Journey into a Dark Heart

A short story set in the heart of Africa and one in which Godel’s Incompleteness Theorems play a central role in the life of one of the characters. The author mentions that this story is about “Love and its conditions on the night of 19 March 1929” (as are the other 7 stories in the collection)

David Rehn was a young, precocious mathematician (“he became a mathematician out of a deep, burning passion for that crystal-clear, purifying algebraic science from which all earthly uncertainty has been distilled.”) till he ran into a young boy at the University of Vienna – Kurt Godel. When the young “Herr Warrum” described his ideas which would shake the world of mathematics at its foundations a few years later, David went into a state of shock and retired from mathematics. As described:

“He had long since also learned to use mathematics as both medicine and stimulant. Whenever he felt down-hearted he could console himself with the scintillating logic of Bertrand Russell, if he were feeling cocky he would read one of the unsuccessful attempts at a geometric trisectioning of the angle and when his mind was in a turmoil he found tranquility and stringency in Euclid’s Elements. But on this particular day, in seeking some salve for his despair, he slipped up. On his desk lay a beautifully bound facsimile of the notes of the French mathematician Galois and, as he had done so often before, David read the young man’s hastily scrawled resume of his momentous work on the solving of irreducible equations and of his burning faith in the future. At the end Galois – then twenty-one years of age -had written: “I have run out of time. I am off to fight a duel.” Then he had risen from his papers and gone out to his death. And suddenly it occurred to David that he was reading of his own undoing. He left Vienna that same evening, firmly resolved never again to have anything to do with mathematics, and those who, later, laughed behind his back at his despair were people who had never understood that so all-embracing is love that to a person in love the very nature of life may be revealed in the smallest details and the verity of life stand or fall on the minutest grain of truth, even that contained within a mathematical proof.”

A long turn of events finds him on a train with some fellow passengers and a terrorist, bounding toward its doom over a bridge far ahead (with a twist in the end). As the passengers muse over their lives and philosophize, the mathematician, facing the supposedly final moments of his life, reminisces how his faith in the certainty of mathematics was shattered by Godel and led him to rethink about the entire edifice of mathematics and even the social sciences which may be teetering under their own collective weight:

“David gazed hopelessly into space. “In actual fact,” he said, “at those times in my life when 1 have been truly afraid I HAVE studied one especially beautiful mathematical theorem, and this has generally brought me some comfort. It has occurred to me that logic seems to contain the very essence of life and indeed that if one were trying to discover the divine plan behind the universe then it is more likely to be found in arithmetic than in the Bible.”

[…]

And yet I am here because I turned my back on mathematics,” he said, “I gave it up because I had a dream. I have been thinking over what you said, Joseph K., sir, about how we live and dream alone and I think I believe that to be wrong. You see, this dream that I had was one, I shared with an entire world. It was the dream of perfect simplicity. I have the feeling that there is, in a way, something wrong with telling you this now, but I will tell you anyway: we dreamed that the world was utterly coherent and simple. Our hoping this was the case had something to do with the fact that,” -David struggled to find the right words - “that mathematics was beginning to resemble the Leaning Tower of Pisa. An enormous structure which is very gradually starting to list to one side, so that there is no telling what one should do. But one goes on hoping.” He stared straight ahead with sadness in his eyes. “It isn’t just in mathematics, it’s the sciences too. It is names such as Boole and Hilbert and Maxwell and Planck, names that mean nothing to you, but all of which have added a few bricks to the pile. And it keeps getting taller and listing further and further. Perhaps it is not only science but the whole world. Just think of the war. So perhaps the Tower of Pisa is not the best metaphor. It is all more like Venice, it is all slowly sinking. So we create a dream, a dream of making sense of this confusion, one coherent theory which might enable us to check the slide into the mud. Not that anyone dares to say it in so many words, but we all know what it is: a longing akin to that which raised the Tower of Babel, a longing to reach all the way to God.”

[…]

“In Vienna,” he continued hesitantly, “I met … someone with a very clear view of things. He is working on a particular theorem, a proposition. When I saw this proposition it seemed to me to shatter my dream. Of course he is not the only one. There have, as I have said, been various indications of what was afoot. But he showed me Venice, he showed me that it is the foundations that are unsound. He has proved - no, he intends to prove that when one is dealing with complex systems, and we humans are complex” - and here he felt himself reddening under the girl’s gaze - “within any complex system there are certain elements which cannot be deduced from its basic characteristics. This may mean that even if we had known every particular of the circumstances surrounding this journey, we would still have been unable to guard against the unpredictable. “This proposition also suggests,” he continued, “that, even when fully aware of our own circumstances, we cannot be certain of avoiding contradictions at a later date. And,” he said, and had to lower his gaze, “life is, as we know, full of contradictory emotions.”

There are many passages which are very lyrically written, and the part where David despairs over the problem of incompleteness built into the foundations of mathematics reminded me of the extremely sad letter full of pathos which Wolfgang Bolyai wrote to his son, Johann, imploring him not to pursue a proof of Euclid’s fifth postulate, citing his own repeated failures in that journey. If one is not easily convinced that David could walk away from his passion at the “mere” thought that foundations of his subject might be crumbling or may even be entirely inconsistent - since Godel’s Theorem says that that sword of Damocles will forever remain hanging over any sufficiently complex mathematics - one should read the heart-wrenching paragraph from that letter (not part of this story but perhaps will give the reader a perspective on David’s mindset):

“You should not investigate the parallels in that way; I know also that path till the end, I also have the measure of this bottomless night: every light, every joy of my life has been extinguished in it. I implore you for God’s sake leave the parallels in peace. You must shy away from it as you would from a dissolute contact. It will bring to an end all your abilities, your health, your peace of mind and your life’s happiness. This bottomless pit will swallow a thousand Newtons, it will never see the light of day and poor mankind will never have something that is pure, not even geometry; it has inflicted a deep and permanent wound in my soul; God forbid that it bites into you so deeply. It robs for one the joy of geometry, for life. I planned to sacrifice myself for truth. I would have been ready to become a martyr if only I could give geometry, purified of this stigma, to mankind. Fearful, enormous work have I done, have by far achieved more than ever achieved before, but I have never found any complete satisfaction. Here, however, the Latin proverb applies: si paulum a summon discessit, vergit ad imum. I have backed out when I found that one cannot reach the bottom of this night from the Earth. Unfortunately, there is no consolation for me and for the entire mankind. Learn from my example.”

I do, however, feel that some statements in the story do not make logical sense (or perhaps David, in his despondent state, is no longer thinking consistently). E.g. David bemoans toward the end:

“This may mean that, even had we known every particular of the circumstances surrounding this journey, we would still have been unable to guard against the unpredictable.”

I doubt any serious mathematician or a half-serious analytic mind thinks in this fashion, as the unpredictability in the world would be omnipresent even in a mathematical world where Godel’s theorems did not exist. The basic concepts of chaos theory ensure that predictability in complex systems in the manner expressed by David is impossible no matter the foundations of mathematics. David should know that the kind of predictability he wants is not precluded by Godel but something far less esoteric.

(I am grateful to Mr. V H Ram from India who very kindly translated the German version of the Bolyai letter for me)