One sentence explanations of mathematical concepts and proofs:
- Triangle inequality: Stopping by Starbucks on your way to class cannot be a shortcut.
- Yoneda lemma: If you know the directions for all the paths from your house to everywhere else, you know where your house is.
- Pigeonhole Principle: Among any three gloves, there must be two for the same hand.
- Intermediate Value Theorem: If you were there a while ago, and now you are here, then there must have been some time that you were exactly halfway.
- Gödel’s Incompleteness Theorems: If the axioms are the rules, and mathematics is the game, Gödel showed that no set of rules could get to every space without being able to go off the board.
- Halting Problem: In a world like the one in the movie Matrix, the Oracle cannot predict the future of all programs correctly because For all such Oracles there exists a program Neo which uses the same method the Oracle uses to see the future and simply avoids Oracle’s prediction from becoming true. e.g. If Oracle predicts Neo will sit, Neo will stand.