As Kent recently intuited, I’ve been rereading Douglas Hofstadter’s GÃ¶del, Escher and Bach recently, in part in hopes that a rereading will illuminate corners that I missed the last time through, and in part because good books in English are very difficult to find here, and prohibitively expensive when I do find them. There are no libraries of which I am aware within a two hour radius of my home, and even if there were, they would not have any books in English. This situation is particularly unhappy because I am and always have been a voracious reader, getting through an average of two or more books a week. Needless to say the tomes that comprise the meager collection I brought with me when we moved here from Sydney are well-thumbed and dog-eared by now.
Anyway, this anecdote from GEB struck me, and I thought I’d share it with you.
Johann Bolyai and Nikolay Lobachevskiy independantly and to all appearances simultaneously discovered non-Euclidean geometry in 1823. Euclidean geometry, of course, is based on five postulates, four elegant and one perhaps a little less so, and had stood proudly for about two thousand years.
The first four postulates :
(1) A straight line segment can be drawn joining any two points.
(2) Any straight line segment can be extended indefinitely in a straight line.
(3) Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center.
(4) All right angles are congruent.
and the fifth, which lacks a little of the concision and elegance of the first four
(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines must inevitably intersect each other on that side if extended far enough.
Over the intervening centuries, dozens of attempts had been made to prove that the fifth postulate was in fact part of ‘four-postulate geometry’, all unsuccessful.
One of the people who had attempted to do so was Bolyai’s father, Wolfgang, who was also a mathematician and a friend of Gauss (who is part of Graham’s mathematical family tree, synchronicitously enough). The elder Bolyai wrote to his son, in an attempt to steer him from the black sinkhole of depair that was Euclid and the Mathematical Life :
This passage astonishes me. Even allowing for floweriness of language, that a man could so deeply feel his life ruined and wasted as a result chasing a mathematical proof somehow sets me back in my seat, a-wondering about how we have changed, or if indeed we have. It may not have a similar effect on you, and if not, I beg your indulgence.