Philosophy of Mathspeaking topic
You cannot paint the inside surface of an infinitely long horn, because that would take infinite paint; yet you can fill the same horn completely with liquid paint, because its volume is finite. How can one surface be both unpaintable and fillable?
— Gabriel's Horn (Torricelli's trumpet)
practice with this topic
Set the timer (5-30 min), take 20 seconds of prep if you like, start talking. Jot your thoughts onto the sticky-note board.
similar topics
- Add up the reciprocals of the natural numbers (1 + 1/2 + 1/3 + ...) and the sum grows to infinity; add up only the reciprocals of the primes and it still grows to infinity. Primes keep thinning out, yet they are enough to overflow the sum. How can such 'rare' numbers be this powerful?
- The digits of pi run on forever without repeating; your birth date, your phone number, and quite possibly the name of your unborn grandchild are hiding somewhere in there. How can a pattern that never repeats contain everything?
- Keep adding positive numbers and you can climb to infinity, but 1 + 1/2 + 1/4 + 1/8 + ... never gets there; it stops at exactly 2. How can adding infinitely many things give a finite answer?
- Why was accepting 'zero' as a number such a struggle throughout history, and why was counting nothingness as a quantity a philosophical leap?
- Among any six people, there must be either three who all know each other or three who are all complete strangers to one another. Even in pure chaos, order is forced to appear. Why can't randomness escape pattern?