Card 1 of 8: Gödel's Incompleteness Theorems
Gödel was 25 when he proved something unsettling about mathematics: any system rich enough to do arithmetic will always contain true statements it can't prove from within.
Bayes never published this in his lifetime — a friend found it in his papers and printed it in 1763. It's the exact formula for how much a new piece of evidence should change what you believe.
In 1936 a 24-year-old at Cambridge asked a disarmingly simple question: can we build a machine that looks at any program and tells us, beforehand, whether it will ever finish running? His answer — no — is the most important negative result in computer science.
Gottlob Frege thought he had finished the foundations of mathematics. Russell found the counterexample in 1901 and mailed it to him in June 1902, as Frege's second volume was going to press: 'the set of all sets that do not contain themselves' — and Frege's system collapsed.
Bernoulli proved in 1713 why your gut misreads streaks: flip a fair coin ten times and 70% heads is common; flip it ten thousand times and 51% heads is rare. Large numbers pull the average back to its true center.
Mandelbrot asked in 1967 'How long is the coast of Britain?' and answered: it depends on your ruler. Zoom in and new bays appear inside bays — the same jaggedness, at every scale.
You pick one of three doors. The host, who knows what's behind each, opens a different losing door and offers you the switch. Intuition screams 50/50. The math says switch and you win two-thirds of the time.
In 1971 Cook isolated a class of problems where checking an answer is easy but finding it seems hard — and proved they are all secretly the same problem. If you solve one fast, you solve them all.