Where theory becomes dopamine
Every message you send carries a measurable amount of surprise. Shannon figured out that information is really about uncertainty reduction. The less likely a message, the more information it carries. That "no way!" moment? That's high entropy at work.
H(X) = -∑ p(x) log&sub2; p(x)
Every interaction is a game. The Nash equilibrium is the point where no player benefits from changing strategy alone. It explains why rational individuals don't always cooperate -- even when cooperation would make everyone better off. The Prisoner's Dilemma isn't just a thought experiment; it's your morning commute.
ui(s*i, s*-i) ≥ ui(si, s*-i)
Tiny differences in initial conditions yield wildly diverging outcomes. That's not randomness -- it's deterministic chaos. The butterfly effect isn't a metaphor; it's a mathematical property of nonlinear dynamical systems. The universe is predictable in theory, impossible in practice.
xn+1 = rxn(1 - xn)
The simplest explanation is usually the right one. Not because the universe prefers simplicity, but because simpler models have fewer ways to be wrong.
Any consistent formal system powerful enough to describe arithmetic contains statements that are true but unprovable. Math has blind spots. Always will.
Your beliefs should update with evidence. Prior times likelihood, divided by marginal. The math of changing your mind rationally.
Disorder always increases. The second law of thermodynamics says the universe tends toward maximum entropy. Your messy room is just physics in action.
A system that can simulate any Turing machine can compute anything computable. Your phone. A game of life. Technically, a sufficiently complex domino chain.
Complex behavior from simple rules. Consciousness from neurons. Markets from trades. Ant colonies from pheromones. The whole exceeds the sum.