mathematics
18 articles
Bayesian Inference: Priors, Posteriors, and Updating Beliefs with Data
How Bayes' theorem works, what prior and posterior distributions mean, the base rate neglect problem in medical testing, the Bayesian vs. frequentist debate, and MCMC computation.
Bayesian Probability: How to Update Beliefs With New Evidence
Learn how Bayesian probability provides a mathematical framework for updating beliefs based on evidence, with applications in medicine, machine learning, and law.
Complexity Theory: Emergence, Edge of Chaos, and Self-Organization
How complex adaptive systems produce emergence in ant colonies and traffic jams, Bak's self-organized criticality, the edge of chaos concept, and Conway's Game of Life cellular automata.
Fermat's Last Theorem: 358 Years From Margin Note to Proof
How Fermat's 1637 claim sat unproven for 358 years until Andrew Wiles secretly spent 7 years connecting elliptic curves, modular forms, and the Shimura-Taniyama-Weil conjecture.
Fractal Geometry: Mandelbrot, Coastlines, and Infinite Complexity
How fractals work, from the Mandelbrot set to the coastline paradox and Hausdorff dimension. Explore self-similarity, Richardson's measurement problem, and fractals in nature and finance.
Game Theory: How Nash Equilibrium Shapes Strategy and Conflict
Explore game theory fundamentals, understand Nash equilibrium through classic examples like the Prisoner's Dilemma, and see how strategic thinking applies to economics and politics.
Gödel's Incompleteness Theorems: The Limits of Mathematical Truth
Gödel's two incompleteness theorems explained, the self-referential proof method, what they mean for formal systems, and their influence on mathematics and computing.
Information Theory and Shannon Entropy: How Information Is Measured
Shannon entropy formula H = -Σp log p explained, with bits of information, channel capacity theorem, Huffman coding for data compression, and error-correcting codes.
Mechanism Design: The Economics of Reverse Game Theory
How mechanism design engineers incentive-compatible rules. Covers the Vickrey auction, Myerson's revelation principle, the 2007 Nobel Prize, and the FCC spectrum auction application.
P vs NP: The Million-Dollar Problem at the Heart of Computer Science
What P vs NP means, examples of P and NP problems, why the question matters, Cook's theorem, and the implications if P equals NP or does not.
Prime Number Distribution: From the Theorem to the Riemann Hypothesis
How primes thin out according to the Prime Number Theorem (π(x) ≈ x/ln x), prime gaps, twin prime conjecture, and the Riemann Hypothesis connection, plus the largest known prime in 2024.
The Prisoner's Dilemma: Cooperation, Betrayal, and Game Theory
How the Prisoner's Dilemma models rational self-interest versus collective benefit, Nash equilibrium, iterated versions, tit-for-tat strategy, and real-world applications.
Simpson's Paradox: When Statistics Lie by Telling the Truth
How Simpson's Paradox reverses statistical trends when data is aggregated, real-world examples from Berkeley admissions and kidney stones, and how to detect and avoid it.
The Birthday Paradox: Why 23 People Make a 50% Match Likely
Why only 23 people are needed for a 50% chance of a shared birthday, the probability calculation, real-world applications in cryptography and hashing, and common misconceptions.
The Mathematical Beauty of Pi: Irrationality, Transcendence, and Computation
The history of pi, proofs of its irrationality and transcendence, record-breaking digit computations, Euler's identity, and pi's appearances across unrelated mathematics.
The Monty Hall Problem: The Probability Puzzle That Divided Mathematicians
The Monty Hall problem explained with conditional probability, Bayes' theorem, simulation results, and why nearly every expert initially got it wrong.
The Traveling Salesman Problem: Optimization's Most Famous Puzzle
How the Traveling Salesman Problem is defined, why it is computationally hard, exact and approximate algorithms, real-world applications, and connection to P vs NP.
Topology: Coffee Cups, Donuts, and the Bridges of Königsberg
How topology studies shape properties preserved under continuous deformation. Covers homeomorphism, the Möbius strip, Klein bottle, Euler's Königsberg bridges, and why a coffee cup equals a donut.