Maths & Science
Notes and explorations in mathematics and science.
- World population density — a flat map of where humanity lives
An equirectangular map of where people actually live: every inhabited cell of a 0.1° grid tinted by its density in people per km², on a logarithmic scale. Built from NASA SEDAC's Gridded Population of the World, adjusted to 2015 UN totals.
- Hairlines — why different people have different hairlines
The hairline is a border drawn before birth: how the frontal follicle field is patterned in the womb, the polygenic genetics behind its shape, the DHT biology that makes it recede, the widow's-peak myth, and why sex and ancestry shift the picture.
- The Prisoner's Dilemma — a reader's summary
Merrill Flood and Melvin Dresher's 1950 RAND game, Albert Tucker's prison-sentence framing, and how Robert Axelrod's 1980 tournaments found a simple rule that beat pure self-interest.
- Amara's Law — a reader's summary
Roy Amara's dictum that we overestimate a technology's short-run effect and underestimate its long-run one, the murky attribution behind the quote, and the Gartner Hype Cycle it inspired.
- The Lindy Effect — a reader's summary
A 1964 essay on comedians' career lengths, the power-law math Mandelbrot and Taleb built under it, and why “it's Lindy” became shorthand for betting on whatever has already lasted.
- Jevons Paradox — a reader's summary
Jevons's 1865 coal-engine observation, the Khazzoom-Brookes formalization, and why cheaper AI inference is argued to raise total compute demand rather than shrink it.
- The Jacobian Conjecture — the 85-year-old problem a machine broke
Keller's 1939 conjecture, why it resisted for eighty-five years, and the single self-checking counterexample — surfaced by an AI — that appears to end it.
- Mathematical map
The major mathematicians placed on two axes — time and influence — linked by known lines of influence.
- Mathematicians & their ideas
Every mathematician from the map, with their main ideas — one card per idea.
- LeCun against the LLM — the world-model bet
Yann LeCun's critique of autoregressive LLMs, the JEPA world-model program, and the mathematics underneath the bet.
- A Mathematical Theory of Communication — a reader's summary
Shannon's 1948 paper: the bit, entropy, channel capacity — and the limits everything digital lives inside.
- Attention Is All You Need — a reader's summary
The 2017 Google paper that introduced the Transformer — history, authors, main ideas, critiques, and impact on everything since.
- P vs NP — history, implications, state of affairs
The central open problem of computer science: is finding an answer fundamentally harder than checking one?
- Distillation — how models copy models, and how you catch them
Soft targets and KL divergence, watermark radioactivity, Cohen's kappa on shared errors — the statistics of proving a model copied a model. Also in Business.
- Scaling Laws for Neural Language Models — a reader's summary
The 2020 OpenAI paper that turned “bigger is better” into a measurable curve — and the debate it started.
- P(doom) — the number the valley can't agree on
Publicly stated estimates span four orders of magnitude. What P(doom) is as a mathematical object, why Aumann's theorem says the spread shouldn't exist, and the consensus that isn't a number.
- The Bitter Lesson — a reader's summary
Richard Sutton's 2019 essay: search and learning, scaled with compute, beat encoded human knowledge — the manifesto behind the scaling era.
- Model Collapse — a reader's summary
What happens, mathematically, when generative models are trained recursively on their own outputs — “Habsburg AI,” the 2024 Nature paper, and how worried the data wall should make us.
- Vibe Coding — a reader's summary
Karpathy's term for building software by prompting a model and accepting its output on faith — where the wager pays off, and where it silently doesn't.