A Bridge Essay I. The Pattern in Ten Stories If you've just read The Axiom Engine, you've experienced something unusual: mathematical theorems as lived constraints. The Oracle tried to predict and failed. The Arbiter tried to satisfy all axioms and collapsed. The Wanderer walked freely and discovered necessity. Each story followed the same arc: Confusion … Continue reading From Axiom Engine to Deferential Realism: How Stories Generate Philosophy
Tag: mathematics
The Axiom Engine: A Phenomenology of Abstract Structures
Prologue We usually treat mathematical structures as things we look at—diagrams on a page, symbols in a line, objects to be manipulated by the intellect. But they are not objects. They are environments. They are the invisible architectures that determine what is possible, what is impossible, and what is necessary. You do not just solve … Continue reading The Axiom Engine: A Phenomenology of Abstract Structures
How a Public School in Florida Built America’s Greatest Math Team
"A single, otherwise unremarkable public high school in Florida has won 13 out of the most recent 14 National Math Championships, a staggeringly successful dynasty for an otherwise average school. It’s accomplished this through treating math competition as any other sport, identifying talent as early as elementary school and developing them over the course of … Continue reading How a Public School in Florida Built America’s Greatest Math Team
Which Computational Universe Do We Live In?
"In 1995, Russell Impagliazzo of the University of California, San Diego broke down the question of hardness into a set of sub-questions that computer scientists could tackle one piece at a time. To summarize the state of knowledge in this area, he described five possible worlds — fancifully named Algorithmica, Heuristica, Pessiland, Minicrypt and Cryptomania — with ascending levels … Continue reading Which Computational Universe Do We Live In?
Cryptography from the Ground Up
"One of the most interesting and useful things computers can do for us is cryptography. We can hide messages, validate identities, and even build entire trustless distributed systems. Cryptography not only defines our modern world, but is a big part of how we will build the world of the future.However, unless you want to dedicate … Continue reading Cryptography from the Ground Up
Math: Ambiguity & Order of Operations
"The real answer, the one I believe any mathematician, physicist, engineer, other number-cruncher would tell you is to make sure your expressions aren’t ambiguous. There’s no extra charge for another set of parentheses. Just toss them in. If you want the answer to be 16, write (8÷2)(2+2). If you want it to be 1, write … Continue reading Math: Ambiguity & Order of Operations
Quantumcomputing For The Very Curious
"This essay explains how quantum computers work. It’s not a survey essay, or a popularization based on hand-wavy analogies. We’re going to dig down deep so you understand the details of quantum computing. Along the way, we’ll also learn the basic principles of quantum mechanics, since those are required to understand quantum computation.Learning this material … Continue reading Quantumcomputing For The Very Curious
Greg Egan and the Permutation Problem
"Then on September 26 of this year, the mathematician John Baez of the University of California, Riverside, posted on Twitter about Houston’s 2014 finding, as part of a series of tweets about apparent mathematical patterns that fail. His tweet caught the eye of Egan, who was a mathematics major decades ago, before he launched an … Continue reading Greg Egan and the Permutation Problem
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