** DID NOT FINISH **
“We are living in a wild geometric boomtown, global in scope. Geometry isn’t out there beyond space and time, it’s right here with us, mixed in with the reasoning of everyday life. Is it beautiful? Yes, but not bare. Geometers see Beauty with its work clothes on.”
“No man can talk well unless he is able first of all to define to himself what he is talking about.” — Gulliver
- What Lincoln took from Euclid was the idea that, if you were careful, you could erect a tall, rock-solid building of belief and agreement by rigorous deductive steps, story by story, on a foundation of axioms no one could doubt: or, if you like, truths one holds to be self-evident.
- “The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It’s that the world is full of non-proofs, and grown-ups need to know the difference.”
- “…like the way master teacher Ben Blum-Smith describes the problem: for students to really feel the fire of math, they have to experience the gradient of confidence—the feeling of moving from something obvious to something not-obvious, pushed uphill by the motor of formal logic.”
“Mathematics is the art of giving the same name to different things.” —Poincaré
- “To analyze Brownian motion and the stock market and mosquito all at once, with the mathematics of the random walk, is to follow Poincaré’s slogan and give the same name to different things.”
- “The ability of a Markov chain to produce something like language gives one pause. Is language just a Markov chain?”
- “And yet, modern-day Markov chains can produce something remarkably like human language. An algorithm like Open AI’s GPT-3 is the spiritual descendant of Shannon’s text machine, only much bigger. The input, instead of being three letters, is a chunk of text hundreds of words long, but the principle is the same…”
- “Our observation about primes isn’t just a fact, it’s a fact with a name: it’s called Fermat’s Little Theorem, after Pierre de Fermat, the first person to write it down. No matter which prime number n you take, however large it may be, 2 raised to the nth power is 2 more than a multiple of n.”