As Kent recently intuited, I’ve been rereading Douglas Hofstadter’s Gödel, 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.
Bitch, moan.
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 :

You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone…I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I acoomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true that si paullum a summo discessit, vergit ad imum. I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind…. I have travelled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness – aut Caesat aut nihil.

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.

Thoughts That, If Not Deep, Are At Least Wide

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  1. I have been reading Hofstatder’s book as well and came across the same passage. I must say that I was struck in the same manner as you. “Jesus Christ!” was my only response. It read like something taken from a story by H.P. Lovecraft or the infamous Necronomicon itself. Kurt Godel himself was an interesting person as well, and his later madness seems also extraordinary. Einstein, it appears, was among the only of the early 20th C. mathemeticians who was emotionally and spiritually prepared for what is to be discovered about the nature of the universe through theoritical scientific method. Most of these men were not prepared to deal with what they discovered. It’s quite apparant to me.

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