differential equations describe oscillations in the populations.

These equations appear to describe historical cycles. The Hudson Bay Company recorded prices of furs that measured lynx and rabbits population. The prices show there were booms and busts in the population.

 

Stable and Unstable Systems

There are dynamically stable and unstable systems, sometimes called robust and fragile systems.

Dynamically stable systems have internal dampening mechanisms that return the system to equilibrium. Unstable systems lack a controlling force and cannot restore themselves to equilibrium.

This has a number of uses. Unstable systems can exceed performance levels of stable systems, but lose stability. This seems obvious when applied to machines. Stable systems are a useful concept in the political sciences as well.

 

Binary Numeral System

We are accustomed to thinking in base 10. We use the numbers 0 through 9 to count. There are other bases we can use instead. We know the ancient Babylonians used base 60.

Binary (or Base Two) requires only two numbers, 0 and 1. Something this minor is the key to creating advanced technologies like computers and understanding neuroscience.

 

The Minimax Theorem

Computers are now the masters of Checkers. The perfect strategy results in a draw. A win only occurs when the losing player made more mistakes than the winning player.

We know that all games of perfect imformation have saddle points. If it has a saddle point, then there is a perfect strategy for each player.