How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

The younger Bolyai kept working on the parallels, and by 1823 he had formed the outline of a solution to the ancient problem. He wrote back to his father, saying: I have discovered such wonderful things that I was amazed, and it would be an everlasting piece of bad fortune if they were lost. When you, my dear Father, see them, you will understand; at present I can say nothing except this: that out of nothing I have created a strange new universe. János Bolyai’s insight was to come at the problem from behind. Rather than try to prove the parallel postulate from the other axioms, he allowed his mind the freedom to wonder: What if the parallel axiom were false? Does a contradiction follow? And he found that the answer was no—that there was another geometry, not Euclid’s but something else, in which the first four axioms were correct but the parallel postulate was not.