Portuguese has a word that is famously difficult to translate: Saudade. It is the shadow left where love meets time, a tender longing for what has gone, what was never ours, and even what we have not yet lost. To call it missing someone feels too light. Homesickness feels too distant. Regret lacks its tenderness. Saudade is closer to the color these feelings leave when they meet at dusk. We long for a person, a stretch of life, a place we cannot return to, or an existence tha

We like to divide judgments into right and wrong. Bayesian probability begins with a less comfortable admission: most of the time, we possess only part of the evidence. Something is true or false. A person is trustworthy or untrustworthy. A choice will succeed or fail. Those categories are satisfyingly clean. Reality rarely cooperates. We often have to judge with incomplete information and then live with the consequences of being wrong. Bayesian probability deals with exactly

The goal of mathematics education should not be merely to make students calculate faster. Nor should it be to march them through algebra, geometry, calculus and a sequence of course titles. The real goal is to help a person build a complete map of mathematics. Calculation is part of mathematics, but it is only one basic skill within mathematical activity. Genuine study also means recognizing mathematical objects, understanding why definitions take the form they do, discoverin

As high-performance GPU clusters become widely deployed, Bayesian probability will gradually move toward the center of artificial intelligence. Frequentist methods will not disappear, but inside an intelligent system they may take a supporting role. A next generation of large models built around Bayesian updating might bring us closer to AGI than parameter growth alone. That is an aggressive claim, but it is not groundless. The central theory of Bayesian methods has existed f

The Bellman equation offers a way to think about life as a sequence of decisions: weigh today's reward, the next state and the options that remain.

A decision is right when it reduces avoidable uncertainty, mismatch, irreversible harm and displaced disorder relative to the path not taken.

A dream about homeomorphic manifolds became a working model of Chinese calligraphy built from topology, geometry, finite-scale ink and brush dynamics.

A practical architecture that separates goals, execution, judgment and human authorization so autonomous AI systems know when they must stop.

An FE-E experiment found smoother gradients and steeper loss slopes without earlier learning, suggesting that stiffness and entropy may work better as observers than controllers.

Six questions about executive power, the courts, political opposition and whether America's legal order can withstand a second Trump presidency.

A short reflection on the East Asian attraction to decline, retreat and withdrawal at the height of success.

A narrow constitutional question raised by Josh Hawley's proposal to decouple American AI capabilities from China.

A time-travel thought experiment for asking whether twenty years of development have actually improved everyday life.

A compact framework for judging natural language by what it can describe across breadth, depth and time.

A calculus metaphor for a familiar puzzle: why someone with more knowledge can still produce work that falls short of an earlier master's.

Beyond the Fields Medal profiles: how molecular collision graphs connect Newtonian reversibility, Boltzmann entropy, cryptographic diffusion and the arrow of time.

A June 2025 reading of Iran's Hormuz threat across oil, crypto, U.S. stocks, the Fed, China's supply chains and A-shares.

Why the Swiss franc surged in 2025, what a strong currency does to Switzerland, and why I see CHF as a hedge rather than a trade.

A May 2025 analysis of Harvard's SEVP lawsuit, its constitutional arguments, and a plan to de-risk higher education beyond U.S. borders.

A blunt look at international and public education in China, the financial risks families take, and why parents matter more than the school label.